Extremal arithmetic–geometric spectral radius of unicyclic graphs

Abstract

Spectral graph theory has been widely used in many fields, including network science, chemistry, physics, biology and sociology. Spectral extremal graph theory aims to minimize or maximize some graph invariants over a class of graphs based on graph matrix. The arithmetic–geometric matrix of a graph G, denoted by AG(G), is a square matrix whose (i, j)-entry is $\frac{d_i+d_j}{2\sqrt{d_id_j}}$ if $v_i$ and $v_j$ are adjacent in G, and 0 otherwise, where $d_i$ is the degree of vertex $v_i$. The largest arithmetic–geometric eigenvalue is the AG spectral radius of G, denoted as $\theta _1(G)$. In this work, we investigate the extremal values on arithmetic–geometric spectral radius of n-vertex unicyclic graphs and characterize the unicyclic graphs that achieve the extremes. We show that, for n-vertex unicyclic graph G, $2=\theta _1(C_n)\le \theta _1(G)\le \theta _1(S_3^+),$ where the lower bound is achieved by $C_n$ and the upper bound is achieved by $S_3^+$ which is obtained by attaching $n-3$ pendant vertices to some fixed vertex of $C_3$.