The existence of a $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$-factor based on the size or the $A_α$-spectral radius of graphs (2412.11580v2)

Abstract: Let $G$ be a connected graph of order $n$. A ${P_{2},C_{3},P_{5},\mathcal{T}(3)}$-factor of $G$ is a spanning subgraph of $G$ such that each component is isomorphic to a member in ${P_{2},C_{3},P_{5},\mathcal{T}(3)}$, where $\mathcal{T}(3)$ is a ${1,2,3}$-tree. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. In this paper, we obtain a lower bound on the size or the $A_{\alpha}$-spectral radius for $\alpha\in[0,1)$ of $G$ to guarantee that $G$ has a ${P_{2},C_{3},P_{5},\mathcal{T}(3)}$-factor, and construct an extremal graph to show that the bound on $A_{\alpha}$-spectral radius is optimal.