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Keywords:
rings of continuous functions; comaximal graph; radius; girth; dominating number; clique number; zero cellularity; $P$-space; almost $P$-space; connected space; regular ring
rings of continuous functions; comaximal graph; radius; girth; dominating number; clique number; zero cellularity; $P$-space; almost $P$-space; connected space; regular ring
Summary:
In this article we study the comaximal graph $\Gamma'_{_2}C(X)$ of the ring $C(X)$. We have tried to associate the graph properties of $\Gamma'_{_2}C(X)$, the ring properties of $C(X)$ and the topological properties of $X$. Radius, girth, dominating number and clique number of the $\Gamma'_{_2}C(X)$ are investigated. We have shown that $2\leq \operatorname{Rad}\Gamma'_{_2}C(X) \leq 3$ and if $|X|> 2$ then $\mathrm{girth } \Gamma'_{_2}C(X)= 3$. We give some topological properties of $X$ equivalent to graph properties of $\Gamma'_{_2}C(X)$. Finally we have proved that $X$ is an almost $P$-space which does not have isolated points if and only if $C(X)$ is an almost regular ring which does not have any principal maximal ideals if and only if $\operatorname{Rad}\Gamma'_{_2}C(X)= 3$.
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