Spectral conditions for path-factors in isolated tough graphs

The isolated toughness of a graph G , denoted by I ( G ) , is defined by I ( G ) = min | S | i ( G − S ) : S ⊆ V ( G ) and i ( G − S ) ≥ 2 or I ( G ) = ∞ if G is complete. A graph G is said to be isolated r -tough if I ( G ) ≥ r . A path-factor of G is a spanning subgraph of G whose components are paths. Let P ≥ k = { P i : i ≥ k ≥ 2 } . A P ≥ k -factor means a path-factor in which every component is a path with at least k vertices. Liu, Lai and Das first introduced the matrix A a ( G ) = a D ( G ) + A ( G ) of G [Spectral results on Hamiltonian problem, Discrete Math. 342 (2019) 1718–1730], where a ≥ 0 is an integer, and D ( G ) and A ( G ) respectively denote the diagonal degree matrix and the adjacency matrix of G . The largest eigenvalue of A a ( G ) , denoted by ρ a ( G ) , is called the A a -spectral radius of G . The largest eigenvalue of the distance matrix D ( G ) , denoted by μ ( G ) , is called the distance spectral radius of G . In this paper, we aim to provide two sufficient conditions with respect to ρ a ( G ) and μ ( G ) to guarantee the existence of P ≥ 2 -factors in graphs. Let G be a connected isolated t 2 t + 1 -tough graph with n vertices, where t ≥ 1 is an integer. Then the following two results hold. (i) If n ≥ max { 6 t + 10 , 2 t 2 + 5 t + 4 } and ρ a ( G ) ≥ ρ a ( K t ∨ ( K n − 3 t − 1 ∪ ( 2 t + 1 ) K 1 ) ) for a ∈ { 0 , 1 } , then G has a P ≥ 2 -factor unless G = K t ∨ ( K n − 3 t − 1 ∪ ( 2 t + 1 ) K 1 ) . (ii) If n ≥ 9 t + 2 and μ ( G ) ≤ μ ( K t ∨ ( K n − 3 t − 1 ∪ ( 2 t + 1 ) K 1 ) ) , then G has a P ≥ 2 -factor unless G = K t ∨ ( K n − 3 t − 1 ∪ ( 2 t + 1 ) K 1 ) .