Characterizing path-factor deleted graphs via $Q$-index and $\mathcal{D}$-index

A graph $G$ has a $\mathcal{P}_{\geq k}$-factor if $G$ has a spanning subgraph $H$ such that every component of $H$ is a path of order at least $k$. A graph $G$ is $\mathcal{P}_{\geq k}$-factor deleted if $G-e$ has a $\mathcal{P}_{\geq k}$-factor for each edge $e \in E(G)$. In this paper, we study the $\mathcal{P}_{\geq 2}$-factor deleted graphs by their $Q$-index (the largest eigenvalue of the signless Laplacian matrix) and $\mathcal{D}$-index (the largest eigenvalue of the distance matrix). Sufficient conditions (in terms of $Q$-index and $\mathcal{D}$-index) to guarantee that a graph $G$ to be a $\mathcal{P}_{\geq 2}$-factor deleted graph are established.