On the $\alpha$-index of graphs with pendent paths

TL;DR

This paper investigates the spectral radius of a family of matrices associated with graphs, generalizing previous results for specific cases, and identifies extremal graphs with maximum spectral radius under certain conditions.

Contribution

It introduces new extremal bounds for the spectral radius of the $ ext{A}_ ext{alpha}$ matrix in graphs, extending known results for special $ ext{alpha}$ values.

Findings

Identifies extremal graphs maximizing spectral radius for given diameter and order.

Generalizes previous bounds for $ ho_0(G)$ and $ ho_{1/2}(G)$.

Provides conditions for equality in the extremal bounds.

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right] $, write $A_{\alpha}\left( G\right) $ for the matrix \[ A_{\alpha}\left( G\right) =\alpha D\left( G\right) +(1-\alpha)A\left( G\right) . \] This paper presents some extremal results about the spectral radius $\rho_{\alpha}\left( G\right) $ of $A_{\alpha}\left( G\right) $ that generalize previous results about $\rho_{0}\left( G\right) $ and $\rho _{1/2}\left( G\right) $. In particular, write $B_{p,q,r}$ be the graph obtained from a complete graph $K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its ends. It is shown that if $\alpha\in\left[ 0,1\right) $ and $G$ is a graph of order $n$ and diameter at least $k,$ then% \[ \rho_{\alpha}(G)\leq\rho_{\alpha}(B_{n-k+2,\lfloor k/2\rfloor,\lceil k/2\rceil}), \] with equality holding if and only if $G=B_{n-k+2,\lfloor k/2\rfloor,\lceil k/2\rceil}$. This result generalizes results of Hansen and Stevanovi\'{c} \cite{HaSt08}, and Liu and Lu \cite{LiLu14}.