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.