A distance spectral radius for star-factors with large components in graphs

Let G be a graph and let m be an integer with m ≥ 2. A {K1,j : m ≤ j ≤ 2m}-factor of a graph G is a spanning subgraph of G, in which every component is isomorphic to an element in {K1,j : m ≤ j ≤ 2m}. For vi, vj ∈ V (G), the distance between vi and vj, denoted by dG(vi, vj) (or dij), is the length of a shortest path from vi to vj. Particularly, dG(vi, vi) = 0 (or dii = 0). The distance matrix of G, denoted by D(G), is the n × n real symmetric matrix in which (i, j)-entry is dG(vi, vj) (or dij). The largest eigenvalue of D(G), denoted by μ(G), is called the distance spectral radius of G. In this paper, it is proved that a connected graph G of order n ≥ 3m2 + 6m + 2 with μ(G) ≤ μ(Km−1 ∨ (Kn−m ∪ K1)) contains a {K1,j : m ≤ j ≤ 2m}-factor unless G = Km−1 ∨ (Kn−m ∪ K1).