The Extremal Values and Rankings of the Gutman Index of k-Caterpillar

Abstract

In this paper, based on the conception of the Gutman index of graphs, we proposed the degree-distance of a graph to a vertex and the vertex-Gutman index. We studied the extremal values and rankings of the vertex-Gutman indices of the k-caterpillar, obtained the distribution of the extremal vertices; we also researched the extremal values and rankings of the Gutman index of the family of k-caterpillar, characterized the corresponding extremal graphs.

The Vertex-gutman Index of k-Caterpillars with Equal Length Hairs

“…Definition 2 [6] Let T be a simple connected graph, uT. The degree-distance of the vertex u to the graph T (denoted as G(u,T)) is defined as From Definition 1 and Definition 2, we can get the relationship among the Gutman index, the degree-distance of a vertex to a graph and the vertex-Gutman index:…”