Graph theoretic representation of rings of continuous functions

Abstract

In this paper, we introduce a graph structure, called zero-set intersection graph ?(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the inter-relationship of cliques of ?(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of ?(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.

On Perfectness of Annihilating-Ideal Graph of $\mathbb{Z}_n$

“…Over the last two decades, various graphs defined on rings has become an interesting topic of research. Various graphs like [1], [3], [4], [5], [6], [7], [8], [9] have been constructed to study the interplay between the graph theoretic and ring-theoretic properties. Interested readers are referred to the following surveys [2], [11] on graphs defined on rings.…”

Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

“…We would like to mention in this context as far as we dig into literature that there are only four papers on graphs having their vertices lying in C(X). See the articles [4], [6], [7] and [8] in this context. In the technical section 2 of this paper we introduce several well-known parameters related to the graph Γ P (X).…”