Complete characterization of path-factor and path-factor covered graphs via Q-index and D-index

Abstract

For $k⩾2$, a ${\mathcal{P}}_{⩾k}$-factor of a graph G is a spanning subgraph F of G such that each component of F is a path with at least k vertices. A graph G is a ${\mathcal{P}}_{⩾k}$-factor covered graph if for each edge e in $E\left(G\right)$, there exists a ${\mathcal{P}}_{⩾k}$-factor containing the edge e. Let $Q\left(G\right)$ and $D\left(G\right)$ be the signless Laplacian matrix and the distance matrix of a graph G, respectively. In this paper, we provide lower bounds for the spectral radius of $Q\left(G\right)$ in an n-vertex connected graph to guarantee that G has a ${\mathcal{P}}_{⩾2}$-factor or is a ${\mathcal{P}}_{⩾2}$-factor covered graph. Furthermore, we establish upper bounds for the spectral radius of $D\left(G\right)$ in an n-vertex connected graph to guarantee that G has a ${\mathcal{P}}_{⩾2}$-factor or is a ${\mathcal{P}}_{⩾2}$-factor covered graph.

Keywords:

• ${\mathcal{P}}_{⩾2}$-factor
• ${\mathcal{P}}_{⩾2}$-factor covered graph
• Q-index
• D-index

AMS subject classifications:

• 05C70
• 05C50

Acknowledgements

The authors are greatly indebted to the referee for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Financially supported by the National Natural Science Foundation of China [grant numbers 12171190, 11671164].