Abstract

In this paper, we will study the chromatic number of a family of Cayley graphs that arise from algebraic constructions. Using Lang–Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of a large family of these graphs. As a corollary we obtain a lower bound for the chromatic number of certain Cayley graphs associated to the ring of n×n matrices over finite fields, establishing a result for the case of SLn parallel to a theorem of Tomon [26] for GLn. Moreover, using Weil's bound for Kloosterman sums we will also prove an analogous result for SL2 over certain finite rings.

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