On the spectral radius of unicyclic and bicyclic graphs with a fixed diameter

TL;DR

This paper investigates the maximum spectral radii of unicyclic and bicyclic graphs with fixed order and diameter, providing characterizations and identifying extremal graphs for these spectral measures.

Contribution

It introduces methods for comparing the $oldsymbol{ ext{alpha}}$-spectral radius and characterizes extremal graphs with maximal spectral radii among unicyclic and bicyclic graphs of given order and diameter.

Findings

Identified graphs with maximal $oldsymbol{ ext{alpha}}$-spectral radius among unicyclic graphs.

Identified graphs with maximal $oldsymbol{ ext{alpha}}$-spectral radius among bicyclic graphs.

Determined the unique bicyclic graph with maximal signless Laplacian spectral radius.

Abstract

The $\alpha$-spectral radius of a connected graph $G$ is the spectral radius of $A_\alpha$-matrix of $G$. In this paper, we discuss the methods for comparing $\alpha$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $\alpha$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.