For any integer \(k\ge 1\) , a graph G is said to be k-factor-critical if \(G-S\) has a perfect matching for each \(S\subseteq V(G)\) with \(|S|=k\) . In this paper, we present a sufficient condition in terms of the number of r-cliques to guarantee a graph with minimum degree at least \(\delta \) to be k-factor-critical, which improves the result of Fan and Lin (Spectral conditions for k-extendability and k-factors of bipartite graphs, arXiv: 2211.09304). For any integer \(k\ge 2,\) a spanning k-tree of a connected graph G is a spanning tree in which every vertex has degree at most k. Neumann–Lara and Rivera–Campo (Combinatorica 11:55–61, 1991) proved that, for an m-connected graph G with \(m\ge 2\) , if its independence number \(\alpha (G)\le (k-1)m+1\) , then G contains a spanning k-tree. Motivated by the above result, we provide tight spectral conditions for an m-connected graph to contain a spanning k-tree.
