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

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 $d$.