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

Abstract

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 $\eta _1(G)<\frac{25+\sqrt{161}}{2}$.