Interplay between various graphs on C(X)ρ

Abstract

[EN] This article focuses on the study of zero-divisor graph ?(C(X)P), annihilator graph AG(C(X)P) and weakly zero-divisor graph W?(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 ?(C(X)P) is 2 and it is not triangulated. Moreover, in this situation, both ?(C(X)P) and AG(C(X)P) are not hypertriangulated and the dominating number of AG(C(X)P) is 2. Furthermore, W?(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 ?(C(X)P) and the Von-Neumann regularity of C(X)P under the assumption that C(X)P?{?{p}:p?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?(C(X)P) being triangulated, hypertriangulated and complemented depend solely on the cardinality of X.