Abstract.

This article focuses on the study of zero-divisor graph $\mathrm{\Gamma }(C{(X)}_{𝒫})$, annihilator graph $AG(C{(X)}_{𝒫})$ and weakly zero-divisor graph $W\mathrm{\Gamma }(C{(X)}_{𝒫})$ on the ring $C{(X)}_{𝒫}$ of all real-valued functions on a topological space $X$ that are continuous outside a member of an ideal $𝒫$ of closed subsets of $X$. We establish that if $C{(X)}_{𝒫}$ properly contains the ring $C(X)$ of real-valued continuous functions on $X$, then the radius of $\mathrm{\Gamma }(C{(X)}_{𝒫})$ is $2$ and it is not triangulated. Moreover, in this situation, both $\mathrm{\Gamma }(C{(X)}_{𝒫})$ and $AG(C{(X)}_{𝒫})$ are not hypertriangulated and the dominating number of $AG(C{(X)}_{𝒫})$ is $2$. Furthermore, $W\mathrm{\Gamma }(C{(X)}_{𝒫})$ fails to be a complete graph under the hypothesis $C{(X)}_{𝒫}⫌C(X)$. We establish a connection between the complemented-ness of $\mathrm{\Gamma }(C{(X)}_{𝒫})$ and the Von-Neumann regularity of $C{(X)}_{𝒫}$ under the assumption that $C{(X)}_{𝒫}\supseteq \{{\chi }_{\{p\}}:p\in X\}$. We realise that any two of these three graphs coincide if and only if $|X|=2$ and in this case, the graphs are complete bipartite. We also note that the phenomena of $W\mathrm{\Gamma }(C{(X)}_{𝒫})$ being triangulated, hypertriangulated and complemented depend solely on the cardinality of $X$.

1. Introduction

Let $X$ be a Tychonoff space and $C(X)$ denote the ring of real-valued continuous functions on $X$. Let the collection of all real-valued functions on $X$ be denoted by ${ℝ}^X$ and for $f\in {ℝ}^X$, the set of points of discontinuities of $f$ by $D_f$. As usual, the closure (interior) of a subset $A$ of $X$ is denoted by $\overline{A}$ (resp. $intA$) in $X$. A collection $𝒫$ of closed subsets of $X$ is said to be an ‘ideal of closed sets’ if it satisfies the following conditions:

1. $𝒫$ is closed under taking finite union.

2. If $A,B$ are closed sets in $X$ such that $A\subseteq B$ and $B\in 𝒫$, then $A\in 𝒫$.

The following collections form ideals of closed sets in $X$:

• ${𝒫}_f=$ the collection of all finite subsets of $X$,

• $𝒦=$ the collection of all closed compact subsets of $X$ and

• ${𝒫}_{nd}=$ the collection of all closed nowhere dense subsets of $X$.

The triplet $(X,\tau ,𝒫)$ is called a $\tau 𝒫$-space ([7]) and the collection, $C{(X)}_{𝒫}=\{f\in {ℝ}^X:\overline{D_f}\in 𝒫\}$ [7] forms a commutative ring with unity, under the operations of pointwise additions and multiplications. Note that the ring $C{(X)}_{𝒫}$ is an abstraction of various known rings of functions.

• If $𝒫=\{\mathrm{\varnothing }\}$, then $C{(X)}_{𝒫}=C(X)$.

• If $𝒫={𝒫}_{nd}$, then $C{(X)}_{𝒫}=T^{\prime }(X)$, which was introduced by Ahmadi Zand in 2010 [2].

• If $𝒫={𝒫}_f$, then $C{(X)}_{𝒫}=C{(X)}_F$, which was introduced by Gharabaghi et. al. in 2018 [9].

• The notation $C{(X)}_K$ is used to denote $C{(X)}_{𝒦}$ in [7].

For $f\in C{(X)}_{𝒫}$, the ‘zero-set’, $Z_{𝒫}(f)$ is the set $\{x\in X:f(x)=0\}$. If $f\in C(X)$, we write $Z(f)$ instead of $Z_{𝒫}(f)$. The collection $\{Z(f):f\in C(X)\}$ is as usual denoted by $Z[X]$ and $\{Z_{𝒫}(f):f\in C{(X)}_{𝒫}\}$ is denoted by $Z_{𝒫}[X]$.

On a commutative ring $R$, Anderson and Livingston defined the zero-divisor graph, $\mathrm{\Gamma }(R)$ on the vertex set $V(R)$, consisting of all non-zero zero-divisors of the ring $R$ and two vertices $x$ and $y$ in $V(R)$ to be adjacent in $\mathrm{\Gamma }(R)$ if $xy=0$ [1]. In 2013, Badawi defined the annihilator graph $AG(R)$ on the same vertex set $V(R)$ such that two vertices $x,y\in V(R)$ are adjacent if $Ann(x)\cup Ann(y)⫋Ann(xy)$ [5], where $Ann(z)$ denotes the collection $\{z^{\prime }\in R:z{z}^{\prime }=0\}$ for $z\in R$. Another graph, called the weakly zero-divisor graph, denoted by $W\mathrm{\Gamma }(R)$, was defined on $V(R)$ by Nikmehr et. al. [16]. Two distinct vertices $x$ and $y$ are adjacent in $W\mathrm{\Gamma }(R)$ if there exist non-zero elements $a\in Ann(x)$ and $b\in Ann(y)$ such that $ab=\mathrm{𝟎}$. It follows that $\mathrm{\Gamma }(R)$ is a spanning subgraph of $AG(R)$ and $AG(R)$ is a spanning subgraph of $W\mathrm{\Gamma }(R)$. Recall that a graph $G_2$ is said to be the spanning subgraph of a graph $G_1$ if they have the same vertex set and whenever two vertices are adjacent in $G_2$, they are also adjacent in $G_1$.

In this article, we wish to study the above mentioned graphs defined on the collection $V(C{(X)}_{𝒫})$ (or simply, $V$) of all non-zero divisors of zero of the ring $C{(X)}_{𝒫}$. In order to do that efficiently, we first recollect some definitions. Let $G$ be a graph defined on a vertex set $V_0$. For $f,g\in V_0$, the distance $d(f,g)$ is the length of the shortest path between $f$ and $g$. The diameter $diam(G)=sup\{d(f,g):f,g\in V_0\}$ and the girth $gr(G)$ is the length of the shortest cycle in $G$. If $G$ has no cycles, $gr(G)=\mathrm{\infty }$. Furthermore, for $f\in V_0$, the associated number, $e(f)$ of $f$ is defined as $max\{d(f,g):g\in V_0\setminus \{f\}\}$. An $f\in V_0$ is said to be a center of $G$ if it has the smallest associated number. The associated number of such a center is called the radius of $G$ and is denoted by $\rho (G)$. $G$ is said to be triangulated if each vertex of $G$ is a vertex of a triangle and hypertriangulated if each edge in $G$ is an edge of a triangle. A subset of $V_0$ is said to be a stable set if no two vertices in the set are adjacent. $G$ is called a bipartite graph if $V_0$ can be decomposed into two stable sets. A bipartite graph is complete if each pair of vertices from distinct stable sets are adjacent. For two vertices $f,g\in V_0$, $𝒄\mathbf{(}𝒇\mathbf{,}𝒈\mathbf{)}$ denotes the length of the shortest cycle (if any) containing $f$ and $g$, and $c(f,g)=\mathrm{\infty }$ if there does not exist any cycle in $G$ containing $f$ and $g$. Two distinct vertices $f,g\in V_0$ are said to be orthogonal (denoted as $𝒇\mathbf{⟂}𝒈$) if $f$ is adjacent to $g$ and there does not exist any $h\in V_0$ which is adjacent to both $f$ and $g$. $G$ shall be called a complemented graph if for each $f\in V_0$, there exists $g\in V_0$ such that $f⟂g$. A complemented graph $G$ is called uniquely complemented if $g,h\in V_0$ are adjacent to the same set of vertices in $G$, whenever $f⟂g$ and $f⟂h$ in $G$ for some $f\in V_0$. A vertex is said to be a universal vertex in $G$ if it is adjacent to all other vertices. A subset $V_1$ of $V_0$ is said to be a dominating set in $G$ if every vertex outside $V_1$ is adjacent to some vertex in $V_1$. The dominating number, $𝒅𝒕\mathbf{(}𝑮\mathbf{)}=\min \{|V_1|:V_1\text{ is a dominating set in }G\}$. The smallest cardinal number $\mathrm{\aleph }$ such that every complete subgraph of $G$ is of cardinality $\le \mathrm{\aleph }$ is said to be the clique number of $G$, and is denoted by $𝝎𝑮$. For more graph theoretic terms, one can refer [6].

We observe that if $R$ has no non-zero divisors of zero, then $V(R)=\mathrm{\varnothing }$ and all the three graphs $\mathrm{\Gamma }(R)$, $AG(R)$ and $W\mathrm{\Gamma }(R)$ are then null-graphs. If $X$ is a topological space with exactly one point, then $C{(X)}_{𝒫}$ is isomorphic to $ℝ$, for any choice of ideal $𝒫$ and hence all the above mentioned graphs on $C{(X)}_{𝒫}$ are null-graphs. So, to avoid this case, we shall always assume $|X|\ge 2$.

Throughout this article, we have made use of the space $X_{𝒫}$, which is generated by $Z_{𝒫}[X]$ as a base for closed sets. First note that if $𝒫=\{\mathrm{\varnothing }\}$, then the space $X_{𝒫}$ is the original space $X$. We observe that the space $X_{𝒫}$ is a $T_3$ space. We also note that $X_{𝒫}$ contains isolated points if either $X$ has isolated points or $C{(X)}_{𝒫}⫌C(X)$. Next we establish a relation between the Goldie dimension of the ring $C{(X)}_{𝒫}$ and the cellularity of the space $X_{𝒫}$. Furthermore, we describe the space of minimal prime ideals of $C{(X)}_{𝒫}$ and establish a condition under which the space is compact. We note that a few results in [3] and [11] can be established as particular cases of some theorems in this section, when we choose $𝒫=\{\mathrm{\varnothing }\}$.

