An Aα-spectral radius for the existence of {P3, P4, P5}-factors in graphs
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2025
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| _version_ | 1866909452289966080 |
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| author | Zhang, Yuli Zhou, Sizhong |
| author_facet | Zhang, Yuli Zhou, Sizhong |
| contents | 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 $0\leqα<\frac{2}{3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_05029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Aα-spectral radius for the existence of {P3, P4, P5}-factors in graphs Zhang, Yuli Zhou, Sizhong Combinatorics 05C50, 05C70, 05C38 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 $0\leqα<\frac{2}{3}$. |
| title | An Aα-spectral radius for the existence of {P3, P4, P5}-factors in graphs |
| topic | Combinatorics 05C50, 05C70, 05C38 |
| url | https://arxiv.org/abs/2501.05029 |