Graph invariant

Filip, P. A., Balaban, T. S., Balaban, A. T. A new approach for devising local graph invariants Derived topological indices with low degeneracy and good correlational ability. J. Math. Chem. 1987, 1, 61-83. 

Basak, S. C., Mills, D., Mumtaz, M. M. Use of graph invariants in the protection of human and ecological health. In Basak, S. C., Balakrishnan, R., Eds., Lecture Notes of the First Indo-US Lecture Series on Discrete Mathematical Chemistry, 2007. 

Optimization of Correlation Weights of Local Graph Invariants... 

Examples of the global graph invariants are represented by the number and size of cycles, and the total number of chemical elements in a molecule. The number of oxygen and/or nitrogen atoms can be (in the first approximation) a measure of... 

Castro EA, Toropov AA, Nesterova AI, Nabiev OM (2004) QSPR modeling aqueous solubility of polychlorinated biphenyls by optimization of correlation weights of local and global graph invariants. CEJC 2 500-523. 

Toropov AA, Roy K (2004) QSPR modeling of lipid-water partition coefficient by optimization of correlation weights of local graph invariants. J. Chem. Inf. Comput. Sci. 44 179-186. 

Toropov AA, Toropova AP (2002) QSPR modeling of complex stability by optimization of correlation weights of the hydrogen bond index and the local graph invariants. Russ. J. Coord. Chem. 28 877-880. 

The distance matrix D(G) of a graph G is another important graph-invariant. Its entries dy, called distances, are equal to the number of edges connecting the vertices i and j on the shortest path between them. Thus, all dy are integers, including du = 1 for nearest neighbours, and, by definition, d = 0. The distance matrix can be derived readily from the adjacency matrix ... 

Diudea, M. V., O. Minailiuc, and A. T. Balaban, Regressive Vertex Degree (New Graph Invariants) and Derived Topological Indices. J. Comput. Chem., 1991 12, 527-535. 

We have already examined a few graph invariants which, according to the above definition, could be included among the topological indices of benzenoid molecules. These are the number of Kekule structures, the eigenvalues, spectral moments, (coefficients of) the characteristic and sextet polynomials, to mention just some. The total 7t-electron energy is surveyed elsewhere in this volume. 

In addition to these, only a limited number of other topological indices of benzenoid molecules have been studied. With a few not too important exceptions, generally valid mathematical results were obtained only for one of them — namely for the Wiener index. Therefore the remaining part of this section is devoted to the Wiener index of benzenoid systems. (Further graph invariants worth mentioning in connection with benzenoids, especially unbranched catacondensed systems, are the Hosoya index [119-121], the Merrifield — Simmons index [122, 123], the modified Hosoya index [38] and the polynomials associated with them.)... 

Lemma Bijective (one-to-one, onto) functions [15] exist between each of the above mentioned sets of graph invariants E,V,H,C- In other words we can write the relation as follows ... 

Subsets of graphs which constitute equivalence classes will be called equivalent graphs and their polynomials, equivalent pol)rnomials It takes a little thought to realize that an equivalence relation (widi transitive closure [50] on a set of n equivalent graphs requires the existence of n mappings of selected graph invariants onto each other Thus for the set S = T,A, B, Pi one must Identify 24 (a4 ) mappings, three of which are... 

If G-, G2, —, are equivalent graphs there must be, by definition, one-to-one oino mappings defined for selected sets of their graph invariants Therefore all such graphs have identical numbers of graph invariants [52]> Furthermore in each case the set of invariants have "equivalent" adjacency relations, e g e- is related to e- (in T) in the same way as v- is related to v- (inV ) Then 0(G, ) s enumerate identical combinations of different objects ... 

By "Graph Invariant" is meant here the selected sets of invariants shown in Table 3-... 

The unique properties of fragment descriptors are related to the fact that (i) any molecular graph invariant i.e., any molecular descriptor or property)... 

Some molecular descriptors, called - determinant-based descriptors, are calculated as the determinant of a - matrix representation of a molecular structure. Moreover, permanents, short- and long-hafnians, calculated on the topological - distance matrix D, were used as graph invariants by Schultz and called per(D) index, shaf(D) index, lhaf(D) index [Schultz et al, 1992 Schultz and Schultz, 1992]. 

Other related molecular descriptors are -+ atomic composition indices, several - information indices and - graph invariants. 

The ATS descriptor is a graph invariant describing how the considered property is distributed along the topological structure. In fact, assuming an additive scheme, the ATS descriptor corresponds to a decomposition of the square molecular property O in different atomic contributions ... 

This is the simplest graph invariant obtained from the -> adjacency matrix A, defined as the number of bonds in the -> molecular graph (7 where multiple bonds are considered as single edges. Bond number is calculated as half the - total adjacency index Ay ... 

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