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).