On [formula omitted]-span and [formula omitted]-span of trees and full binary trees

Abstract

The sum of distances between all pairs of vertices (denoted by σ(⋅) and called the Wiener index) and the number of subtrees (denoted by F(⋅) and called the subtree index) of a graph G are two representative graph invariants that have been extensively studied. The “local” version of these graph invariants (i.e. sum of distances from a given vertex, called the distance of the vertex, and the number of subtrees containing such a vertex, called the local subtree index of the vertex) have been studied. The distance of a vertex v in a tree T, denoted by σT(v), attains its minimum at one or two adjacent vertices called the centroid while the maximum σT(v) occurs at one or more leaves. On the other hand, the local subtree index, denoted by FT(v), attains its maximum at one or two adjacent vertices called the subtree core and the minimum FT(v) occurs at one ore more leaves. In this paper we study the difference between the values of σT(v) at a centroid vertex and a leaf, called the σ-span, and similarly the F-span for the difference in values of the local subtree index at the subtree core and at a leaf. Among trees and full binary trees (trees in which each vertex has degree 1 or 3) on a given number of vertices we study the maximum and minimum possible values of the σ-span and F-span. The extremal structures corresponding to some of these extremal values are also presented. Some unsolved problems are also discussed and proposed as open questions.