Spectral radius and spanning trees of graphs

Sufficient conditions for spanning trees with constrained leaf distance in a graph

“…Fan, Goryainov, Huang and Lin [5] established sufficient conditions for the existence of a spanning k-tree in a connected graph with fixed order in terms of ρ(G) and q(G), respectively. Zheng, Huang and Wang [13] provided a tight spectral radius condition for the existence of a spanning k-ended tree in a t-connected graph, which generalizes a result of Ao, Liu and Yuan [1], etc. We will not list them all here, but will not focus primarily on those related to a spanning tree with constrained leaf distance in the following.…”

“…Recently, in view of Theorem 1, Ao, Liu and Yuan [1] established the following tight spectral conditions for the existence of a spanning tree with leaf degree at most k in a connected graph.…”

“…Chen, X. Lv, J. Li and S.-J. Xu Motivated by [1,7], it is natural and interesting to ask "whether or not there are some spectral conditions to ensure the existence of a spanning tree with leaf distance at least 4 in a graph G?" Inspired by the ideas from O [8] and using the typical spectral techniques and Theorem 3, we also establish the following spectral conditions (involving ρ(G) or q(G)) for the existence of a spanning tree with leaf distance at least 4 in a graph G with minimum degree δ.…”