Spectral and Sharp Sufficient Conditions for Graphs to Admit a Strong Star Factor

Let G be a graph. An odd [ 1 , k ] -factor of a graph G is a spanning subgraph H of G such that d e g H ( v ) is odd and 1 ⩽ d e g H ( v ) ⩽ k for every v ∈ V ( G ) where k is a positive odd integer. We call a spanning subgraph H of a graph G a strong star factor if every component of H is isomorphic to an element of the stars K 1 , 1 , K 1 , 2 , ⋯ , K 1 , r and is an induced subgraph of G where r ⩾ 2 is an integer. In a { K 1 , 1 , K 1 , 2 , C m : m ⩾ 3 } -factor of G , each component is isomorphic to a member in { K 1 , 1 , K 1 , 2 , C 3 , C 4 ⋯ , C m } . A graph G is a strong star factor deleted graph if G − e has a strong star factor for each edge e of G . In this paper, through the typical spectral techniques, we obtain the respective necessary and sufficient conditions defining a strong star factor deleted graph, an odd [ 1 , k ] -factor deleted graph, and a { K 1 , 1 , K 1 , 2 , C m : m ⩾ 3 } -factor deleted graph. We determine a lower bound on the size to guarantee that G is a { K 1 , 1 , K 1 , 2 , C m : m ⩾ 3 } -factor deleted graph. We establish the upper bound of the signless Laplacian spectral radius (resp. the spectral radius) and the lower bound of the distance signless Laplacian spectral radius (resp. the distance spectral radius) to determine whether G admits a strong star factor. Furthermore, by constructing extremal graphs, we show that all the bounds obtained in this contribution are the best possible.