Extremal $ABS$ Spectral Radius in Bicyclic and Bipartite Unicyclic Graphs

TL;DR

This paper identifies bipartite unicyclic graphs and bicyclic graphs with extremal $ABS$ spectral radii, providing a detailed spectral characterization of these graph classes.

Contribution

It determines the graphs with the largest and second largest $ABS$ spectral radii within bipartite unicyclic and bicyclic graph classes.

Findings

Bipartite unicyclic graphs with maximum $ABS$ spectral radius identified.

Bicyclic graphs with the top two $ABS$ spectral radii characterized.

Spectral extremal properties of these graphs established.

Abstract

The ABS spectral radius of a graph G is defined as the largest eigenvalue of its $ABS$ matrix. Motivated by recent studies on this parameter, in this paper, we determine the bipartite unicyclic graphs that attain the largest $ABS$ spectral radius. Furthermore, we characterize the bicyclic graphs that attain the largest and the second largest $ABS$ spectral radii.