On the ABS Spectral Radius of Trees and Unicyclic Graphs

Abstract

Characterization of extremal graphs with respect to the spectral radius of the adjacency matrix weighted by topological indices is one of the interesting problems in the study of graph matrices. Recently, attempts have been made to unify the study of the spectral radius of the adjacency matrix weighted by topological indices. As a result, extremal trees and unicyclic graphs for the spectral radius of several topological matrices have been characterized. One of the topological matrices that do not come under the unified approaches studied so far is the atom-bound sum connectivity matrix $\mathcal{AS}(G)$ of a graph $G$. Until now, it is known that among all trees with $n$ vertices, the star graph has maximum spectral radius with respect to the matrix $\mathcal{AS}(G)$ (called $\mathcal{ABS}$ spectral radius), whereas the path graph has minimum $\mathcal{ABS}$ spectral radius. Motivated by these works, in this paper, we determine trees with second-largest, third-largest, fourth-largest, and fifth-largest $\mathcal{ABS}$ spectral radius. We show that among all unicyclic graphs on $n$ vertices, the graph $S_{n}+e$ has maximum $\mathcal{ABS}$ spectral radius. Also, we obtain an upper bound for the $\mathcal{ABS}$ spectral radius of unicyclic graphs with girth at least $5$.