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.
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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