Spanning Trees with Few Peripheral Branch Vertices

Let T be a tree, a vertex of degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. The set of leaves of T is denoted by L(T) and the set of branch vertices of T is denoted by B(T). For two distinct vertices u, v of T, let P-T [u, v] denote the unique path in T connecting u and v. Let T be a tree with B(T) 6 not equal empty set, for each vertex x is an element of L(T), set y(x) is an element of B(T) such that (V (P-T [x, y(x)])\{y(x}))boolean AND B(T) not equal empty set. We delete V (P-T [x; y(x)])\{y(x)} from T for all x is an element of L(T). The resulting graph is a subtree of T and is denoted by R Stem(T). It is called the reducible stem of T. A leaf of R Stem(T) is called a peripheral branch vertex of T. In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph G to have a spanning tree with few peripheral branch vertices.