Distance signless Laplacian spectral radius for the existence of path-factors in graphs (Q6562884)

Let \(G\) be a finite graph of order \(n\) with the vertex set \(\{v_1,v_2,\ldots,v_n\}\) and let \(F\) be a spanning subgraph of \(G\). Then, \(F\) is called a path factor if every component of \(F\) is a path of order at least 2. A \(P_{\geq k}\)-factor means a path factor in which every component admits order at least \(k (k \geq 2)\). Let \(Q(G)\) denote the distance signless Laplacian matrix of \(G\) and \(\eta_1(G)\) denote the spectral radius of \(Q(G)\). In this article, the authors present a distance signless Laplacian spectral radius condition to guarantee the existence of a \(P_{\geq 2}\)-factor in a graph \(G\). In this direction, for a connected graph \(G\) of order \(n\geq 4\), the authors prove the following:\N\begin{itemize}\N\item \(G\) admits a \(P_{\geq 2}\)-factor for \(n\neq7\) if \(\eta_1(G) < \theta(n)\), where \(\theta(n)\) is the largest root of the equation \N\[\Nx^3 - (5n - 3)x^2 + (8n^2 - 23n + 48)x- 4n^3 + 22n^2 - 74n + 80 = 0;\N\]\N\item \(G\) admits a \(P_{\geq 2}\)-factor for \(n=7\) if \(\eta_1(G) < \frac{25+\sqrt{161}}{2}\).\N\end{itemize}