Arithmetic-Geometric spectral radii of Unicyclic graphs

TL;DR

This paper investigates the spectral radii of the arithmetic-geometric matrix for unicyclic graphs, identifying those with the smallest and top four largest spectral radii among graphs of order at least five.

Contribution

It determines the unicyclic graphs with the smallest and four largest arithmetic-geometric spectral radii for graphs of order n ≥ 5, filling a gap in spectral graph theory.

Findings

Identified the unicyclic graph with the smallest spectral radius.

Determined the top four unicyclic graphs with the largest spectral radii.

Provided bounds and characterizations for 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 $\displaystyle \frac{d_{v_{i}}+d_{v_{j}}}{2\sqrt{d_{v_{i}}d_{v_{j}}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$, denoted by $\rho_{ag}(G)$, is the largest eigenvalue of the arithmetic-geometric matrix $A_{ag}(G)$. In this paper, the unicyclic graphs of order $n\geq5$ with the smallest and first four largest arithmetic-geometric spectral radii are determined.