Geodesic
Comaximal graph of $C(X)$
Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 3, pp. 353-364.

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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 $\operatorname{Rad}\Gamma'_{_2}C(X)= 3$.
DOI : 10.14712/1213-7243.2015.178
Classification : 54C40
Keywords: rings of continuous functions; comaximal graph; radius; girth; dominating number; clique number; zero cellularity; $P$-space; almost $P$-space; connected space; regular ring 
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Badie, Mehdi. Comaximal graph of $C(X)$. Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 3, pp. 353-364. doi : 10.14712/1213-7243.2015.178. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.178/

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