Characterization of commuting graphs of finite groups having small genus

In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only K-8 (sic) 9K(1), K-8 (sic) 5K(2), K-8 (sic) 3K(4), K-8 (sic) 9K(3), K-8 (sic) 9(K-1 boolean OR 3K(2)), 3K(6) and 3K(6) (sic) 4K(4) (sic) 6K(2) can be realized as commuting graphs of finite groups, where (sic) and boolean OR 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-Vukicevic Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group (Z3 x Z3) (sic) Q(8), that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs.