Abstract. 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. (doi: 10.5562/cca2294)

In 1972, within a study of the structure-dependency of total π -electron energy (E), it was shown that 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 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.

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In 1993 two novel distance-based topological indices were put forward. In the case of acyclic molecular graphs both are equal to the Wiener index, but both differ from it if the graphs contain cycles. One index is defined (Mohar, B.; Babić, D.; Trinajstić, N. J. Chem. Inf. Comput. Sci. 1993, 33, 153−154) in terms of eigenvalues of the Laplacian matrix, whereas the other is conceived (Klein, D. J.; Randić, M. J. Math. Chem. 1993, 12, 81−95) as the sum of resistances between all pairs of vertices, assuming that the molecule corresponds to an electrical network, in which the resistance between adjacent vertices is unity. Eventually, the former quantity was named quasi-Wiener index and the latter Kirchhoff index. We now demonstrate that the quasi-Wiener and Kirchhoff indices of all graphs coincide.

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In this paper we report on the properties of the matching polynomial a(G) of a graph G. We present a number of recursion formulas for a(G), from which it follows that many families of orthogonal polynomials arise as matching polynomials of suitable families of graphs. We consider the relation between the matching and characteristic polynomials of a graph. Finally, we consider results which provide information on the zeros of a(G).

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