On the Commuting Graph of Semidihedral Group

Abstract

The commuting graph Δ(G) of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if xy=yx. In this paper, first we discuss some properties of Δ(G). We determine the edge connectivity and the minimum degree of Δ(G) and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of Δ(G) are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of Δ(G) are equal. Further, we investigate the commuting graph of the semidihedral group SD8n. In this connection, we discuss various graph invariants of Δ(SD8n) including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of Δ(SD8n).