Novel graph distance matrix

Randić, Milan; Pisanski, Tomaž; Novič, Marjana; Plavšić, Dejan

doi:10.1002/jcc.21461

Cited by 19 publications

(55 citation statements)

References 45 publications

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“…In the last section, we traced the origin of the expansion coefficients of the Green’s function back to Sachs graphs. In this section, we will show an alternative way to obtain the coefficients using Newton’s identities . To this end, we again begin with eq , the most general expression for the characteristic polynomial in this manuscript where p 0 = 1 by definition.…”

Section: Where Do the Coefficients In The Power Expansion Come From?mentioning

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: Where Do the Coefficients In The Power Expansion Come From?mentioning

confidence: 99%

Quantum Interference, Graphs, Walks, and Polynomials

Tsuji

Estrada

Movassagh³

et al. 2018

Chem. Rev.

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“…For zethrenes and perylenes, we should, however, be aware of the significant shortcomings of single-reference treatments, which can have drastic effects on predictions of molecular conduction, for example . Other authors have suggested that some wholesale “renormalization” of the MO results may be needed . Here, we can simply note that there is a well-defined class of benzenoids (those with fixed bonds) for which the results of pseudo-π and HL methods are likely to deviate from those of pure CC models.…”

Section: Resultsmentioning

confidence: 94%

“…This simple function defines the entire Randić current map in the case of a catafused benzenoid. Interestingly, though, Randić himself has recently advocated weighting by area, effectively converting Model R to Model CKCDA up to scaling …”

Section: Resultsmentioning

confidence: 99%

Ring-Current Maps for Benzenoids: Comparisons, Contradictions, and a Versatile Combinatorial Model

et al. 2020

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As a key diagnostic property of benzenoids and other polycyclic hydrocarbons, induced ring current has inspired diverse approaches for calculation, modeling, and interpretation. Grid-based methods include the ipsocentric ab initio calculation of current maps, and its surrogate, the pseudo-π model. Graph-based models include a family of conjugated-circuit (CC) models and the molecular-orbital Huckel-London (HL) model. To assess competing claims for physical relevance of derived current maps for benzenoids, a protocol for graph-reduction and comparison was devised. Graph reduction of pseudo-π grid maps highlights their overall similarity to HL maps, but also reveals systematic differences. These are ascribed to unavoidable pseudo-π proximity limitations for benzenoids with short nonbonded distances, and to poor continuity of pseudo-π current for classes of benzenoids with fixed bonds, where single-reference methods can be unreliable. Comparison between graph-based approaches shows that the published CC models all shadow HL maps reasonably well for most benzenoids (as judged by L 1 -, L 2 -, and L ∞ -error norms on scaled bond currents), though all exhibit physically implausible currents for systems with fixed bonds. These comparisons inspire a new combinatorial model (Model W) based on cycle decomposition of current, taking into account the two terms of lowest order that occur in the characteristic polynomial. This improves on all pure-CC models within their range of applicability, giving excellent adherence to HL maps for all Kekulean benzenoids, including those with fixed bonds (halving the rms discrepancy against scaled HL bond currents, from 11% in the best CC model, to 5% for the set of 18 360 Kekulean benzenoids on up to 10 hexagonal rings). Model W also has excellent performance for open-shell systems, where currents cannot be described at all by pure CC models (4% rms discrepancy against scaled HL bond currents for the 20112 non-Kekulean benzenoids on up to 10 hexagonal rings). Consideration of largest and next-to-largest matchings is a useful strategy for modeling and interpretation of currents in Kekulean and non-Kekulean benzenoids (nanographenes).

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“…This index was originally defined as the sum of path distances between any two carbons in a saturated acyclic hydrocarbon. Since it's inception this index has been generalized to a variety of structures as well as used in Quantitative structureactivity relationship (QSAR) regression models [FCH01,ME05,RPNP10,YWH04]. Some indices related to Wiener's work are the first and second multiplicative Zagreb indices [GRTW75], respectively…”

Section: Introductionmentioning

confidence: 99%

Generalised Multiplicative Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems

Kulli

Stone

Wang

et al. 2017

Zeitschrift Für Naturforschung A

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Many types of topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices are explored during past recent years. Among degree based topological indices, Zagreb indices are the oldest one and studied well. In the paper, we define a generalized multiplicative version of these indices and compute exact formulas for Polycyclic Aromatic Hydrocarbons and Jagged-Rectangle Benzenoid Systems.

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Novel graph distance matrix

Cited by 19 publications

References 45 publications

Quantum Interference, Graphs, Walks, and Polynomials

Quantum Interference, Graphs, Walks, and Polynomials

Ring-Current Maps for Benzenoids: Comparisons, Contradictions, and a Versatile Combinatorial Model

Generalised Multiplicative Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems

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