Eigenvalues and component factors in graphs

Abstract

For a set $\mathcal{H}$ of connected graphs, an $\mathcal{H}$-factor of $G$ is a spanning subgraph $F$ of $G$ if each component of $F$ is isomorphic to an element of $\mathcal{H}.$ Kano, Lu and Yu [Electron. J. Combin. 26 (2019) P4.33] provided a good characterization based on an isolated vertex condition for the existence of a $\{K_{1,1},K_{1,2},\ldots,K_{1,k},$ $\mathcal{T}(2k+1)\}$-factor in graphs. Motivated by the above elegant result, we in this paper focus on the existence of a $\{K_{1,1},K_{1,2},\ldots,K_{1,k},$ $\mathcal{T}(2k+1)\}$-factor in graphs from perspective of eigenvalues. By adopting a crucial technique due to Tait [J. Combin. Theory Ser. A 166 (2019) 42-58] and combining typical spectral methods and structural analysis, we present tight sufficient conditions in terms of the spectral radius and the distance spectral radius for a graph to contain a $\{K_{1,1},K_{1,2},\ldots,K_{1,k},$ $\mathcal{T}(2k+1)\}$-factor, respectively.