Chromatic number and clique number of subgraphs of regular graph of matrix algebras

Let R be a ring and X subset of R be a non-empty set. The regular graph of X, Gamma(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Gamma(GL(n)(F)) finite? In this paper, we show that if G is a soluble sub-group of GL(n)(F), then x (Gamma(G)) < infinity. Also, we show that for every field F, chi (Gamma(M-n(F))) = chi (Gamma(M-n(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Gamma(< A >), where < A > denotes the subgroup generated by A is an element of GL(n)(F). (C) 2011 Elsevier Inc. All rights reserved.