Distance signless Laplacian spectral radius for the existence of path-factors in graphs

Let G be a connected graph of order n, where n is a positive integer. A spanning subgraph F of G is called a path-factor if every component of F is a path of order at least 2. A P->= k-factor means a path-factor in which every component admits order at least k (k >= 2). The distance matrix D(G) of G is an n x n real symmetric matrix whose (i, j)-entry is the distance between the vertices v(i) and v(j). The distance signless Laplacian matrix Q(G) of G is defined by Q(G) = Tr(G) + D(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue eta(1)(G) of Q( G) is called the distance signless Laplacian spectral radius of G. In this paper, we aim to present a distance signless Laplacian spectral radius condition to guarantee the existence of a P->= 2-factor in a graph and claim that the following statements are true: (i) G admits a P->= 2-factor for n >= 4 and n not equal 7 if eta(1)(G) < theta(n), where theta(n) is the largest root of the equation x(3) - (5n - 3)x(2) + (8n(2) - 23n + 48)x - 4n(3) + 22n(2) - 74n + 80 = 0; (ii) G admits a P->= 2-factor for n = 7 if eta(1)(G) < 25+ root 161/2.