Extremal trees with respect to functions on adjacent vertex degrees

The question of finding extremal structures with respect to various graph indices has received much attention in recent years. Among these graph indices, many are defined on adjacent vertex degrees and maximized or minimized by the same extremal structure. We consider a function defined on adjacent degrees of a tree, T, to be f(x; y) and the connectivity function associated with f, (Equation presented). We first introduce the extremal tree structures, with a given degree sequence, that maximize or minimize such functions under certain conditions. When a partial ordering, called \majorization", is defined on the degree sequences of trees on n vertices, we compare the extremal trees of different degree sequences π and π′. As a consequence many extremal results follow as immediate corollaries. Our finding provides a uniform way of characterizing the extremal structures with respect to a class of graph invariants. We also briey discuss the applications to specific indices.