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 \(P_{\ge k}\)-factor means a path-factor in which every component admits order at least k (\(k\ge 2\)). The distance matrix \({\mathcal {D}}(G)\) of G is an \(n\times n\) real symmetric matrix whose (i, j)-entry is the distance between the vertices \(v_i\) and \(v_j\). The distance signless Laplacian matrix \({\mathcal {Q}}(G)\) of G is defined by \({\mathcal {Q}}(G)=Tr(G)+{\mathcal {D}}(G)\), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue \(\eta _1(G)\) of \({\mathcal {Q}}(G)\) 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 \(P_{\ge 2}\)-factor in a graph and claim that the following statements are true: (i) G admits a \(P_{\ge 2}\)-factor for \(n\ge 4\) and \(n\ne 7\) 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 \(P_{\ge 2}\)-factor for \(n=7\) if \(\eta _1(G)<\frac{25+\sqrt{161}}{2}\).