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This study identifies properties and uses of the permanental polynomial of adjacency matrixes of unweighted chemical graphs. Coefficients and zeroes of the polynomial for several representative structures are provided, and their properties are discussed. A computer program for calculating the permanental polynomial from the adjacency matrix is also described.

A quantitative structure-activity relationship (QSAR) model relating electrotopological state (E-state) indices and mutagenic potency was previously described by Cash [Mutat. Res. 491 (2001) 31-37] using a data set of 95 aromatic amines published by Debnath et al. [Environ. Mol. Mutagen. 19 (1992) 37-52]. Mutagenic potency was expressed as the number of Salmonella typhimurium TA98 revertants per nmol (LogR). Earlier work on the development of QSARs for the prediction of genotoxicity indicated that numerous methods could be effectively employed to model the same aromatic amines data set, namely, Debnath et al.; Maran et al. [Quant. Struct.-Act. Relat. 18 (1999) 3-10]; Basak et al. [J. Chem. Inf. Comput. Sci. 41 (2001) 671-678]; Gramatica et al. [SAR QSAR Environ. Res. 14 (2003) 237-250]. However, results obtained from external validations of those models revealed that the effective predictivity of the QSARs was well below the potential indicated by internal validation statistics (Debnath et al., Gramatica et al.). The purpose of the current research is to externally validate the model published by Cash using a data set of 29 aromatic amines reported by Glende et al. [Mutat. Res. 498 (2001) 19-37; Mutat. Res. 515 (2002) 15-38] and to further explore the potential utility of using E-state sums for the prediction of mutagenic potency of aromatic amines.

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.

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