Section 3 of this article is devoted to the study of the zero-divisor graph $\mathrm{\Gamma }(C{(X)}_{𝒫})$ (or simply, $\mathrm{\Gamma }$) of the ring $C{(X)}_{𝒫}$. In particular, we calculate the diameter, girth and radius of the graph. It has been noted in this section that the radius of the graph $\mathrm{\Gamma }$ is $2$ if $C{(X)}_{𝒫}⊋C(X)$. Moreover, we have observed that if $C{(X)}_{𝒫}⊋C(X)$, then $\mathrm{\Gamma }$ is neither triangulated nor hypertriangulated. This brings out a contrast between the zero-divisor graph of $C(X)$ and $\mathrm{\Gamma }$ (see Example 3.23). We note that $\mathrm{\Gamma }$ is a complete bipartite graph if $|X|=2$. We further observe that the clique number of $\mathrm{\Gamma }$ and the Goldie dimension of $C{(X)}_{𝒫}$ are equal. In particular, we explicitly show that the clique number of $\mathrm{\Gamma }(C{(X)}_F)$, $\mathrm{\Gamma }(T^{\prime }(X))$ and $\mathrm{\Gamma }(C{(X)}_K)$ are equal and equal to the cardinality of $X$. We next establish that the compactness of the space of minimal prime ideals of $C{(X)}_{𝒫}$ is equivalent to the complemented-ness of the graph $\mathrm{\Gamma }$. We further recall a condition equivalent to the Von-Neumann regularity of the ring $C{(X)}_{𝒫}$ (see [7]) to establish that if $C{(X)}_{𝒫}\supseteq \{{\chi }_{\{p\}}:p\in X\}$, then the complemented-ness of the graph $\mathrm{\Gamma }$ is equivalent to the Von-Neumann regularity of the ring $C{(X)}_{𝒫}$.

Section 4 of this article deals with the annihilator graph $AG(C{(X)}_{𝒫})$ of the ring $C{(X)}_{𝒫}$. For simplicity, we write $AG$ instead of $AG(C{(X)}_{𝒫})$ when there is no ambiguity. We realise that $AG$ coincides with $\mathrm{\Gamma }$ if and only if $|X|=2$ and then and only then $AG$ is a complete bipartite graph. Moreover, we note that the distance between any two non-adjacent vertices is $2$ and it follows that the diameter of the graph is $2$. We further establish that $AG$ is hypertriangulated if and only if $C{(X)}_{𝒫}=C(X)$ and $X$ has no isolated points. However, $AG$ is triangulated if and only if $|X|\ge 3$. We also observe a correspondence between the dominating number of $AG$ and the phenomenon of $AG$ being hypertriangulated.

The final section involves the weakly zero-divisor graph of the ring $C{(X)}_{𝒫}$. We denote it by $W\mathrm{\Gamma }$, instead of $W\mathrm{\Gamma }(C{(X)}_{𝒫})$. Just as we have seen in case of $\mathrm{\Gamma }$ and $AG$, $W\mathrm{\Gamma }$ is also a complete bipartite graph, which coincides with $\mathrm{\Gamma }$ and $AG$ if and only if $|X|=2$. We are able to describe the distance between two vertices, diameter of $W\mathrm{\Gamma }$ and the dominating number of the graph. We further show that $W\mathrm{\Gamma }$ is a complete graph if and only if $C{(X)}_{𝒫}=C(X)$ and $X$ has no isolated points. In particular, whenever $C{(X)}_{𝒫}$ properly contains $C(X)$, $W\mathrm{\Gamma }$ is not a complete graph. We realise that $W\mathrm{\Gamma }$ is triangulated if and only if $|X|\ge 3$ and this is equivalent to the hypertriangulation of $W\mathrm{\Gamma }$ as well. Moreover, the above statements are equivalent to saying that $W\mathrm{\Gamma }$ is not complemented. Finally, we describe some of these results for the particular case of $C(X)$.

2. Prerequisites on the ring $C{(X)}_{𝒫}$ and the space $X_{𝒫}$

We initiate this section with the recollection that the set of all zero-sets of functions in $C{(X)}_{𝒫}$, denoted by $Z_{𝒫}[X]$, forms a base for closed sets for the topology ${\tau }_{𝒫}$ on $X$, which is finer than the original topology on $X$ [8]. In short we denote the pair $(X,{\tau }_{𝒫})$ as $X_{𝒫}$. For a subset $A$ of $X$, $c{l}_{X_{𝒫}}A$ ($in{t}_{X_{𝒫}}A$) denotes the closure (resp. interior) of $A$ in $X_{𝒫}$. We first observe a topological property of the space $X_{𝒫}$ which shall be useful throughout this article.

Theorem 2.1.

$X_{𝒫}$ is a $T_3$-topological space and the collection $\{in{t}_{X_{𝒫}}{Z}_{𝒫}(f):f\in C{(X)}_{𝒫}\}$ forms a base for open sets for the space $X_{𝒫}$.

Proof 2.2.

Suppose $K$ is a closed set in $X_{𝒫}$ and $x_0\notin K$. Then there exists $f\in C{(X)}_{𝒫}$ such that $x_0\in X\setminus Z_{𝒫}(f)\subseteq X\setminus K$. Without loss of generality assume that $f(x_0)>0$ and let $r=\frac{f(x_0)}{3}$. Define and $V=\{x\in X:f(x)>r\}$. Then $U$ and $V$ are disjoint basic open sets in $X_{𝒫}$ such that $x_0\in V$ and $K\subseteq U$. This ensures that $X_{𝒫}$ is a $T_3$-space.

Moreover, there exist $g,h\in C{(X)}_{𝒫}$ such that $U=X\setminus Z_{𝒫}(g)$ and $V=X\setminus Z_{𝒫}(h)$. Also since $x_0\in V\subseteq X\setminus U\subseteq X\setminus K$, it follows that $x_0\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq X\setminus K$.

The following observation helps us make certain compelling deductions regarding the rings $C{(X)}_F$, $T^{\prime }(X)$ and $C{(X)}_K$.

Observation 2.3.

The assumption that the characteristic function ${\chi }_{\{p\}}\in C{(X)}_{𝒫}$ for each $p\in X$ (for eg.: $C{(X)}_F$, $C{(X)}_K$, $T^{\prime }(X)$, etc) ensures that $X\setminus \{p\}=Z_{𝒫}({\chi }_{\{p\}})\in Z_{𝒫}[X]$, which is a base for closed sets $X_{𝒫}$ and so $X_{𝒫}$ is the discrete space. In this context,

Next, we note that often the existence of even a single function of the form ${\chi }_{\{p\}}$, for some $p\in X$, in $C{(X)}_{𝒫}$ describes certain behaviours of graphs on the ring $C{(X)}_{𝒫}$. Such an existence is clearly guaranteed if $X$ has an isolated point. Moreover, we have the following useful lemma.

Lemma 2.4.

$C{(X)}_{𝒫}$ contains ${\chi }_{\{p\}}$ for some $p\in X$ if and only if either $X$ has an isolated point or there exists a non-isolated point $p\in X$ with $\{p\}\in 𝒫$. Moreover, the latter holds if and only if $C(X)⫋C{(X)}_{𝒫}$. In particular, a point $p\in X$ is isolated in the space $X_{𝒫}$ if and only if either $p$ is an isolated point in $X$ or $\{p\}\in 𝒫$.

Proof 2.5.

If $C(X)\ni {\chi }_{\{p\}}$, for some $p\in X$, then $p$ is an isolated point in $X$. If ${\chi }_{\{p\}}\in C{(X)}_{𝒫}\setminus C(X)$, then $p$ is a non-isolated point in $X$ and so $D_{{\chi }_{\{p\}}}=\{p\}$. Therefore, $\{p\}\in 𝒫$. The converse is straightforward.

Moreover, if $\{p\}\in 𝒫$, where $p$ is non-isolated, then ${\chi }_{\{p\}}\in C{(X)}_{𝒫}\setminus C(X)$. Conversely let $f\in C{(X)}_{𝒫}\setminus C(X)$. Then $\mathrm{\varnothing }\ne D_f\subseteq \overline{D_f}\in 𝒫$. Let $p\in D_f$, then it follows that $p$ is a non-isolated point in $X$ and $\{p\}\in 𝒫$. The rest is immediate.

Recall that the cellularity $c(X)$ of a space $X$ is defined as the smallest cardinal number $\mathrm{\aleph }(\ge {\mathrm{\aleph }}_0)$ such that every family of pairwise disjoint non-empty open subsets of $X$ has cardinality $\le \mathrm{\aleph }$ and a cardinal number $\mathrm{\aleph }$ is called the Goldie dimension, $dim(R)$ of a commutative ring $R$ if a direct sum of non-zero ideals in $R$ consists of $m$ terms where $m\le \mathrm{\aleph }$.

We now realise that the Goldie dimension, $dim(C{(X)}_{𝒫})$ of $C{(X)}_{𝒫}$ is closely related to the cellularity of the space $X_{𝒫}$. This relation shall be useful in discussing the clique number of the zero-divisor graph of $C{(X)}_{𝒫}$.

Theorem 2.6.

For a topological space $X$, $dim(C{(X)}_{𝒫})=c(X_{𝒫})$.

Proof 2.7.

Let ${\displaystyle \underset{\alpha \in \mathrm{\Lambda }}{⨁}}{I}_{\alpha }$ be a direct sum of non-zero ideals in $C{(X)}_{𝒫}$. Note that for $\alpha ,\beta \in \mathrm{\Lambda }$ with $\alpha \ne \beta$, $f_{\alpha }{f}_{\beta }=\mathrm{𝟎}$ as $I_{\alpha }\cap I_{\beta }=\{\mathrm{𝟎}\}$. Thus, the collection $\{X\setminus Z_{𝒫}(f_{\alpha }):\alpha \in \mathrm{\Lambda }\}$ is a family of pairwise disjoint open sets in $X_{𝒫}$. Thus, we have $c(X_{𝒫})\ge |\mathrm{\Lambda }|$ and hence $c(X_{𝒫})\ge dim(C{(X)}_{𝒫})$.

Next suppose $\{G_i:i\in I\}$ is a family of pairwise disjoint open subsets of $X_{𝒫}$. Then for each $i\in I$, there exists $f_i\in C(X)$ such that $\mathrm{\varnothing }\ne X\setminus Z_{𝒫}(f_i)\subseteq G_i$. Set for each $i\in I$. We claim that the family $\{I_i:i\in I\}$ of non-zero ideals in $C{(X)}_{𝒫}$, constitutes an independent family. In other words, for $i\in I$, $I_i\cap {\displaystyle \sum _{j\in I\setminus \{i\}}}{I}_j=\mathrm{\varnothing }$. Indeed, if $f\in I_i\cap {\displaystyle \sum _{j\in I\setminus \{i\}}}{I}_j$, then $f=f_{i}h={\displaystyle \sum _{k=1}^m}{f}_{j_k}{h}_k$, where $j_1,j_2,\mathrm{\cdots },j_m\in I\setminus \{i\}$ and $h,h_k\in C{(X)}_{𝒫}$ for $k=1,2,\mathrm{\cdots },m$. This ensures that $f^2=(f_{i}h)({\displaystyle \sum _{k=1}^m}{f}_{j_k}{h}_k)={\displaystyle \sum _{k=1}^m}{f}_i{f}_{j_k}h{h}_k=\mathrm{𝟎}$, since $f_i{f}_j=\mathrm{𝟎}$ for $i\ne j$ (as $G_i\cap G_j=\mathrm{\varnothing }$ for $i\ne j$). Thus, we have a direct sum of ideals in $C{(X)}_{𝒫}$, ${⨁}_{i\in I}{I}_i$ and so $|I|\le dim(C{(X)}_{𝒫})$. It follows that $c(X_{𝒫})\le dim(C{(X)}_{𝒫})$.

Next, we wish to discuss the compactness of the space of minimal prime ideals of $C{(X)}_{𝒫}$.

We initiate this discussion by revisiting some notations and results regarding minimal prime ideals and the space of minimal prime ideals of a commutative ring $R$.

Lemma 2.8.

The intersection of all minimal prime ideals in $R$ is the set of all nilpotent elements in $R$.

Lemma 2.9 ([12, Lemma 3.1]).

A prime ideal $P$ of $R$ is minimal if and only if for each $x\in P$, there exists $a\in R\setminus P$ such that $ax$ is a nilpotent element in $R$.

We borrow the notation, $𝒫(R)$ from [11] to denote the family of all minimal prime ideals of $R$.

Notations 2.10.

• For any subset $S$ of $R$, the hull $h(S)$ of $S$ is the collection of all minimal prime ideals of $R$ that contain the set $S$.

• For a subfamily $ℒ$ of $𝒫(R)$, the kernel $k(ℒ)$ is the intersection of all members of $ℒ$.

It is easy to verify that $𝒫(R)$ becomes a topological space with the understanding that a set $ℒ\subseteq 𝒫(R)$ is closed if $ℒ=h(k(ℒ))$.

The annihilator of a subset $S$ of $R$, $Ann(S)$ is defined as $Ann(S)=\{y\in R:xy=0\text{ for all }x\in S\}$. A ring $R$ is said to be reduced if it does not contain any non-zero nilpotents. Clearly, any subring of ${ℝ}^X$ is a reduced ring. A reduced ring $R$ is said to satisfy the annihilator condition, and is said to be an a.c. ring if for $x,y\in R$, there exists $z\in R$ such that $Ann(x)\cap Ann(y)=Ann(z)$ [4]. We observe that for $f,g\in C{(X)}_{𝒫}$, $Ann(f)\cap Ann(g)=Ann(f^2+g^2)$. Therefore, $C{(X)}_{𝒫}$ is an a.c. ring.

Theorem 2.11 ([11, Theorem 2.3]).

In a reduced ring $R$, for $a\in R$, $h(a)=𝒫(R)\setminus h(Ann(a))$.

Theorem 2.12 ([11, Theorem 2.7]).

If $R$ is a reduced ring and $S\subseteq R$, we have:

$$Ann(S)=kh(Ann(S))=\bigcap \{P\in 𝒫(R):Ann(S)\subseteq P\}$$

Lemma 2.13 ([11, Lemma 3.1]).

For $x,y\in R$, where $R$ is a reduced ring, $Ann(Ann(x))=Ann(y)$ if and only if $h(x)=h(Ann(y))$.

Theorem 2.14 ([11, Theorem 3.4]).

The following conditions are equivalent for a reduced ring $R$.

1. $𝒫(R)$ is compact and $R$ is an a.c. ring.

2. For $x\in R$, there exists $y\in R$ such that $Ann(Ann(y))=Ann(x)$. (Equivalently, For $x\in R$, there exists $y\in R$ such that $h(y)=h(Ann(x))$.)

We use the above results to see when is $𝒫(C{(X)}_{𝒫})$ compact.

Theorem 2.15.

The space $𝒫(C{(X)}_{𝒫})$ is compact if and only if for each $f\in C{(X)}_{𝒫}$ there exists $g\in C{(X)}_{𝒫}$ such that $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $in{t}_{X_{𝒫}}(Z_{𝒫}(f)\cap Z_{𝒫}(g))=\mathrm{\varnothing }$.

Proof 2.16.

Let us suppose that $𝒫(C{(X)}_{𝒫})$ is compact. Since $C{(X)}_{𝒫}$ is an a.c. ring, it follows from Theorem 2.14 that for $f\in C{(X)}_{𝒫}$, there exists $g\in C{(X)}_{𝒫}$ such that $h(g)=h(Ann(f))$. Let $P\in 𝒫(C{(X)}_{𝒫})$ be such that $Ann(f)\subseteq P$. Since $h(Ann(f))=h(g)$, $P\in h(g)$ and it follows that $g\in P$. By Theorem 2.12, $g\in Ann(f)$ and hence $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$. Next note that $h(g)=h(Ann(f))$ implies that $h(f)\cap h(g)=\mathrm{\varnothing }$ since otherwise there exists a minimal prime ideal $P$ of $C{(X)}_{𝒫}$ containing both $f$ and $g$. As $g\in P$, $Ann(f)\subseteq P$. But it follows from Lemma 2.9 that there exists $t\in Ann(f)\setminus P$ which is a not possible. Again, $h(f)\cap h(g)=\mathrm{\varnothing }$ implies that $h(f^2+g^2)=\mathrm{\varnothing }$. Indeed, if $P\in h(f^2+g^2)$, then $f^2+g^2\in P$. By Lemma 2.9, there exists $t\notin P$ such that $(f^2+g^2)t=\mathrm{𝟎}$ and thus, $ft=gt=\mathrm{𝟎}$. By Theorem 2.11 we get that $f,g\in P$ and so $P\in h(f)\cap h(g)$. By Theorem 2.11, $h(Ann(f^2+g^2))=𝒫(C{(X)}_{𝒫})\setminus h(f^2+g^2)$ where $h(f^2+g^2)=\mathrm{\varnothing }$. So we have $h(Ann(f^2+g^2))=𝒫(C{(X)}_{𝒫})$. Thus, $Ann(f^2+g^2)$ is contained in the intersection of all minimal prime ideals of $R$. It then follows from Lemma 2.8 and the fact that $C{(X)}_{𝒫}$ is a reduced ring that $Ann(f^2+g^2)=\{\mathrm{𝟎}\}$. It is then clear that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f^2+g^2)=\mathrm{\varnothing }$ which directly implies that $in{t}_{X_{𝒫}}(Z_{𝒫}(f)\cap Z_{𝒫}(g))=\mathrm{\varnothing }$.

Conversely let $f\in C{(X)}_{𝒫}$. Then there exists $g\in C{(X)}_{𝒫}$ such that $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $in{t}_{X_{𝒫}}(Z_{𝒫}(f)\cap Z_{𝒫}(g))=\mathrm{\varnothing }$. It is sufficient to show that $h(g)=h(Ann(f))$ and the rest will follow from Theorem 2.15. Indeed, $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ implies that $fg=\mathrm{𝟎}$, that is $g\in Ann(f)$ and so $h(Ann(f))\subseteq h(g)$. Next see that $in{t}_{X_{𝒫}}(Z_{𝒫}(f)\cap Z_{𝒫}(g))=\mathrm{\varnothing }$ implies that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f^2+g^2)=\mathrm{\varnothing }$ and so $Ann(f^2+g^2)=\{\mathrm{𝟎}\}$. It follows that $Ann(f^2+g^2)$ is contained in all minimal prime ideals of $R$ and hence $h(Ann(f^2+g^2))=𝒫(C{(X)}_{𝒫})$. It is now evident from Theorem 2.11 that $h(f^2+g^2)=\mathrm{\varnothing }$. Now, if $P\in h(g)$, then $P\notin h(f)$. Indeed, if $P\in h(f)\cap h(g)$, then $f,g\in P$ and so $f^2+g^2\in P$ which contradicts that $h(f^2+g^2)=\mathrm{\varnothing }$. Finally, it follows from Theorem 2.11 that $P\in h(Ann(f))$. Thus, $h(g)\subseteq h(Ann(f))$. This completes the proof.

