On the Commuting Graph of Semidihedral Group

Abstract

The commuting graph $$\varDelta (G)$$ 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 $$\varDelta (G)$$ . We determine the edge connectivity and the minimum degree of $$\varDelta (G)$$ and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of $$\varDelta (G)$$ are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of $$\varDelta (G)$$ are equal. Further, we investigate the commuting graph of the semidihedral group $$SD_{8n}$$ . In this connection, we discuss various graph invariants of $$\varDelta (SD_{8n})$$ including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $$\varDelta (SD_{8n})$$ .