The extremal spectral radius of generalized block graphs

Abstract

A block graph is a graph in which every block is a complete graph. Denote by Kn,q the set of block graphs with n vertices and all blocks on q+1 vertices for every q≥2. Recently, Zhao and Liu (2023) determined the minimum spectral radius of graphs in Kn,q, which verified a conjecture posed by Conde et al. (2022). Replacing the complete graph by a general block or a cycle, we define a generalized block graph or a cycle tree, respectively. Let Bn,q (resp. Cn,q) be the set of generalized block graphs (resp. cycle trees) on n vertices and in which each block is of order q+1 for q≥2. In this paper, we obtain the maximum/minimum spectral radius of a graph in Cn,q. Furthermore, we determine the extremal graph attaining the minimum spectral radius among graphs in Bn,q, which coincides with that in Cn,q.