Arithmetic-Geometric spectral radii of Unicyclic graphs

Abstract

Let $d_{v_i}$ be the degree of the vertex $v_i$ of $G$. The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\frac{d_{v_i}+d_{v_j}}{2\sqrt{d_{v_i}{d}_{v_j}}}$ if the vertices $v_i$ and $v_j$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$, denoted by $\ddot{I}{}_{ag}(G)$, is the largest eigenvalue of the arithmetic-geometric matrix $A_{ag}(G)$. In this paper, the unicyclic graphs of order $n\ge 5$ with the smallest and first four largest arithmetic-geometric spectral radii are determined.