Annihilator graph of the ring $C_\mathscr{P}(X)$

TL;DR

This paper introduces the annihilator graph of the ring $C_ ext{P}(X)$, explores its properties, and establishes connections with other graphs, providing insights into graph coloring and isomorphisms in the context of rings of continuous functions.

Contribution

It defines the annihilator graph for $C_ ext{P}(X)$, compares it with related graphs, and develops algorithms for coloring and analyzing graph isomorphisms, linking graph properties to the underlying space.

Findings

$AG(C_ ext{P}(X))$ coincides with zero divisor graphs when $|X_ ext{P}| extless=2$.

$AG(C_ ext{P}(X))$ and $G(C_ ext{P}(X))$ share similar graph parameters.

An algorithm for coloring $G(C_ ext{P}(X))$ is formulated, enabling finite coloring of certain infinite graphs.

Abstract

In this article, we introduce the annihilator graph of the ring $C_\mathscr{P}(X)$, denoted by $AG(C_\mathscr{P}(X))$ and observe the effect of the underlying Tychonoff space $X$ on various graph properties of $AG(C_\mathscr{P}(X))$. $AG(C_\mathscr{P}(X))$, in general, lies between the zero divisor graph and weakly zero divisor graph of $C_\mathscr{P}(X)$ and it is proved that these three graphs coincide if and only if the cardinality of the set of all $\mathscr{P}$-points, $X_\mathscr{P}$ is $\leq 2$. Identifying a suitable induced subgraph of $AG(C_\mathscr{P}(X))$, called $G(C_\mathscr{P}(X))$, we establish that both $AG(C_\mathscr{P}(X))$ and $G(C_\mathscr{P}(X))$ share similar graph theoretic properties and have the same values for the parameters, e.g., diameter, eccentricity, girth, radius, chromatic number and clique number. By choosing the ring $C_\mathscr{P}(X)$ where $\mathscr{P}$ is the ideal of all finite subsets of $X$ such that $X_\mathscr{P}$ is finite, we formulate an algorithm for coloring the vertices of $G(C_\mathscr{P}(X))$ and thereby get the chromatic number of $AG(C_\mathscr{P}(X))$. This exhibits an instance of coloring infinite graphs by just a finite number of colors. We show that any graph isomorphism $\psi : AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ maps $G(C_\mathscr{P}(X))$ isomorphically onto $G(C_\mathscr{Q}(Y))$ as a graph and a graph isomorphism $\phi : G(C_\mathscr{P}(X)) \rightarrow G(C_\mathscr{Q}(Y))$ can be extended to a graph isomorphism $\psi : AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ under a mild restriction on the function $\phi$. Finally, we show that atleast for the rings $C_\mathscr{P}(X)$ with finitely many $\mathscr{P}$-points, so far as the graph properties are concerned, the induced subgraph $G(C_\mathscr{P}(X))$ is a good substitute for $AG(C_\mathscr{P}(X)$.