$C^*(X)$ from a graphical point of view

Abstract

In this paper, we define a graph called the chain-link graph $\gamma(C^{^*}(X))$ on the ring $C^{^*}(X)$ of all real-valued, bounded, continuous functions defined over a Tychonoff space $X$. We briefly study some aspects like connectedness, diameter, radius, cycles, chords, dominating sets etc. of $\gamma(C^{^*}(X))$ and some of its subgraphs. We also inspect the relation between the ideals of $C^{^*}(X)$ and the cliques of $\gamma(C^{^*}(X))$ and finally provide a characterization for all maximal cliques of $\gamma(C^{^*}(X))$. In the sequel, we prove that there are at least $2^c$ many different maximal cliques which are never graph isomorphic to each other. Moreover, we inquire about the topological and algebraic notions linked to the neighbourhood of a vertex of the graph. We then observe the correspondence between graph isomorphisms on $\gamma(C^{^*}(X))$, ring isomorphisms on $C^{^*}(X)$ and homeomorphims on $X$ when the topology of $X$ is suitably chosen. Finally, we note down some facts about the zero-set intersection graph $\Gamma(C(X))$ analogous to our discourse upon $\gamma(C^{^*}(X))$.