Extremal trees, unicyclic and bicyclic graphs with respect to $p$-Sombor spectral radii

Abstract

For a graph $G=(V,E)$ and $v_i\in V$, denote by $d_{v_i}$ (or $d_i$ for short) the degree of vertex $v_i$. The $p$-Sombor matrix ${\text{S}}_{\text{p}}(G)$ ($p\ne 0$) of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $(d_i^p+d_j^p{)}^{\frac{1}{p}}$ if the vertices $v_i$ and $v_j$ are adjacent, and 0 otherwise. The $p$-Sombor spectral radius of $G$, denoted by $\ddot{I}({\text{S}}_{\text{p}}(G))$, is the largest eigenvalue of the $p$-Sombor matrix ${\text{S}}_{\text{p}}(G)$. In this paper, we consider the extremal trees, unicyclic and bicyclic graphs with respect to the $p$-Sombor spectral radii. We characterize completely the extremal graphs with the first three maximum Sombor spectral radii, which answers partially a problem posed by Liu et al. in [MATCH Commun. Math. Comput. Chem. 87 (2022) 59-87].