The next corollary follows immediately using Observation 2.3.

Corollary 2.17.

If $C{(X)}_{𝒫}$ contains the collection $\{{\chi }_{\{p\}}:p\in X\}$, then the space $𝒫(C{(X)}_{𝒫})$ is compact if and only if for each $f\in C{(X)}_{𝒫}$ there exists $g\in C{(X)}_{𝒫}$ such that $Z_{𝒫}(f)=X\setminus Z_{𝒫}(g)$.

3. Zero-divisor graph of $C{(X)}_{𝒫}$

The objective of this section is to discuss the zero-divisor graph $\mathrm{\Gamma }(C{(X)}_{𝒫})$ of the ring $C{(X)}_{𝒫}$. We first characterise the zero-divisors of the ring $C{(X)}_{𝒫}$.

Theorem 3.1.

An element $f\in C{(X)}_{𝒫}\setminus \{\mathrm{𝟎}\}$ is a divisor of zero in the ring $C{(X)}_{𝒫}$ if and only if $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\ne \mathrm{\varnothing }$.

Proof 3.2.

Suppose there exists $g\in C{(X)}_{𝒫}\setminus \{\mathrm{𝟎}\}$ such that $fg=\mathrm{𝟎}$. Then $\mathrm{\varnothing }\ne X\setminus Z_{𝒫}(g)\subseteq Z_{𝒫}(f)$ and so $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\ne \mathrm{\varnothing }$. Conversely let $x\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$. Then there exists $h\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and so $fh=\mathrm{𝟎}$ where $h\ne \mathrm{𝟎}$.

The above theorem ensures that the vertex set $V$ of the three graphs which are under study in this article is given by:

$$V=\{f\in C{(X)}_{𝒫}:\mathrm{\varnothing }\ne in{t}_{X_{𝒫}}{Z}_{𝒫}(f)⫋X\}.$$

We aim to shed light on the distance between two vertices in $\mathrm{\Gamma }$. To achieve this, we first establish a lemma.

Lemma 3.3.

For $f,g\in V$, there exists a vertex adjacent to both $f$ and $g$ if and only if $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$.

Proof 3.4.

Let $x\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$. Then there exists $h\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$. This ensures that $h\in V$ and $h$ is adjacent to both $f$ and $g$. Conversely let there exist $h\in V$ adjacent to both $f$ and $g$. Then it follows that $\mathrm{\varnothing }⫋X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$.

The next theorem can be proven with the help of the above lemma and by following the steps in the proof of Lemma 1.2 in [4].

Theorem 3.5.

For $f,g\in V$,

1. $d(f,g)=1$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$,

2. $d(f,g)=2$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)\ne X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$,

3. $d(f,g)=3$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)\ne X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$.

In order to calculate the diameter and girth of $\mathrm{\Gamma }$, we need another lemma.

Lemma 3.6.

If $X$ contains (at least) three distinct elements $x,y$ and $z$, then there exists $f\in C{(X)}_{𝒫}$ such that .

Proof 3.7.

Let $A=\{x,y\}$, $B=\{y,z\}$ and $C=\{z,x\}$. Then $x\in X\setminus B$, which is open in $X$, and hence in $X_{𝒫}$. Thus, there exists $g\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(g)\subseteq X\setminus B$. Choose $f_x=\frac{1}{g(x)}g$. Then $f_x(x)=1$ and $f_x(B)=\{0\}$. Similarly, there exist $f_y,f_z\in C{(X)}_{𝒫}$ such that $f_y(y)=2$, $f_z(z)=3$ and $f_y(C)=0=f_z(A)$. Finally, take $f=f_x+f_y+f_z-\mathrm{𝟏}$.

Theorem 3.8.

If $X$ is a topological space which contains at least three points, then the $diam(\mathrm{\Gamma })=3$ and $gr(\mathrm{\Gamma })=3$.

Proof 3.9.

Let $X$ contain three distinct elements $a,b$ and $c$. Then by Lemma 3.6, there exists $f\in C{(X)}_{𝒫}$ such that . Choose $r,s\in ℝ$ such that . We then have and $c\in \{x\in X:f(x)>s\}\subseteq \{x\in X:f(x)\ge s\}$ where $\{x\in X:f(x)\le r\}$ and $\{x\in X:f(x)\ge s\}$ (resp. and $\{x\in X:f(x)>s\}$) are zero-sets (resp. cozero-sets) of functions in $C{(X)}_{𝒫}$. Therefore, there exist $f_1,g_1\in C{(X)}_{𝒫}$ such that $Z_{𝒫}(f_1)=\{x\in X:f(x)\le r\}$ and $Z_{𝒫}(g_1)=\{x\in X:f(x)\ge s\}$ and we have $Z_{𝒫}(f_1)\cap Z_{𝒫}(g_1)=\mathrm{\varnothing }$ and $b\notin Z_{𝒫}(f_1)\cup Z_{𝒫}(g_1)$. It follows from Theorem 3.5 that $d(f_1,g_1)=3$ and hence $diam(\mathrm{\Gamma })=3$.

Employing the steps used in the proof of Lemma 3.6, there exists $h\in C{(X)}_{𝒫}$ such that $h(x)=-1$, $h(y)=0$ and $h(z)=1$. Set $Z_{𝒫}(f_2)=\{t\in X:h(t)\le \frac{1}{2}\}$ and $Z_{𝒫}(g_2)=\{t\in X:h(t)\ge \frac{1}{2}\}$ for some $f_2,g_2\in C{(X)}_{𝒫}$. Then $y\in Z_{𝒫}(f_2)\cap Z_{𝒫}(g_2)$ which ensures that $Z_{𝒫}(f_2^2+g_2^2)\ne \mathrm{\varnothing }$. Thus, $f_2,g_2,f_2^2,g_2^2,f_2^2+g_2^2$ are all non-zero zero-divisors in $C{(X)}_{𝒫}$. Thus, there exists $k\in C{(X)}_{𝒫}$ such that $k(f_2^2+g_2^2)=\mathrm{𝟎}$ and so $k{f}_2=\mathrm{𝟎}=k{g}_2$. Also see that $X\setminus Z_{𝒫}(f_2)\subseteq Z_{𝒫}(g_2)$ and so $f_2{g}_2=\mathrm{𝟎}$. Therefore, $k,f_2,g_2$ constitutes a circle in $\mathrm{\Gamma }$. Thus, $gr(\mathrm{\Gamma })=3$.

When $|X|=2$, the graph $\mathrm{\Gamma }$ is of a particular form.

Theorem 3.10.

The following statements are equivalent.

1. $|X|=2$.

2. $\mathrm{\Gamma }$ is a bipartite graph.

3. $\mathrm{\Gamma }$ is a complete bipartite graph.

Proof 3.11.

We note that if $X=\{p,q\}$, then the vertex set $V=P\bigsqcup Q$, where $P=\{c{\chi }_{\{p\}}:c\in ℝ\setminus \{0\}\}$ and $Q=\{c{\chi }_{\{q\}}:c\in ℝ\setminus \{0\}\}$. It follows that $P$ and $Q$ are stable sets and each vertex in $P$ is adjacent to each vertex in $Q$. This ensures that $\mathrm{\Gamma }$ is a complete bipartite graph.

Moreover, if $|X|\ge 3$, then it follows from Theorem 3.8 that $\mathrm{\Gamma }$ contains a triangle and thus it cannot be a bipartite graph.

The following corollary is therefore immediate.

Corollary 3.12.

If $X$ is a space with exactly two points, then $diam(\mathrm{\Gamma })=2$ and $gr(\mathrm{\Gamma })=\mathrm{\infty }$.

The associated number of each vertex is crucial data when calculating the radius of a graph. Keeping this in mind, we calculate the associated number of each vertex of $\mathrm{\Gamma }$.

Theorem 3.13.

For $f\in V$,

Proof 3.14.

Let $f\in V$ and $X\setminus Z_{𝒫}(f)=\{p\}$. Let $g\in V\setminus \{f\}$. If $x\in Z_{𝒫}(g)$, then $fg=\mathrm{𝟎}$ and hence $d(f,g)=1$. Again if $x\notin Z_{𝒫}(g)$, then $Z_{𝒫}(g)\subseteq Z_{𝒫}(f)$. This shows that $x\notin Z_{𝒫}(f)\cup Z_{𝒫}(g)$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$. It follows from Theorem 3.5 that $d(f,g)=2$. Therefore, $e(f)=2$.

Now, let there exist distinct points $x$ and $y$ in $X\setminus Z_{𝒫}(f)$. Then $Z_{𝒫}(f)\cup \{x\}$ is a closed set in $X_{𝒫}$ and $y\notin Z_{𝒫}(f)\cup \{x\}$. Thus, by Theorem 2.1, there exists $h\in C{(X)}_{𝒫}$ such that $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq Z_{𝒫}(h)\subseteq X\setminus (Z_{𝒫}(f)\cup \{x\})$. This ensures that $x\notin Z_{𝒫}(f)\cup Z_{𝒫}(h)$ and $Z_{𝒫}(f)\cap Z_{𝒫}(h)=\mathrm{\varnothing }$. Therefore, we have from Theorem 3.5 that $d(f,g)=3$ and so $e(f)=3$.

Remark 3.15.

