Some Results on the Existence of Spanning k-ended Trees in a t-connected Graph

Abstract

For a graph G of order n, let ϱ ^a₁ (G) be the spectral radius of A_a(G):= aD(G) + A(G) for a ≥ 0, where A(G) and D(G) are the adjacency matrix and the degree matrix of G, respectively. In this paper, we investigate the relationship between ϱ ^a₁ (G) and the spanning trees with bounded leaves in a t-connected graph G. We provide an upper bound for ϱ ^a₁ (G) to ensure the existence of a spanning k-ended tree in a t-connected graph. Our result extends the corresponding known results on a = 0 and a = 1, offering a unified framework for exploring the existence of a spanning k-ended tree in a t-connected graph. Additionally, we establish sufficient conditions to ensure a spanning k-ended tree exists in a t-connected graph based on its algebraic connectivity, nullity and energy, respectively.