Commuting graph of non-abelian groups of order $p^n$ with center having $p^{n-2}$ elements

Let $G$ be a group. The commuting graph $\zeta (G,V)$ of a group $G$ has a vertex set $V\subseteq G,$ two vertices are connected with an edge if the corresponding elements commute in $G\mathrm{.}$ In this article, we study the commuting graphs for the non-abelian $p$-groups of order $p^n$ with center size $p^{n-2}\mathrm{.}$ We study detour distance, metric dimension, resolving polynomials, and the spectral properties for these graphs.