We realise that if $C{(X)}_{𝒫}$ contains at least one $f$ such that it is zero everywhere except exactly a single point, then $\rho (\mathrm{\Gamma })=2$. Recall that if $𝒫$ contains all singleton subsets of $X$ (e.g.: ${𝒫}_f$ and $𝒦$), then $C{(X)}_{𝒫}$ contains functions of the form ${\chi }_{\{p\}}$. In such cases, $\rho (\mathrm{\Gamma })=2$. In particular, $\rho (\mathrm{\Gamma }(C{(X)}_F))=2=\rho (\mathrm{\Gamma }(C{(X)}_K))$ Moreover, recall that the ring $T^{\prime }(X)=C{(X)}_{{𝒫}_{nd}}$ contains functions of the form ${\chi }_{\{p\}}$ for each $p\in X$, even though ${𝒫}_{nd}$ may not contain all singleton subsets of $X$. In conclusion, we have $\rho (\mathrm{\Gamma }(T^{\prime }(X)))=2$.

The above remark prompts us to realise that the existence of ${\chi }_{\{p\}}$ in $C{(X)}_{𝒫}$, for even a single $p\in X$ induces the radius of $\mathrm{\Gamma }$ to be $2$. The following corollary is an immediate consequence of this apprehension and can be deduced using Lemma 2.4.

Corollary 3.16.

If $C(X)⫋C{(X)}_{𝒫}$, then $\rho (\mathrm{\Gamma })=2$.

In the context of the zero-divisor graph being triangulated (resp. hypertriangulated), we first recall that following result established by Azarpanah and Motamedi in [4].

Theorem 3.17 ([4, Proposition 2.1]).

Suppose that $|X|>1$.

1. $\mathrm{\Gamma }(C(X))$ is a triangulated graph if and only if $X$ has no isolated points.

2. $\mathrm{\Gamma }(C(X))$ is a hypertriangulated graph if and only if $X$ is a connected middle $P$-space, i.e., if every non-empty zero set $Z\in Z[X]$ of the form $Z=E\cap F$, where $E,F\in Z[X]$ with $E\cup F=X$, has a non-empty interior.

We now realise when is a vertex in $\mathrm{\Gamma }$ a part of a triangle.

Theorem 3.18.

Let $f\in V$. Then $f$ is a vertex in a triangle if and only if $|X|\ge 3$ and $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$.

Proof 3.19.

Suppose that $|X|\ge 3$ and $x$ and $y$ are two distinct points in $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$. Then by Theorem 2.1, there exists $g\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(g)\subseteq X\setminus in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\setminus \{y\}$. It is evident that $g\in V$ and $fg=\mathrm{𝟎}$. Also, $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ and so by Lemma 3.3, there exists a vertex $h\in V$ such that $h$ is adjacent to both $f$ and $g$. Therefore, $f$, $g$ and $h$ form a triangle.

Conversely, let $f$ be a vertex of a triangle. Then it follows from Theorem 3.10 that $|X|\ge 3$. Now let $g,h\in V$ be such that $fg=\mathrm{𝟎}=gh=hf$. Then $(X\setminus Z_{𝒫}(g))\cup (X\setminus Z_{𝒫}(h))\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$. Since $gh=\mathrm{𝟎}$, $(X\setminus Z_{𝒫}(g))\cap (X\setminus Z_{𝒫}(h))=\mathrm{\varnothing }$ and hence $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$.

We achieve something more general and concrete when $C{(X)}_{𝒫}$ contains $C(X)$ properly.

Theorem 3.20.

If $C(X)⫋C{(X)}_{𝒫}$, then $\mathrm{\Gamma }$ is neither triangulated nor hypertriangulated.

Proof 3.21.

Since $C(X)⫋C{(X)}_{𝒫}$, by Lemma 2.4, there exists a non-isolated $p\in X$ such that $\{p\}\in 𝒫$ and hence $f={\chi }_{\{p\}}$ and $g={\chi }_{{}_{X\setminus \{p\}}}$ are in $V$. It follows from Theorem 3.18 that $g$ is not a vertex of a triangle and the edge joining $f$ and $g$ is not an edge of a triangle.

In this context, we must make the following remark.

Remark 3.22.

It is immediate from Theorem 3.20 that for $\mathrm{\Gamma }$ to be triangulated and/or hypertriangulated, $C{(X)}_{𝒫}$ must be equal to $C(X)$. Moreover, we can make the following conclusions by implementing Theorem 3.17.

1. $\mathrm{\Gamma }$ is triangulated if and only if $C{(X)}_{𝒫}=C(X)$ and $X$ has no isolated points.

2. $\mathrm{\Gamma }$ is hypertriangulated if and only if $C{(X)}_{𝒫}=C(X)$ and $X$ is a connected middle $P$-space.

In particular, if $X$ is a connected middle $P$-space with no isolated points, then note that $\mathrm{\Gamma }(C(X))$ is both triangulated and hypertriangulated but $\mathrm{\Gamma }$ is neither triangulated nor hypertriangulated, when $C{(X)}_{𝒫}⫌C(X)$. We state an example to support our remark.

Example 3.23.

Let $s$ denote the set of all $\{0,1\}$-valued transfinite sequences and the set of all upper elements of $s$ is denoted by $Q$. Then the Dedekind completion, $R$ of $Q$ (see [10]) is a connected almost $P$-space (and hence a middle $P$-space) (see [13]). Thus, $\mathrm{\Gamma }(C(R))$ is both triangulated and hypertriangulated; and if $𝒫⫌\{\mathrm{\varnothing }\}$, then $\mathrm{\Gamma }(C{(R)}_{𝒫})$ is neither triangulated nor hypertriangulated.

In particular, $\mathrm{\Gamma }(C{(R)}_F)$ (See [14]) (resp. $\mathrm{\Gamma }(T^{\prime }(R))$ and $\mathrm{\Gamma }(C{(R)}_K)$) is neither triangulated nor hypertriangulated.

Next we discuss the length of a cycle in $\mathrm{\Gamma }$ containing two specific vertices $f,g\in V$, which can be proved by following closely the proof of Proposition 2.2 in [4].

Theorem 3.24.

For a topological space $X$ and $f,g\in V$, the following assertions are true.

1. $c(f,g)=3$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$.

2. $c(f,g)=4$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)\ne X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$ or $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$.

3. $c(f,g)=6$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)\ne X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$.

At this point we recall Observation 2.3 to obtain the following remark.

Remark 3.25.

If $\{{\chi }_{\{p\}}:p\in X\}\subseteq C{(X)}_{𝒫}$ and $f,g\in C{(X)}_{𝒫}$, we have

1. $c(f,g)=3$ if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $Z_{𝒫}(f)\cap Z_{𝒫}(g)\ne \mathrm{\varnothing }$.

2. $c(f,g)=4$ if and only if either $Z_{𝒫}(f)\cup Z_{𝒫}(g)\ne X$ and $Z_{𝒫}(f)\cap Z_{𝒫}(g)\ne \mathrm{\varnothing }$ or $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $Z_{𝒫}(f)\cap Z_{𝒫}(g)=\mathrm{\varnothing }$. In particular, if $X\setminus Z_{𝒫}(f)=Z_{𝒫}(g)$, then $c(f,g)=4$.

3. $c(f,g)=6$ if and only if $Z_{𝒫}(f)⫋X\setminus Z_{𝒫}(g)$ or $Z_{𝒫}(g)⫋X\setminus Z_{𝒫}(f)$.

We now realise that clique number of $\mathrm{\Gamma }$ coincides with the cellularity of $X_{𝒫}$, and hence also with $dim(C{(X)}_{𝒫})$.

Theorem 3.26.

For a $\tau 𝒫$-space $(X,\tau ,𝒫)$, $c(X_{𝒫})=\omega \mathrm{\Gamma }=dim(C{(X)}_{𝒫})$.

Proof 3.27.

Suppose $H$ is a complete subgraph of $\mathrm{\Gamma }$. Then for $f,g\in H$, $fg=0$ and so $X\setminus Z_{𝒫}(f)\cap X\setminus Z_{𝒫}(g)=\mathrm{\varnothing }$. Hence the family, $\{X\setminus Z_{𝒫}(f):f\in H\}$ is a collection of pairwise disjoint open sets in $X_{𝒫}$. It follows that $|H|\le c(X_{𝒫})$ and since $H$ is an arbitrary complete subgraph of $\mathrm{\Gamma }$, $\omega \mathrm{\Gamma }\le c(X_{𝒫})$

Conversely let $\mathscr{H}$ be a collection of pairwise disjoint non-empty open subsets of $X_{𝒫}$. For $A\in \mathscr{H}$ and $p\in A$, there exists $h_1\in C{(X)}_{𝒫}$ such that $p\in X\setminus Z_{𝒫}(h_1)\subseteq A$. Fix $h_A=\frac{1}{h_1(p)}{h}_1$. Then $h_A(p)=1$ and $h_A(X\setminus A)=\{0\}$. Let $H=\{h_A:A\in \mathscr{H}\}$. Then each member of $H$ is a non-zero zero-divisor in $C{(X)}_{𝒫}$ and for each $A,B\in \mathscr{H}$, $f_A{f}_B=\mathrm{𝟎}$ as $A\cap B=\mathrm{\varnothing }$. Therefore, $H$ is a complete subgraph of $\mathrm{\Gamma }$. Thus, we have $|\mathscr{H}|=|H|\le \omega \mathrm{\Gamma }$ and as $\mathscr{H}$ is an arbitrary collection of pairwise disjoint non-empty open subsets of $X_{𝒫}$, $c(X_{𝒫})\le \omega \mathrm{\Gamma }$. The last equality follows directly from Theorem 2.6.

The following remark is obvious from Observation 2.3.

Remark 3.28.

If $C{(X)}_{𝒫}$ contains the collection $\{{\chi }_{\{p\}}:p\in X\}$, then $dim(C{(X)}_{𝒫})=\omega \mathrm{\Gamma }$ is nothing but the cardinality of $X,|X|$.

