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

TL;DR

The paper studies when a connected graph $G$ of order $n\ge25$ must contain a $\{P_3,P_4,P_5\}$-factor, through the lens of the $A_{\alpha}$-spectral radius $\lambda_{\alpha}(G)$ with $0\le\alpha<\tfrac{2}{3}$. It introduces $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ and proves that if $\lambda_{\alpha}(G) \ge \lambda_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$ then $G$ has a $\{P_3,P_4,P_5\}$-factor, unless $G$ is the extremal graph $K_1\vee(K_{n-2}\cup K_1)$. The proof leverages a Kano–Lu–Yu style obstruction condition, equitable partitions, and quotient matrices to bound $\lambda_{\alpha}$ and employs interlacing to derive contradictions across three subcases determined by $n$ relative to $s=|S|$. A corollary extends the result to a $P_{\geq3}$-factor. The results connect spectral radius bounds with structural spanning subgraphs, enriching spectral conditions for graph factors under the $A_{\alpha}$ framework. Mathematical threshold $\alpha<\tfrac{2}{3}$ and the extremal graph $K_1\vee(K_{n-2}\cup K_1)$ play central roles.

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 $A_α$-matrix of $G$ as $A_α(G)=αD(G)+(1-α)A(G)$ [V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107], where $α\in[0,1]$, $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_α(G)$, denoted by $λ_α(G)$, is called the $A_α$-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 $λ_α(G)\geqλ_α(K_1\vee(K_{n-2}\cup K_1))$, where $α$ be a real number with $0\leqα<\frac{2}{3}$.