On the sum of the largest $A_{\alpha}$-eigenvalues of graphs

TL;DR

This paper establishes bounds on the sum of the largest $A_{\alpha}$-eigenvalues of graphs, generalizing known results for adjacency and signless Laplacian matrices, and characterizes extremal graphs.

Contribution

It provides new bounds and characterizations for the sum of the largest $A_{\alpha}$-eigenvalues, unifying previous spectral graph results.

Findings

Derived several upper and lower bounds for $S_k(A_{\alpha}(G))$.

Characterized extremal graphs for specific cases.

Explored graph operations affecting $S_k(A_{\alpha}(G))$.

Abstract

For every real $0\leq \alpha \leq 1$, Nikiforov defined the $A_{\alpha}$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. The eigenvalues of $A_{\alpha}(G)$ are called the $A_{\alpha}$-eigenvalues of $G$. Let $S_k(A_{\alpha}(G))$ be the sum of $k$ largest $A_{\alpha}$-eigenvalues of $G$. In this paper, we present several upper and lower bounds on $S_k(A_{\alpha}(G))$ and characterize the extremal graphs for certain cases, which can be regard as a common generalization of the sum of $k$ largest eigenvalues of adjacency matrix and signless Laplacian matrix of graphs. In addition, some graph operations on $S_k(A_{\alpha}(G))$ are presented.