In particular, $dim(C{(X)}_F)=|X|=dim(C{(X)}_K)=dim(T^{\prime }(X))$ and $\omega \mathrm{\Gamma }(C{(X)}_F)=|X|=\omega \mathrm{\Gamma }(C{(X)}_K)=\omega \mathrm{\Gamma }(T^{\prime }(X))$.

We now proceed to discuss when is $\mathrm{\Gamma }$ a complemented graph. Recall that two vertices $f,g\in V$ are orthogonal if and only if they are adjacent and there is no vertex $h\in V$ which is adjacent to both $f$ and $g$. It follows now from Lemma 3.3 and Theorem 3.5(i) that $f,g\in V$ are orthogonal if and only if $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$. The following theorem is now immediate.

Theorem 3.29.

$\mathrm{\Gamma }$ is a complemented graph if and only if for each $f\in V$, there exists $g\in \mathrm{\Gamma }$ such that $Z_{𝒫}(f)\cup Z_{𝒫}(g)=X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$.

The following corollary is now immediate with the assistance of Theorem 2.15.

Corollary 3.30.

The zero-divisor graph $\mathrm{\Gamma }$ of the ring $C{(X)}_{𝒫}$ is a complemented graph if and only if the space of minimal prime ideals $𝒫(C{(X)}_{𝒫})$, of $C{(X)}_{𝒫}$ is compact.

An interesting characterisation of $C{(X)}_{𝒫}$ as a Von-Neumann regular ring can be achieved using the above theories, provided $C{(X)}_{𝒫}$ contains the collection $\{{\chi }_{\{p\}}:p\in X\}$. A ring $R$ is said to be Von-Neumann regular if for each $a\in R$, there exists $x\in R$ such that $a=axa$. We first recall the following result.

Theorem 3.31 ([7, Theorem 5.7]).

A $\tau 𝒫$-space, $(X,\tau ,𝒫)$ is a $𝒫P$-space (i.e., $C{(X)}_{𝒫}$ is a Von-Neumann regular ring) if and only if for $f\in C{(X)}_{𝒫}$, there exists $g\in C{(X)}_{𝒫}$ with $Z_{𝒫}(f)=X\setminus Z_{𝒫}(g)$.

The next observation follows directly by using the above discussions and Observation 2.3.

Theorem 3.32.

Consider a ${\tau }_{𝒫}$-space $(X,\tau ,𝒫)$ such that $C{(X)}_{𝒫}\supseteq \{{\chi }_{\{p\}}:p\in X\}$. Then the following statements are equivalent.

1. The zero-divisor graph $\mathrm{\Gamma }$ of $C{(X)}_{𝒫}$ is a complemented graph.

2. The space of minimal prime ideals $𝒫(C{(X)}_{𝒫})$ of $C{(X)}_{𝒫}$ is compact.

3. $C{(X)}_{𝒫}$ is a Von-Neumann regular ring.

Remark 3.33.

Since the rings $C{(X)}_F$ [9], $C{(X)}_K$ [7] and $T^{\prime }(X)$ [2] all contain the collection $\{{\chi }_{\{p\}}:p\in X\}$, Theorem 3.32 is valid for each of the aforementioned rings. Moreover, the ring $T^{\prime }(X)$ is a Von-Neumann regular ring, for any topological space $X$ and hence it follows that $\mathrm{\Gamma }(T^{\prime }(X))$ is always a complemented graph and the space of minimal prime ideals of $T^{\prime }(X)$ is compact, for any topological space $X$.

4. Annihilator graph on $C{(X)}_{𝒫}$

In this section, we discuss the annihilator graph on the ring $C{(X)}_{𝒫}$. For simplicity, we write $AG$ instead of $AG(C{(X)}_{𝒫})$ when there is no ambiguity. Note that $\mathrm{\Gamma }$ is a spanning subgraph of $AG$. We begin with a lemma, which was established by us in [8] and sketch its proof to make this article self-contained.

Lemma 4.1.

Let $f,g\in C{(X)}_{𝒫}$. Then $in{t}_{X_{𝒫}}(Z_{𝒫}(f))\subseteq in{t}_{X_{𝒫}}(Z_{𝒫}(g))$ if and only if $Ann(f)\subseteq Ann(g)$.

Proof 4.2.

Suppose $in{t}_{X_{𝒫}}(Z_{𝒫}(f))\subseteq in{t}_{X_{𝒫}}(Z_{𝒫}(g))$ and $h\in Ann(f)$. Then we have $X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}(Z_{𝒫}(f))\subseteq in{t}_{X_{𝒫}}(Z_{𝒫}(g))\subseteq Z_{𝒫}(g)$ and so $h\in Ann(g)$. Conversely let $x\in in{t}_{X_{𝒫}}(Z_{𝒫}(f))$. Then there exists $h\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}(Z_{𝒫}(f))$ which implies that $h\in Ann(f)\subseteq Ann(g)$ and so $x\in X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}(Z_{𝒫}(g))$.

Now, we provide a way to check when two vertices in $AG$ are adjacent to each other.

Theorem 4.3.

Let $f,g\in V$, then $f$ and $g$ are adjacent in $AG$ if and only if $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)⊈in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)⊈in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$.

Proof 4.4.

Let $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$. Then it can be easily observed that $Ann(fg)=Ann(f)\cup Ann(g)$. Indeed, if $h\in Ann(fg)$, then $X\setminus Z(gh)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq Z_{𝒫}(g)\subseteq Z_{𝒫}(fg)$, which is possible only if $Z_{𝒫}(gh)=X$, that is $gh=\mathrm{𝟎}$ and $h\in Ann(g)\subseteq Ann(f)\cup Ann(g)$. Therefore, $f$ and $g$ are non-adjacent. Analogously, it can be shown that if $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$, then also $f$ and $g$ are non-adjacent.

Conversely let $f$ and $g$ be non-adjacent. Then $Ann(f)\cup Ann(g)=Ann(fg)$. Since the union of two ideals is an ideal if and only if one contains the other, it follows that $Ann(f)\subseteq Ann(g)$ or $Ann(g)\subseteq Ann(f)$. The conclusion then follows from Lemma 4.1.

The next question that we address is when the graph $AG$ equals $\mathrm{\Gamma }$.

Theorem 4.5.

The graphs $AG$ and $\mathrm{\Gamma }$ coincide if and only if $|X|\le 2$.

Proof 4.6.

We first aim to show that if $X$ contains three distinct points, then $AG$ and $\mathrm{\Gamma }$ are unequal as graphs. Let $x,y,z$ be three distinct points in $X$. Then, by Theorem 2.1, there exist $f,g\in C{(X)}_{𝒫}$ such that $\{x,y\}\subseteq X\setminus Z_{𝒫}(f)\subseteq c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(f))\subseteq X\setminus \{z\}$ and $\{x,z\}\subseteq X\setminus Z_{𝒫}(g)\subseteq c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(g))\subseteq X\setminus \{y\}$. It then follows that $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\setminus in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and $z\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\setminus in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$. Therefore, $f$ and $g$ are adjacent in $AG$, but non-adjacent in $\mathrm{\Gamma }$, as $x\notin (Z_{𝒫}(f)\cup Z_{𝒫}(g))$. Now if $X=\{p,q\}$, then $V=P\bigsqcup Q$, where $P=\{c{\chi }_{\{p\}}:c\in ℝ\setminus \{0\}\}$ and $Q=\{d{\chi }_{\{q\}}:d\in ℝ\setminus \{0\}\}$. See that $Ann(c{\chi }_{\{p\}})=Ann({\chi }_{\{p\}})$ for any $c\in ℝ\setminus \{0\}$ and hence $P$ is a stable set in $AG$. Analogously, $Q$ is a stable set in $AG$. The rest follows from Theorem 3.10 and the fact that $\mathrm{\Gamma }$ is a spanning subgraph of $AG$.

The next corollary follows immediately using Theorem 3.10.

Corollary 4.7.

$AG$ is a complete bipartite graph if and only if $|X|=2$.

Our next aim is to compute the diameter of $AG$. We first establish a result describing the existence of a vertex adjacent to two distinct vertices in $AG$.

Theorem 4.8.

Let $f,g\in V$. Then there exists a vertex adjacent to both $f$ and $g$ in $AG$ if and only if either $fg\ne \mathrm{𝟎}$ on $X$ or $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$ and $|in{t}_{X_{𝒫}}{Z}_{𝒫}(g)|\ge 2$.

Proof 4.9.

First suppose that $fg\ne \mathrm{𝟎}$ and choose $x\in X\setminus Z_{𝒫}(fg)$. By Theorem 2.1, there exists $h\in C{(X)}_{𝒫}$ such that $x\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq X\setminus Z_{𝒫}(fg)⫋X$. It is clear that $h\in V$ and is adjacent to both $f$ and $g$ in $AG$. Now, let $fg=\mathrm{𝟎}$ and $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$ and $|in{t}_{X_{𝒫}}{Z}_{𝒫}(g)|\ge 2$. If $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$, then there exists a vertex adjacent to $f$ and $g$ in $\mathrm{\Gamma }$, and hence in $AG$. If not, then there exist distinct points $x,y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and $z,w\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$. By Theorem 2.1, there exists $h\in C{(X)}_{𝒫}$ such that $\{x,z\}\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq X\setminus \{y,w\}$. It follows that $h\in V$ and is adjacent to both $f$ and $g$.

Finally, assume that $fg=\mathrm{𝟎}$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=\{x\}$. Then $Z_{𝒫}(g)=X\setminus \{x\}$, which is open in $X_{𝒫}$ and so $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=X\setminus \{x\}$. Let $h\in V$ be adjacent to $g$ in $AG$. Then $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)⊈in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$ which implies that $x\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$. But $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=\{x\}\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$, which implies that $f$ and $h$ are non-adjacent.

