Proof and disproof of conjectures on spectral radii of coclique extension of cycles and paths

Abstract

A coclique extension of a graph H is a graph G obtained from H by replacing each vertex of H by a coclique, where vertices of G coming from different vertices of H are adjacent if and only if the original vertices are adjacent in H. Let Mn(H) be the set of graphs with order n, which are the coclique extensions of H. In this paper, we discuss the minimum spectral radius in Mn(Pk) and the maximum spectral radius in Mn(Ck), where Pk and Ck are the path of order k and the cycle of order k, respectively. Then we disprove a conjecture on the minimum spectral radius in Mn(Pk) and confirm a conjecture on the maximum spectral radius in Mn(Ck), which are given by Monsalve and Rada (2021) [4].