An A{\alpha}-spectral radius for the existence of {P3, P4, P5}-factors in graphs

TL;DR

This paper establishes spectral conditions based on the $A_{\alpha}$-spectral radius for the existence of certain path factors in connected graphs of order at least 25, identifying a specific extremal graph case.

Contribution

It introduces spectral criteria using the $A_{\alpha}$-matrix to determine the presence of {P3, P4, P5}-factors in graphs, extending spectral graph theory applications.

Findings

Graphs with $\lambda_{\alpha}(G)$ above a threshold have {P3, P4, P5}-factors.

Characterization of extremal graph $K_1\vee(K_{n-2}\cup K_1)$ as the boundary case.

Spectral condition holds for $0 \leq \alpha < \frac{2}{3}$.

Abstract

Let $G$ be a connected graph of order $n$ with $n\geq25$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. Nikiforov introduced the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ [V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107], where $\alpha\in[0,1]$, $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha}(G)$, denoted by $\lambda_{\alpha}(G)$, is called the $A_{\alpha}$-spectral radius of $G$. In this paper, it is proved that $G$ has a $\{P_3,P_4,P_5\}$-factor unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $\lambda_{\alpha}(G)\geq\lambda_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$, where $\alpha$ be a real number with $0\leq\alpha<\frac{2}{3}$.