We obtain the following result as a corollary.

Corollary 4.10.

The distance between two non-adjacent vertices in $AG$ is 2.

Proof 4.11.

Let $f,g\in V$ be non-adjacent in $AG$. Then they are also non-adjacent in $\mathrm{\Gamma }$. Therefore, $fg\ne \mathrm{𝟎}$ on $X$. It follows from Theorem 4.8 that there exists $h\in V$, which is adjacent to both $f$ and $g$. It follows that $d(f,g)=2$.

It is also evident from Theorem 4.8 that if $f\in V$, then $d(f,2f)=2$ and it follows from Corollary 4.10 that $e(f)\le 2$ for any $f\in V$. Consequently, the following conclusions can be drawn about the graph $AG$.

Theorem 4.12.

In the graph $AG$, the following assertions hold.

1. $e(f)=2$ for any $f\in V$.

2. $\rho (AG)=2$

3. Each vertex $f\in V$ is a center.

4. $diam(AG)=2$

Again, if $f$ and $g$ are vertices of an edge in $AG$ which is not part of a triangle, then by Theorem 4.8, $fg=\mathrm{𝟎}$ on $X$ and either $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ or $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ is a singleton set. In light of these arguments, we establish the following result.

Theorem 4.13.

$AG$ is hypertriangulated if and only if $C{(X)}_{𝒫}=C(X)$ and $X$ has no isolated points.

Proof 4.14.

Suppose that $C(X)⫋C{(X)}_{𝒫}$. Then there exists a point $p$ in $X$ such that $\{p\}\in 𝒫$. Therefore, ${\chi }_{\{p\}}$ and ${\chi }_{{}_{X\setminus \{p\}}}$ constitutes an edge in $AG$ and by Theorem 4.8, this edge is not part of a triangle. Now, let $p$ be an isolated point in $X$. Then ${\chi }_{\{p\}}$ and ${\chi }_{{}_{X\setminus \{p\}}}$ are in $C(X)$ and again ${\chi }_{\{p\}}$ and ${\chi }_{{}_{X\setminus \{p\}}}$ constitutes an edge which is not a part of a triangle.

Conversely let $AG$ be not hypertriangulated. Then there exists an edge joining vertices $f$ and $g$ which is not part of a triangle. Then by Theorem 4.8, $fg=\mathrm{𝟎}$ and either $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ or $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ is a singleton set. Without loss of generality let $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=\{p\}.$ Then there exists a basic open set $X\setminus Z_{𝒫}(h)$ in $X_{𝒫}$ such that $p\in X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=\{p\}$ and so $X\setminus Z_{𝒫}(h)=\{p\}$. But $h=h(p){\chi }_{\{p\}}$ and so ${\chi }_{\{p\}}\in C{(X)}_{𝒫}$. Therefore, either $p$ is an isolated point in $X$ or ${\chi }_{\{p\}}\in C{(X)}_{𝒫}\setminus C(X)$.

The next result, which was already established in [15], also follows from Theorem 4.13.

Corollary 4.15.

$AG(C(X))$ is hypertriangulated if and only if $X$ has no isolated points.

We now address when is $AG$ a triangulated graph.

Theorem 4.16.

The graph $AG$ is triangulated if and only if $|X|\ge 3$.

Proof 4.17.

Suppose that $|X|\ge 3$ and let $f\in V$. If $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ is a singleton set, $\{x\}$, then there exist distinct points $y,z\in X\setminus \{x\}$. By Theorem 2.1, there exist $g,h\in C{(X)}_{𝒫}$ such that $z\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq X\setminus \{x,y\}$ and so $g\in V$ with $f$ and $g$ adjacent in $AG$. Analogously, there exists $h\in V$ with $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq X\setminus \{x,z\}$ which is adjacent to $f$ as well. Moreover, $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\setminus in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ and $z\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\setminus in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$, which ensures that $g$ and $h$ are also adjacent. Thus, $f$, $g$ and $h$ constitutes a triangle in $AG$. Now, if $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$, then by Theorem 3.18 $f$ is a vertex of a triangle in $\mathrm{\Gamma }$ and the rest follows from the fact that $\mathrm{\Gamma }$ is a spanning subgraph of $AG$. The converse follows from Corollary 4.7.

We next address the question of complementation of the graph $AG$.

Theorem 4.18.

The following assertions are true for the graph $AG$.

1. A pair of vertices $f,g\in V$ are orthogonal if and only if $fg=\mathrm{𝟎}$ on $X$ and either $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ or $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ is singleton.

2. $AG$ is a complemented graph if and only if $|X|\le 3$.

Proof 4.19.

1. If $f⟂g$, then $f$ and $g$ are adjacent and there does not exist any vertex adjacent to both. By Theorem 4.8, $fg=\mathrm{𝟎}$ and either $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ or $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ is singleton. Conversely let $f,g\in V$ be such that $fg=\mathrm{𝟎}$ and either $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ or $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ is singleton. Therefore, $f$ and $g$ are adjacent in $\mathrm{\Gamma }$, and hence in $AG$ and by Theorem 4.8, there is no vertex adjacent to both $f$ and $g$.

2. By Corollary 4.7, $AG$ is complemented if $|X|=2$. Let $|X|=3$ and $f\in V$. Then either $Z_{𝒫}(f)$ or $X\setminus Z_{𝒫}(f)$ is singleton. It follows that ${\chi }_{{}_{Z_{𝒫}(f)}}\in V$ and by the above result, $f⟂{\chi }_{{}_{Z_{𝒫}(f)}}$. Finally, let $|X|\ge 4$ and $x,y,z,w$ be four distinct points in $X$. By Theorem 2.1, there exists $f\in V$ such that $\{x,y\}\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq Z_{𝒫}(f)\subseteq X\setminus \{z,w\}$. If possible let there exist $g\in V$ such that $f⟂g$. Then $X\setminus Z_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ which implies that $\{z,w\}\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$. This contradicts that $f⟂g$.

Theorem 4.20.

$\{f,g\}\subseteq V$ is a dominating set in $AG$ if and only if $fg=\mathrm{𝟎}$ on $X$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$.

Proof 4.21.

Suppose $\{f,g\}$ constitutes a dominating set in $AG$. If $fg\ne \mathrm{𝟎}$, then $fg\in V$ and is non-adjacent to both $f$ and $g$. If $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$, then $f^2+g^2\in V$ and is non-adjacent to both $f$ and $g$.

Conversely let there exist $h\in V$ which is non-adjacent to both $f$ and $g$ in $AG$. Then there are four possibilities:

1. $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$

2. $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$

3. $in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$

4. $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$

The first three possibilities contradict the fact that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\mathrm{\varnothing }$. Now, assume the fourth possibility. Since $fg=\mathrm{𝟎}$, $X\setminus Z_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$ and $X\setminus Z_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$. Thus, $X\setminus Z_{𝒫}(f)\cup X\setminus Z_{𝒫}(g)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$ and so, $X\setminus Z_{𝒫}(h)\subseteq X\setminus in{t}_{X_{𝒫}}{Z}_{𝒫}(h)\subseteq Z_{𝒫}(f)\cap Z_{𝒫}(g)$. Therefore, $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cap in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$, which contradicts the hypothesis.

We see that if $C(X)⫋C{(X)}_{𝒫}$, then there exists a point $p\in X$ such that ${\chi }_{\{p\}},{\chi }_{{}_{X\setminus \{p\}}}\in C{(X)}_{𝒫}$ and the same thing happens when $p$ is an isolated point in $X$. This observation leads to the following corollary.

Corollary 4.22.

$dt(AG)=2$ if either $C(X)⫋C{(X)}_{𝒫}$ or $X$ has an isolated point.

The following corollary follows from Lemma 3.3, Theorem 4.20 and Corollary 4.22.

Corollary 4.23.

$dt(AG)=2$ if and only if $\mathrm{\Gamma }$ is not hypertriangulated.

5. Weakly zero-divisor graph on $C{(X)}_{𝒫}$

The aim of this section is to study the weakly zero-divisor graph $W\mathrm{\Gamma }$ of $C{(X)}_{𝒫}$. The following observation has been discussed in the introduction, in a more general setting. We state it explicitly for ease of reference.

Observation 5.1.

If $\le$ denotes the spanning subgraph relation between two graphs over the same vertex set, then we have:

$$\mathrm{\Gamma }\le AG\le W\mathrm{\Gamma }.$$

We next realise that the weakly zero-divisor graph and zero-divisor graph of $C{(X)}_{𝒫}$ coincide if and only if $X$ contains exactly two points. To see this, we first check what happens when $|X|=2$.

Theorem 5.2.

If $|X|=2$, $W\mathrm{\Gamma }$ is a complete bipartite graph and in this case, the graphs $W\mathrm{\Gamma }$, $AG$ and $\mathrm{\Gamma }$ are equal.

Proof 5.3.

If $X=\{p,q\}$, then $V=P\bigsqcup Q$, where $P=\{c{\chi }_{\{p\}}:c\in ℝ\setminus \{0\}\}$ and $Q=\{c{\chi }_{\{q\}}:c\in ℝ\setminus \{0\}\}$. Note that for any $f\in P$, $Ann(f)=Q\cup \{\mathrm{𝟎}\}$ and for $g\in Q$, $Ann(g)=P\cup \{\mathrm{𝟎}\}$ and thus $P$ and $Q$ are stable sets in $W\mathrm{\Gamma }$. Therefore, $W\mathrm{\Gamma }$ is a complete bipartite graph. Moreover, it has been noted in Theorem 3.10 and Corollary 4.7 that the same structure is exhibited by $\mathrm{\Gamma }$ and $AG$ as well. Therefore, the three graphs coincide.

