Characterization of commuting graphs of finite groups having small genus

Abstract

In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only \(K_8 \sqcup 9K_1\), \(K_8 \sqcup 5K_2\), \(K_8 \sqcup 3K_4\), \(K_8 \sqcup 9K_3\), \(K_8\sqcup 9(K_1 \vee 3K_2)\), \(3K_6\) and \(3K_6 \sqcup 4K_4 \sqcup 6K_2\) can be realized as commuting graphs of finite groups, where \(\sqcup\) and \(\vee\) stand for disjoint union and join of graphs respectively. As consequences of our results we also show that for any finite non-abelian group \(G\) if the commuting graph of \(G\) (denoted by \(\Gamma_c(G)\)) is double-toroidal or  triple-toroidal then \(\Gamma_c(G)\) and its complement satisfy Hansen-Vukičević Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group \((\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8\), that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs.