Arithmetic-Geometric Spectral Radius of Trees and Unicyclic Graphs

TL;DR

This paper investigates the arithmetic-geometric spectral radius of trees and unicyclic graphs, establishing bounds and characterizing extremal graphs, with implications for spectral graph theory.

Contribution

It provides new bounds for the spectral radius of these graphs and characterizes the extremal structures achieving these bounds.

Findings

Spectral radius bounds for trees and unicyclic graphs are established.

Extremal graphs for the bounds are identified as paths, stars, cycles, and specific unicyclic graphs.

The bounds depend on graph order and structure, with precise equality conditions.

Abstract

The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{i}+d_{j}}{2\sqrt{d_{i}d_{j}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$, denoted by $\rho_{ag}(G)$, is the largest eigenvalue of the arithmetic-geometric matrix $A_{ag}(G)$. Let $S_{n}$ be the star of order $n\geq3$ and $S_{n}+e$ be the unicyclic graph obtained from $S_{n}$ by adding an edge. In this paper, we prove that for any tree $T$ of order $n\geq2$, $\displaystyle 2\cos\frac{\pi}{n+1}\leq\rho_{ag}(P_{n})\leq\rho_{ag}(T)\leq\rho_{ag}(S_{n})=\frac{n}{2},$ with equality if and only if $T\cong P_{n}$ for the lower bound, and if and only if $T\cong S_{n}$ for the upper bound. We also prove that for any unicyclic graph $G$ of order $n\geq3$, $\displaystyle 2=\rho_{ag}(C_{n})\leq\rho_{ag}(G)\leq\rho_{ag}(S_{n}+e),$ the lower (upper, respectively) bound is attained if and only if $T\cong C_{n}$ ($T\cong S_{n}+e$, respectively) and $\displaystyle\rho_{ag}(S_{n}+e)<\frac{n}{2}$ for $n\geq7$.