On the $\alpha$-spectral radius of graphs

TL;DR

This paper investigates the properties and bounds of the $eta$-spectral radius of graphs, introducing new extremal graph characterizations and analyzing how structural modifications affect this spectral measure.

Contribution

It provides new upper bounds, characterizations of extremal graphs, and analyzes the impact of graph modifications on the $eta$-spectral radius, expanding understanding of spectral graph theory.

Findings

Upper bounds for $eta$-spectral radius in various graph classes

Unique extremal trees with second maximum and maximum $eta$-spectral radius

Behavior of $eta$-spectral radius under pendant edge relocation

Abstract

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $\alpha$-spectral radius of $G$ is the largest eigenvalue of $A_{\alpha}(G)$. We give upper bounds for $\alpha$-spectral radius for unicyclic graphs $G$ with maximum degree $\Delta\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $\alpha$-spectral radius among trees, and the unique tree with maximum $\alpha$-spectral radius among trees with given diameter. For a graph with two pendant paths at a vertex or at two adjacent vertex, we prove results concerning the behavior of the $\alpha$-spectral radius under relocation of a pendant edge in a pendant path. We also determine the unique graphs such that the difference between the maximum degree and the $\alpha$-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.