Note that a complete bipartite graph is uniquely complemented. This gives us the following corollary.

Corollary 5.4.

If $|X|=2$, $W\mathrm{\Gamma }$ is a uniquely complemented graph.

The following corollary follows from Observation 5.1, Theorem 5.2 and Theorem 4.5.

Corollary 5.5.

The graphs $W\mathrm{\Gamma }$ and $\mathrm{\Gamma }$ coincide if and only if $|X|=2$.

In order to explore the condition under which $AG$ coincides with $W\mathrm{\Gamma }$, we establish the adjacency relation between two vertices in $W\mathrm{\Gamma }$.

Theorem 5.6.

Two distinct vertices $f,g\in V$ are adjacent in $W\mathrm{\Gamma }$ if and only if $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cup in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ contains at least two points.

Proof 5.7.

First note that since $f$ and $g$ are zero-divisors, $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\ne \mathrm{\varnothing }$ and $in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\ne \mathrm{\varnothing }$. Now, assume that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cup in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ contains exactly one point $p$. Then $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\{p\}$. If possible let there exist $h\in Ann(f)\setminus \{\mathrm{𝟎}\}$ and $k\in Ann(g)\setminus \{\mathrm{𝟎}\}$ such that $hk=\mathrm{𝟎}$. Then $\mathrm{\varnothing }\ne X\setminus Z_{𝒫}(h)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and $\mathrm{\varnothing }\ne X\setminus Z_{𝒫}(k)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ where $X\setminus Z_{𝒫}(h)\cap X\setminus Z_{𝒫}(k)=\mathrm{\varnothing }$; which is impossible as $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ is a singleton set. Therefore, no such $h$ and $k$ can exist and so $f$ and $g$ are not adjacent.

Let us now suppose that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cup in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ contains at least two distinct points. Choose $x\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ such that $x\ne y$. By Theorem 2.1, there exist $h,k\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(h)\subseteq c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(h))\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\setminus \{y\}$. Again $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\setminus c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(h))$ and so there exists $k\in C{(X)}_{𝒫}$ such that $y\in X\setminus Z_{𝒫}(k)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)\setminus c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(h))$. Therefore, we have $h\in Ann(f)\setminus \{\mathrm{𝟎}\}$, $k\in Ann(g)\setminus \{\mathrm{𝟎}\}$ and $hk=\mathrm{𝟎}$. Hence, $f$ and $g$ are adjacent vertices.

Corollary 5.8.

If $f\in V$ is such that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ contains at least two distinct points, then $f$ is adjacent to all other vertices in $W\mathrm{\Gamma }$. That is, $f$ is a universal vertex.

By using Theorem 5.6, we realise when the graph $W\mathrm{\Gamma }$ coincides with the graph $AG$.

Theorem 5.9.

The graphs $AG$ and $W\mathrm{\Gamma }$ coincide if and only if $|X|\le 2$.

Proof 5.10.

Suppose $x,y,z$ are three distinct points in $X$. By Theorem 2.1, there exists $f\in C{(X)}_{𝒫}$ such that $\{x,y\}\subseteq X\setminus Z_{𝒫}(f)\subseteq c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(f))\subseteq X\setminus \{z\}$. Moreover, $Z_{𝒫}(f)\cup \{y\}$ is a closed set in $X_{𝒫}$, which misses $x$. Again, by Theorem 2.1, there exists $g\in C{(X)}_{𝒫}$ such that $x\in X\setminus Z_{𝒫}(g)\subseteq c{l}_{X_{𝒫}}(X\setminus Z_{𝒫}(g))\subseteq X\setminus (Z_{𝒫}(f)\cup \{y\})$. Therefore, $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\subseteq Z_{𝒫}(f)\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ and so $f$ and $g$ are not adjacent in $AG$. However, $\{y,z\}\subseteq in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cup in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$, where $y\ne z$ and so $f$ and $g$ adjacent in $W\mathrm{\Gamma }$. Thus, $AG$ is a proper (spanning) subgraph of $W\mathrm{\Gamma }$. The rest follows from Observation 5.1 and Corollary 5.5.

Observe that if $f,g\in V$ are such that $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=in{t}_{X_{𝒫}}{Z}_{𝒫}(g)=\{p\}$ for some $p\in X$, then $p$ is an isolated point in $X_{𝒫}$. Therefore, there exists a basic open set $X\setminus Z_{𝒫}(h)$ for some $h\in C{(X)}_{𝒫}$ such that $X\setminus Z_{𝒫}(h)=\{p\}$ and hence $fh=gh=\mathrm{𝟎}$ on $X$. Therefore, $h$ is adjacent to both $f$ and $g$ in $W\mathrm{\Gamma }$. In this case, $d(f,g)=2$. Consequently, we have the following result.

Theorem 5.11.

For two distinct vertices $f,g\in V$,

Corollary 5.12.

Moreover, it follows from Corollary 5.8 that if there exists $f\in C{(X)}_{𝒫}$ with $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$, then $\{f\}$ is a dominating set. If no such $f$ exists, then $|X|=2$ and hence $W\mathrm{\Gamma }$ is a complete bipartite graph. Thus, we have the following result.

Theorem 5.13.

The dominating number of $W\mathrm{\Gamma }$ is given by:

We now establish when this graph is a complete graph.

Theorem 5.14.

$W\mathrm{\Gamma }$ is a complete graph if and only if $X$ has no isolated points and $C{(X)}_{𝒫}=C(X)$.

Proof 5.15.

If $p\in X$ is an isolated point in $X$, then $f={\chi }_{{}_{X\setminus \{p\}}}$ and $2f$ are in $V$ and are non-adjacent. Now, let $f\in C{(X)}_{𝒫}\setminus C(X)$. Then by Lemma 2.4, $\{p\}\in 𝒫$. This implies that the functions $f={\chi }_{{}_{X\setminus \{p\}}}$ and $2f$ are in $V$ and are non-adjacent. Therefore, $W\mathrm{\Gamma }$ is not a complete graph. If $X$ has no isolated points and $C{(X)}_{𝒫}=C(X)$, then for each $f\in V$, $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ consists of at least two points and so is adjacent to all other vertices. Therefore, $W\mathrm{\Gamma }$ is a complete graph.

The weakly zero-divisor graphs of the well-known rings $C{(X)}_F$, $C{(X)}_K$ and $T^{\prime }(X)$ are not complete. On the other hand, if $X$ is a connected topological space, then $W\mathrm{\Gamma }(C(X))$ is a complete graph.

Moreover, the following corollary is immediate from Remark 3.22 and Theorem 4.13.

Corollary 5.16.

The following statements are equivalent.

1. $X$ has no isolated points and $C{(X)}_{𝒫}=C(X)$.

2. $\mathrm{\Gamma }$ is triangulated.

3. $AG$ is a hypertriangulated graph.

4. $W\mathrm{\Gamma }$ is a complete graph.

Next we proceed to discuss when is $W\mathrm{\Gamma }$ a triangulated graph.

Theorem 5.17.

The following assertions are equivalent.

1. $|X|\ge 3$

2. $W\mathrm{\Gamma }$ is triangulated.

3. $W\mathrm{\Gamma }$ is hypertriangulated.

Proof 5.18.

If $|X|=2$, then we have seen that $W\mathrm{\Gamma }$ is a complete bipartite graph, and so is neither triangulated nor hypertriangulated.

Let $|X|\ge 3$ and $f\in V$. If $|in{t}_{X_{𝒫}}{Z}_{𝒫}(f)|\ge 2$, then $f$, $2f$ and $3f$ constitute a triangle. Suppose $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)=\{x\}$, for some $x\in X$. Choose $y,z\in X$ such that $x,y,z$ are distinct. Then there exist $g,h\in C{(X)}_{𝒫}$ such that $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ and $z\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$, by Theorem 2.1 and Theorem 5.6, $f$, $g$ and $h$ constitute a triangle. Again see that if $f,g\in V$ form an edge, then $in{t}_{X_{𝒫}}{Z}_{𝒫}(f)\cup in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ contains atleast two points. If $x\in in{t}_{X_{𝒫}}{Z}_{𝒫}(f)$ and $y\in in{t}_{X_{𝒫}}{Z}_{𝒫}(g)$ with $x\ne y$, then choose $z\in X\setminus \{x,y\}$ and $h\in C{(X)}_{𝒫}$ such that $z\in in{t}_{X_{𝒫}}{Z}_{𝒫}(h)$. It then follows that $f$, $g$ and $h$ are vertices of a triangle.

As a hypertriangulated graph cannot be complemented, the following result follows from Corollary 5.4.

Corollary 5.19.

$W\mathrm{\Gamma }$ is (uniquely) complemented if and only if $|X|\le 2$.

The following conclusions can be drawn about the weakly zero-divisor graph $W\mathrm{\Gamma }(C(X))$ of the ring $C(X)$.

Theorem 5.20.

The following assertions hold for a Tychonoff space $X$.

1. Two vertices $f,g$ in $W\mathrm{\Gamma }(C(X))$ are adjacent if and only if $intZ(f)\cup intZ(g)$ contains at least two distinct points.

2. A vertex $f\in W\mathrm{\Gamma }(C(X))$ is a universal vertex if and only if $intZ(f)$ contains at least two distinct points.

3. For $f,g\in W\mathrm{\Gamma }(C(X))$, .

5. $W\mathrm{\Gamma }(C(X))$ is a complete graph if and only if $X$ has no isolated points.

6. The following statements are equivalent.

   1. $W\mathrm{\Gamma }(C(X))$ is triangulated.

   2. $W\mathrm{\Gamma }(C(X))$ is hypertriangulated.

   3. $W\mathrm{\Gamma }(C(X))$ is not complemented.

   4. $|X|\ge 3$.