The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

In 1972, within a study of the structure-dependency of total $$\pi $$
 -electron energy (
 $${\mathcal {E}}$$
 ), it was shown that $${\mathcal {E}}$$
 depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton. In the same paper, also the sum of cubes of degrees of vertices of the molecular graph was shown to influence $${\mathcal {E}}$$
 , but this topological index was never again investigated and was left to oblivion. We now establish a few basic properties of this “forgotten topological index” and show that it can significantly enhance the physico-chemical applicability of the first Zagreb index.

A forgotten topological index

The degree of a vertex of a molecular graph is the number of first neighbors of this vertex. A large number of molecular-graph-based structure descriptors (topological indices) have been conceived, depending on vertex degrees. We summarize their main properties, and provide a critical comparative study thereof.

/pdf/degree-based-topological-indices-5asctyg1ke.pdf

Degree-Based Topological Indices

The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, communicability, and betweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network's adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks.

http://www.cs.utexas.edu/users/naga86/research/network_properties.pdf

Network Properties Revealed through Matrix Functions

Inspired by recent work on the atom-bond connectivity (ABC) index we propose here a new topological index, augmented Zagreb index (AZI). The tight upper and lower bounds for chemical trees are obtained. Moreover, it has been shown that among all trees the star has the minimum AZI value. Characterizing trees with maximal augmented Zagreb index remains an open problem for future research.

Augmented Zagreb index

Research on the topological indices based on end-vertex degrees of edges has been intensively rising recently. Randic index, one of the best-known topological indices in chemical graph theory, is belonging to this class of indices. In this paper, we introduce a novel topological index based on the end-vertex degrees of edges and its basic features are presented here. We named it as geometrical-arithmetic index (GA).

Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges

This survey outlines results on graphs extremal with respect to distance–based indices, with emphasis on the Wiener index, hyper–Wiener index, Harary index, Wiener polarity index, reciprocal complementary Wiener index, and terminal Wiener index.

Boris Furtula

Papers

A forgotten topological index

Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges

Augmented Zagreb index

A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices

Atom-bond connectivity index of trees