Overall molecular descriptors. 3. Overall Zagreb indices

This paper develops further the concept of overall characterization of molecular topology, which is based on calculation of a given graph-invariant for all subgraphs of molecular graph. The new approach defines a cumulative topological descriptor, and an ordered series of terms (eth-order descriptor), which present the sum of the graph-invariant values for all subgraphs having the same number of edges. Alternatively, the terms in the series may be further partitioned, in the manner of molecular connectivity concept of Randic, Kier, and Hall, into contributions of path, cluster, and path-cluster type of subgraphs. The previous publications on the novel approach were based on the simplest graph-invariants - the sum of entries of the adjacency matrix and the distance matrix. Overall connectivity and overall Wiener index were thus defined, along with their respective series of e-order terms. The present study makes use of two other simple functions of vertex degrees, the first and second Zagreb indices. The overall versions of these two indices, very recently constructed, are analyzed in detail. Their potential applicability is verified by deriving multilinear regression models of ten physicochemical properties of alkanes, and comparing them to the results obtained by molecular connectivity and overall connectivity indices.