A note on the distance spectra of co-centralizer graphs

Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph ${\mathrm{\Gamma }}_{cent}(G)$, whose vertex set consists of proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are identical [6]. We call the complement of the centralizer graph as the co-centralizer graph. In this paper, we investigate the distance, distance (signless) Laplacian spectra of co-centralizer graphs of some classes of finite non-abelian groups, and obtain some conditions on a group so that the co-centralizer graph is distance, distance (signless) Laplacian integral.