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 $ is a real number with $ 0\leq\alpha < \frac{2}{3} $.
Citation: Yuli Zhang, Sizhong Zhou. An $ A_{\alpha} $-spectral radius for the existence of $ \{P_3, P_4, P_5\} $-factors in graphs[J]. AIMS Mathematics, 2025, 10(7): 15497-15511. doi: 10.3934/math.2025695
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 $ is a real number with $ 0\leq\alpha < \frac{2}{3} $.
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Yuli Zhang, Sizhong Zhou. An $ A_{\alpha} $-spectral radius for the existence of $ \{P_3, P_4, P_5\} $-factors in graphs[J]. AIMS Mathematics, 2025, 10(7): 15497-15511. doi: 10.3934/math.2025695
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