Graph potentials method and its application for chemical information processing

Golender, V. E.; Drboglav, V. V.; Rosenblit, A. B.

doi:10.1021/ci00032a004

Cited by 53 publications

(41 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted that vertices 1 and 2 for C 32 , as well as vertices 7,8,9,10,13,14,43,44,45,46,49, 50 for C 78 are achiral (marked as Ø is following tables) because there are at least two topologically equivalent terminal vertices in respective simplexes (Fig. 8).…”

Section: Results: the Assessment Of Equivalence/nonequivalence Of Thementioning

confidence: 99%

“…Calculations of second-order potentials are presented in [13], have been performed by using components of the C vector equal to the reciprocal values of the diagonal entries of the G matrix. Despite the fact that the method of graph potentials is able to recognize nonequivalent vertices, it is a classic example of iterative heuristic method that is characterized by exponential complexity that makes it unable for application in case of fullerene studies.…”

Section: Test Case: Nonequivalence Of Vertices In Triphenylene Derivamentioning

confidence: 99%

“…Several papers reported methods of evaluation of equivalence or nonequivalence of vertices in graphs [9][10][11][12][13][14][15][16][17][18]. These methods are helpful: (1) to evaluate the substitutions of automorphism groups of graphs; (2) to evaluate the isomorphism (or nonisomorphism) for two given graphs; (3) to describe graph using the canonical numbering of vertices; (4) to find classes of equivalence.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficacy of topological informational potentials for analysis of nonequivalent atoms in molecular graphs: the case of chiral fullerenes

et al. 2016

View full text Add to dashboard Cite

In this study, the topological informational field was applied for the assessment of equivalence or nonequivalence of atoms and molecular fragments in structurally homogeneous chiral fullerenes. Two types of so-called topological informational potentials were calculated using noniterative heuristic method. Application of methods that allow estimating the nonequivalence of similar atoms could be a breakthrough for the description of highly homogenous structures. It was shown that method, which takes into account multiple paths in the molecular graph, is the most efficient. Results of equivalence/nonequivalence analysis allow to differentiate sources of molecule's chirality.

show abstract

Section: Results: the Assessment Of Equivalence/nonequivalence Of Thementioning

confidence: 99%

Section: Test Case: Nonequivalence Of Vertices In Triphenylene Derivamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficacy of topological informational potentials for analysis of nonequivalent atoms in molecular graphs: the case of chiral fullerenes

et al. 2016

View full text Add to dashboard Cite

show abstract

“…Then, by (14), f n3 = 2f n1 = 4 = 4 1 . Consequently, using (14), (11) and induction, we obtain that…”

Section: Corollary 2 the Entries Of The Doubly Stochastic Matrixmentioning

confidence: 99%

Spanning forests and the golden ratio

Chebotarev

2008

Discrete Applied Mathematics

View full text Add to dashboard Cite

For a graph G, let f ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f ij 's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F = (f ij ) n×n /f , where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F −1 = I + L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to −1 or to 1 − −1 , where is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as "golden introverts" and "golden extroverts," respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.

show abstract

“…For example, all elements of the adjacency matrix, the Laplace matrix, or the matrix of distances can be used as IS's. To solve the problem of isomorphism and automorphism of the molecular structures, the method of potentials [26] uses the matrices G = (I + L),…”

Section: Introductionmentioning

confidence: 99%

Laplace Spectra of Orgraphs and Their Applications

Agaev

Chebotarev

2005

Autom Remote Control

View full text Add to dashboard Cite

The Laplace matrix is the matrix L = ( ij ) ∈ R n×n with nonpositive off-diagonal elements and zero row sums. A weighted orgraph corresponds to each Laplace matrix, its properties being closely related to the algebraic properties of the Laplace matrix. The normalized Laplace matrix L is the Laplace matrix where − 1 n ij 0 for all i = j. The paper was devoted to the spectrum of the Laplace matrices and to the relationship between the spectra of the Laplace and stochastic matrices. The normalized Laplace matrices were proved to be semiconvergent. It was established that the multiplicity of the eigenvalue 0 of the matrix L is equal to the in-forest dimension of the corresponding orgraph, and the multiplicity of the eigenvalue 1 is one less than the in-forest dimension of the complementary orgraph. The spectra of the matrices L belong to the intersection of two circles of radius 1 − 1/n centered at the points 1/n and 1 − 1/n, respectively. Additionally, the domain that comprises them is included in the intersection of two angles (defined in the paper) with vertices 0 and 1 and the band |Im (z)| 1 2n cot π 2n (at the limit |Im(z)| < 1 π ). A polygon with all points being the eigenvalues of the normalized n-order Laplace matrices was constructed.

show abstract

Graph potentials method and its application for chemical information processing

Cited by 53 publications

References 17 publications

Efficacy of topological informational potentials for analysis of nonequivalent atoms in molecular graphs: the case of chiral fullerenes

Efficacy of topological informational potentials for analysis of nonequivalent atoms in molecular graphs: the case of chiral fullerenes

Spanning forests and the golden ratio

Laplace Spectra of Orgraphs and Their Applications

Contact Info

Product

Resources

About