On "the matching polynomial of a polygraph"

In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: \ud 1.\ud Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraghs.\ud 2.\ud Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half

Ring-current maps for benzenoids : comparisons, contradictions, and a versatile combinatorial model

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 Hückel-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 L1-, L2-, 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)

The Structure of 4-Clusters in Fullerenes

Fullerenes can be considered to be either molecules of pure carbon or the trivalent plane graphs with all hexagonal and (exactly 12) pentagonal faces that models these molecules. Since carbon atoms have valence 4 and our models have valence 3, the edges of a perfect matching are doubled to bring the valence up to 4 at each vertex. The edges in this perfect matching are called a Kekule structure and the hexagonal faces bounded by three Kekule edges are called benzene rings. A maximal independent (disjoint) set of benzene rings for a given Kekule structure is called a Clar set, and the maximum possible size of a Clar set over all Kekule structures is the Clar number of the fullerene. For any patch of hexagonal faces in the fullerene away from all pentagonal faces, there is a perfect Kekule structure: a Kekule structure for which the faces of an independent set of benzene rings are packed together as tightly as possible. Starting with such a patch and extending it as far as possible results in a perfect Kekule structure except for isolated regions, called clusters, containing the pentagonal faces. It has been shown that clusters must contain even numbers of pentagonal faces. It has also been shown that the Kekule structure of the patch can be extended into each of these clusters to give a full Kekule structure. However, these Kekule extensions will not admit as tightly packed benzene rings as in the patch external to the clusters. A basic problem in computing the Clar number of a fullerene is to make these extensions in a way that maximizes the number of benzene rings in each cluster. The simplest case, that of 2-clusters, has been completely solved. This thesis is devoted to developing a complete understanding of the Clar structures of 4-clusters

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Counterexamples to Ferromagnetic Ordering of Energy Levels

The Heisenberg ferromagnet has symmetry group ${\rm SU}(2)$. The property known as ferromagnetic ordering of energy levels (FOEL) states that the minimum energy eigenvalue among eigenvectors with total spin $s$ is monotone decreasing as a function of $s$. While this property holds for certain graphs such as open chains, in this note we demonstrate some counterexamples. We consider the spin 1/2 model on rings of length $2n$ for $n=2,3,...,8$, and show that the minimum energy among all spin singlets is less than or equal to the minimum energy among all spin triplets, which violates FOEL. This also shows some counterexamples to the "Aldous ordering" for the symmetric exclusion process. We also review some of the literature related to these examples.Comment: We corrected an earlier misinterpretation we made of a famous result of Sutherland, which an anonymous referee corrected us on. 29 page