Interplay between various graphs on C(X)ρ

The first author is immensely grateful for the award of research fellowship provided by the University Grants Commission, New Delhi (NTA Ref. No. 221610014636).

Abstract:

This article focuses on the study of zero-divisor graph Γ(C(X)_P), annihilator graph AG(C(X)_P) and weakly zero-divisor graph WΓ(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 Γ(C(X)_P) is 2 and it is not triangulated. Moreover, in this situation, both Γ(C(X)_P) and AG(C(X)_P) are not hypertriangulated and the dominating number of AG(C(X)_P) is 2. Furthermore, WΓ(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 Γ(C(X)_P) and the Von-Neumann regularity of C(X)_P under the assumption that C(X)_P⊇{χ_{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 WΓ(C(X)_P) being triangulated, hypertriangulated and complemented depend solely on the cardinality of X.

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