Commuting graph of non-abelian groups of order p4 with center having p elements

Let Formula: see text be a non-abelian group and Formula: see text We define the commuting graph Formula: see text with vertex set Formula: see text and two distinct vertices Formula: see text and Formula: see text of Formula: see text are connected by an edge when they commute in Formula: see text. In this article, certain distance and detour distance properties of commuting graph of non-abelian groups of order Formula: see text with center having Formula: see text elements, where Formula: see text is an odd prime have been studied. Metric dimensions, resolving polynomials and spectral properties of such graphs have also been effectively computed.