The graph-parameters mainly studies vertex degree and distance between unordered vertices. There are many graph-parameters, and this paper mainly focuses on the Gutman index. For the sake of discussion, this paper puts forward the concept of “contribution” based on the relationship between Gutman index of graph and vertex-Gutman index. On this basis, we discussed the Gutman index of unicyclic graph with pendent edges in odd and even parts, and gave the corresponding extremal graphs, which lays a foundation for the study of the Gutman index of other cycle graphs.
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.
As a topological index closely related to the physical, chemical and biological properties of compounds, graph indexes have been concerned and studied by scholars. This paper defined the concept of " Level-Contribution " from the view of degree and distance. On the basis of this, we studied the Vertex-Gutman-Index and Graph-Gutman-Index of the concurrent and double circle graph, and obtained the extremal values of the Vertex-Gutman of the vertices and extremal graphs of the Graph-Gutman-Index of the concurrent and double circle graph.
Abstract. The Newkome resinol is a kind of synthetic macromolecule compounds which could be described by a graph with 253 vertices and 36 leaves. Based on the structure of Newkome resinol, a kind of trees (called the Newkome-type tree) is defined in this paper. According to the recursive structure of the Newkome-type trees, the vertex-degree-distance and the degree-distance of the graph are studied, and the analytic expression of the vertex-degree-distance of all vertices and the degree-distance of the graph are obtained.
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