Properties and derivation of the fourth and sixth coefficients of the characteristic polynomial of molecular graphs?New graphical invariants

1985 ◽  
Vol 68 (2) ◽  
pp. 107-123 ◽  
Author(s):  
Jerry Ray Dias
Keyword(s):  
Characteristic Polynomial ◽  
Molecular Graphs ◽  
Graphical Invariants

Graph Theory in Chemistry II. Graph-Theoretical Description of Heteroconjugated Molecules

1975 ◽  
Vol 30 (12) ◽  
pp. 1696-1699 ◽  
Author(s):  
A. Graovac ◽  
O. E. Polansky ◽  
N. Trinajstić ◽  
N. Tyutyulkov
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Theoretical Description ◽  
Weighted Graphs ◽  
Conjugated Molecules ◽  
Molecular Graphs

Abstract The Sachs' formula which relates the structure of a (vertex-and edge) -weighted graph and its characteristic polynomial is given. Weighted graphs are used to represent conjugated molecules containing the variety of heteroatoms and heterobonds. Vertices and edges in such molecular graphs have different weights following the deviation of the corresponding (Hiickel) Coulomb and resonance integrals from the standard (benzene) values.

Facile calculations of select eigenvalues and the characteristic polynomial of small molecular graphs containing heteroatoms

1987 ◽  
Vol 65 (4) ◽  
pp. 734-739 ◽  
Author(s):  
Jerry Ray Dias
Keyword(s):  
Characteristic Polynomial ◽  
Theoretical Determination ◽  
Molecular Graphs ◽  
Universal Expression

Determination of common eigenvalues by embedding smaller subgraphs onto larger ones is illustrated. A nearly universal expression of the sixth coefficient of the characteristic polynomial valid for all conjugated hydrocarbon polyenes is presented. Its use in a graph theoretical determination of the Hückel MO characteristic polynomial of molecular graphs possessing heteroatoms is detailed.

Higher-Order and Mixed Discrete Derivatives such as a Novel Graph- Theoretical Invariant for Generating New Molecular Descriptors

2019 ◽  
Vol 19 (11) ◽  
pp. 944-956 ◽  
Author(s):  
Oscar Martínez-Santiago ◽  
Yovani Marrero-Ponce ◽  
Ricardo Vivas-Reyes ◽  
Mauricio E.O. Ugarriza ◽  
Elízabeth Hurtado-Rodríguez ◽  
...  
Keyword(s):  
Molecular Descriptors ◽  
3D Qsar ◽  
Higher Order ◽  
Molecular Structures ◽  
Qsar Modeling ◽  
Statistical Parameters ◽  
Molecular Graphs ◽  
New Family ◽  
Mixed Derivatives ◽  
Discrete Derivative

Background: Recently, some authors have defined new molecular descriptors (MDs) based on the use of the Graph Discrete Derivative, known as Graph Derivative Indices (GDI). This new approach about discrete derivatives over various elements from a graph takes as outset the formation of subgraphs. Previously, these definitions were extended into the chemical context (N-tuples) and interpreted in structural/physicalchemical terms as well as applied into the description of several endpoints, with good results. Objective: A generalization of GDIs using the definitions of Higher Order and Mixed Derivative for molecular graphs is proposed as a generalization of the previous works, allowing the generation of a new family of MDs. Methods: An extension of the previously defined GDIs is presented, and for this purpose, the concept of Higher Order Derivatives and Mixed Derivatives is introduced. These novel approaches to obtaining MDs based on the concepts of discrete derivatives (finite difference) of the molecular graphs use the elements of the hypermatrices conceived from 12 different ways (12 events) of fragmenting the molecular structures. The result of applying the higher order and mixed GDIs over any molecular structure allows finding Local Vertex Invariants (LOVIs) for atom-pairs, for atoms-pairs-pairs and so on. All new families of GDIs are implemented in a computational software denominated DIVATI (acronym for Discrete DeriVAtive Type Indices), a module of KeysFinder Framework in TOMOCOMD-CARDD system. Results: QSAR modeling of the biological activity (Log 1/K) of 31 steroids reveals that the GDIs obtained using the higher order and mixed GDIs approaches yield slightly higher performance compared to previously reported approaches based on the duplex, triplex and quadruplex matrix. In fact, the statistical parameters for models obtained with the higher-order and mixed GDI method are superior to those reported in the literature by using other 0-3D QSAR methods. Conclusion: It can be suggested that the higher-order and mixed GDIs, appear as a promissory tool in QSAR/QSPRs, similarity/dissimilarity analysis and virtual screening studies.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

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