Characterizing tricyclic graphs with pendant vertices having largest $A_{\alpha}$-spectral radius

TL;DR

This paper identifies the tricyclic graph with the maximum $A_{\alpha}$-spectral radius among graphs with pendant vertices and provides a spectral condition to exclude tricyclic structures.

Contribution

It characterizes the unique extremal graph with the largest $A_{\alpha}$-spectral radius for $\alpha \in [\frac{1}{2}, 1)$ among tricyclic graphs with pendant vertices.

Findings

Identifies the graph with the largest $A_{\alpha}$-spectral radius among tricyclic graphs with pendant vertices.

Provides a spectral condition to determine the absence of tricyclic structures.

Extends spectral graph theory to characterize extremal graphs based on $A_{\alpha}$-spectral radius.

Abstract

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1]. \end{equation*} The $A_{\alpha}$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_{\alpha}$-spectral radius for $\alpha \in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices.