We consider construction of a set of smaller 4 x 4 matrices to represent DNA primary sequences which are based on enumeration of all 64 triplets of nucleic acids bases. The leading eigenvalue from the constructed matrices has been selected as an invariant for construction of a vector to characterize DNA. Additional invariants considered of the derived condensed matrices of DNA include a 64-component vector, the components of which consist of ordered triplets XYZ, with X, Y, Z = A, C, G, T. Construction of similarity/dissimilarity tables based on different invariants for a set of sequences of DNA belonging to the first exon of the beta-globin gene of eight species illustrates the utility of newly formulated invariants for DNA.

We report some properties of new graph matrices which appear to offer novel graph invariants of potential interest in structureproperty studies. The matrices are constructed by generalizing Wiener's procedure for evaluation of Wiener numbers in alkanes. Among the invariants considered we particularly examined the sequences generated by summing the entries in the matrix for vertices at the same distance from one another. These numbers may be viewed as "higher" Wiener numbers in analogy with "higher" connectivity indices.We have listed the higher Wiener numbers of alkanes up to n = 9 carbon atoms and also report several recursions for the construction of these invariants for selected families of acyclic graphs. Briefly, we have outlined how the Wiener matrix can be extended to cyclic systems, while in the concluding comments we have outlined an extension of the Wiener matrix to molecules having heteroatoms. The significance of the matrices as a source of graph invariants is precisely in this possibility to go beyond simple models of molecular graphs and extend graph invariants of interest to molecules having different kinds of atoms.

We report on some properties of Wiener matrices as illustrated for alkanes. We start by considering the problem of graph reconstruction from Wiener matrix. Next, we consider the eigenvalues of the Wiener matrices. We also consider the row sums of the Wiener matrices, from which an index JJ, similar to the connectivity index and Balaban's J index, was constructed. The new invariants were tested for suitability for the structure-property relationship by analyzing regressions of novel invariants for selected properties of octanes.