Tree with maximum non-self-centrality number among all trees of fixed order and maximum degree

Abstract

Let G be a simple and connected graph. The eccentricity of a vertex x in G is denoted by e(x). The non-self-centrality number of the graph G is defined as N(G)=∑x≠y|e(x)−e(y)|, where the summation is taken over all unordered pairs of vertices of G. Xu et al. (2016) proposed a problem to determine the extremal trees with maximum non-self-centrality number among all trees of order n with maximum fixed degree Δ, (3<Δ≤n−1). In this paper, we address this problem and find a tree with largest non-self-centrality number in the family of trees of order n with maximum fixed degree Δ, (3<Δ≤n−1).