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

Dias, Jerry Ray

doi:10.1139/v87-124

Cited by 23 publications

(14 citation statements)

References 8 publications

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“…Such molecular graphs can be represented by vertex- and edge-weighted graphs . Dias established a way of obtaining the characteristic polynomial for a molecular graph perturbed by a heteroatom from the characteristic polynomial for the isoconjugate graph (a graph in which the heteroatom is replaced with the carbon atom) and that for the graph obtained upon deletion of heteroatom vertex with its adjacent edges . In this context, the isoconjugate graph corresponds to an unperturbed system.…”

Section: Hückel Mo Perturbation Theory and Its Impact On Qimentioning

confidence: 99%

Quantum Interference, Graphs, Walks, and Polynomials

Tsuji

Estrada

Movassagh³

et al. 2018

Chem. Rev.

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In this paper we explore quantum interference in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix orHückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are cancelled by others, leading to another class of quantum interference. For non-alternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for non-alternants, from cancellation of odd and even-length walk terms. We report some progress, but not a complete resolution of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. And we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.

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Section: Hückel Mo Perturbation Theory and Its Impact On Qimentioning

confidence: 99%

Quantum Interference, Graphs, Walks, and Polynomials

Tsuji

Estrada

Movassagh³

et al. 2018

Chem. Rev.

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“…This procedure is specific for the acyclic polynomial, without an equivalent version for the characteristic polynomials in the general case. 26 Use of eqn ( 6) until all the cycles have been opened leads to an expression for P ac (x) as a weighted sum of characteristic polynomials (for open-chain fragments, the acyclic and characteristic polynomials are equal). Since characteristic polynomials have a strong chemical relevance, it is clear that eqn (6) is of not only practical, but also of fundamental importance as it provides TRE with a definite, if indirect chemical interpretation.…”

Section: Graph-theoretical Backgrounds Of Trementioning

confidence: 99%

The chemical roots of the matching polynomial

Chauvin

Lepetit

Fowler

et al. 2010

Phys. Chem. Chem. Phys.

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The interpretation of topological resonance energy (TRE), the mathematical measure of energetic chemical aromaticity at the Hückel level of theory, is revisited by providing a concise analysis of the matching polynomial of a chemical graph. Whereas the matching (or acyclic) polynomial is not a properly characteristic polynomial in general, it is the arithmetic mean of the characteristic polynomials of the corresponding Hückel and Möbius chemical graphs. For unicyclic graphs, the roots of the matching polynomial (serving to calculate the TRE values) can be graphically approximated from generalized Frost-Musulin-Zimmermann diagrams, the results becoming exact for annulenic monocycles. The matching polynomial is also shown to be a weighted sum of characteristic polynomials of subgraphs: the whole graph itself and the exocyclic graphs obtained by suppression of individual circuits. The performance of these ring twisting and ring masking processes is compared to that of the previously proposed ring opening process. Further prospects for the interpretation and approximation of TRE are discussed.

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“…For the unique C 20 fullerene, the first eight coefficients have the same absolute values. Balasubramanian, 19 adapting work published by Dias,20 showed that these coefficients for characteristic polynomials of all C n fullerenes depend only on n. His formulas may also apply to the permanental polynomials, but that has yet to be determined.…”

Section: Resultsmentioning

confidence: 99%