On numbers of vertices of maximum degree in the spanning trees of a graph

Abstract

For a connected graph G, let $\text{T}\text{(G)}$ be the set of all spanning trees of G and let n_Δ(G) be the number of vertices of maximum degree in G. In this paper we show that if G is a cactus or a connected graph with p vertices and p + 1 edges, then the set $\text{{n}{}_{\mathrm{\Delta }}\text{(T) : T ϵ}\text{T}\text{(G)}}$ has at most one gap, that is, it is a set of consecutive integers or it is the union of two sets each of which consists of consecutive integers.