Graph theoretical invariants

According to Graph Theory, a molecule may be represented by its skeletal molecular graph which, from a mathematical point of view, is the union of a set of points, symbolising the atoms other than hydrogens, and a set of lines, symbolising bonds. Its properties can be then expressed in terms of graph-theoretical invariants, Njj, which have been defined as "the number of distinct ways in which skeleton i... 

From the edge adjacency matrix, a graph-theoretical invariant analogous to the -> Randic connectivity index was derived by Estrada [Estrada, 1995a] it is called edge connectivity index, denoted by e, and defined as ... 

Araujo, O. and Morales, D.A. (1996b). An Alternative Approach to Orthogonal Graph Theoretical Invariants. Chem.Phys.Lett., 257, 393-396. 

Basak, S.C. (1990). A Nonempirical Approach to Predicting Molecular Properties Using Graph-Theoretic Invariants. In Practical Applications of Quantitative Structure-Activity Relationships (QSAR) in Environmental Chemistry and Toxicology (Karcher, W. and Devillers, J., eds.), Kluwer, Dordrecht (The Netherlands), pp. 83-103. 

Estrada, E. (1995c). Graph Theoretical Invariant of Randic Revisited. J.Chem.InfComput.Sci., 35,1022-1025. 

Ivanciuc, O. and Balaban, A.T. (1994a). Design of Topological Indices. Part 5. Precision and Error in Computing Graph Theoretic Invariants for Molecules Containing Heteroatoms and Multiple Bonds. MATCH (Comm.Math.Comp.Chem.), 30,117-139. 

Morales, D.A. and Araujo, O. (1993). On the Search for the Best Correlation Between Graph Theoretical Invariants and Physicochemical Properties. J.Math.Chem., 13, 95-106. 

Randic, M., Hansen, P.J. and Jurs, RC. (1988a). Search for Useful Graph Theoretical Invariants of Molecular Structure. J.Chem.Inf.ComputSci., 28,60-68. 

The Bertz index incorporates skeletal complexity by representing the molecular structure as a graph (a skeletal molecular graph that represents all nonhydrogen atoms) that will have properties that are expressed as so-called graph theoretical invariants. Bertz uses information theory to incorporate the effects of symmetry, and also accounts for molecular size and the presence of heteroatoms in his general index of complexity. 

Padron, J.A., Carrasco, R. and PeI16n, R.F. (2002) Molecular descriptor based on a molar refractivity partition using Randic-type graph-theoretical invariant./. Pharm. Pharmaceut. Sci., 5, 258—265. 

E. Estrada, Graph theoretical invariant of Randic revisited, J. Chem. Inf. Comput. Sci. 35 (1995c) 1022-1025. 

These kind of graphs are called twin graphs (Hosoya et al., 1994, 2001), because they possess, besides identical distance-spectra and consequently identical distance-polynomials, identical characteristic polynomials and their spectra (3.8801, 1.3557, 0.7732, 0.4773, -0.7376, -1.2464, -2.0953, -2.4069), identical matching polynomials and their spectra, and many identical graph-theoretical invariants. 

O. Araujo and D. A. Morales, An alternative approach to orthogonal graph theoretical invariants, Chem. Phys. Lett. 257 (1996) 393-396. 

T. Okuyama, Y. Miyashita, S. Kanaya, H. Katsumi, S.-I. Sasaki, and M. Randid, Computer-assisted structure taste studies of sulfamates by pattern-recognition method using graph theoretical invariants, J. Comput. Chem. 6 (1988) 636-646. 

---