Spanning trees and even integer eigenvalues of graphs

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\lambda\equiv2\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\lambda\equiv0\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if $\tau(G)=2^ts$ with $s$ odd, then the multiplicity of any even integer eigenvalue of $Q(G)$ is at most $t+1$. Among other things, we prove that if $L(G)$ or $Q(G)$ has an even integer eigenvalue of multiplicity at least $2$, then $\tau(G)$ is divisible by $4$. As a very special case of this result, a conjecture by Zhou et al. [On the nullity of connected graphs with least eigenvalue at least $-2$, Appl. Anal. Discrete Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows.Comment: Final version. To appear in Discrete Mat

An elementary chromatic reduction for gain graphs and special hyperplane arrangements

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for chromatic functions of gain graphs. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page

Omni-conducting and omni-insulating molecules

The source and sink potential model is used to predict the existence of omni-conductors (and omni-insulators): molecular conjugated π systems that respectively support ballistic conduction or show insulation at the Fermi level, irrespective of the centres chosen as connections. Distinct, ipso, and strong omni-conductors/omni-insulators show Fermi-level conduction/insulation for all distinct pairs of connections, for all connections via a single centre, and for both, respectively. The class of conduction behaviour depends critically on the number of non-bonding orbitals (NBO) of the molecular system (corresponding to the nullity of the graph). Distinct omni-conductors have at most one NBO; distinct omni-insulators have at least two NBO; strong omni-insulators do not exist for any number of NBO. Distinct omni-conductors with a single NBO are all also strong and correspond exactly to the class of graphs known as nut graphs. Families of conjugated hydrocarbons corresponding to chemical graphs with predicted omni-conducting/insulating behaviour are identified. For example, most fullerenes are predicted to be strong omni-conductors

Renormalization group-like proof of the universality of the Tutte polynomial for matroids

In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended abstract.Comment: 12 pages, 3 figures, conference proceedings, 25th International Conference on Formal Power Series and Algebraic Combinatorics, Paris, France, June 201