A Penrose polynomial for embedded graphs

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem

Evaluations of topological Tutte polynomials

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of medial graphs; show that $R(G)$ counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of $R(G)$. We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of $P(G)$ and $R(G)$.Comment: V2: major revision, several new results, and improved expositio

Phase Transitions in the Ising Model

This paper investigates the Ising model, a model conceived by Ernst Ising to model ferromagnetism. This paper presents a historical analysis of a model which brings together aspects of graph theory, statistical mechanics, and linear algebra. We will illustrate the model and calculate the probability of individual states in the one dimensional case. We will investigate the mathematical relationship between the energy and temperature of the model, and, using the partition function of the probability equation, show that there are no phase transitions in the one dimensional case. We endeavor to restate these proofs with greater clarity and explanation in order for them to be more accessible to other undergraduates

Interlace Polynomials for Multimatroids and Delta-Matroids

We provide a unified framework in which the interlace polynomial and several related graph polynomials are defined more generally for multimatroids and delta-matroids. Using combinatorial properties of multimatroids rather than graph-theoretical arguments, we find that various known results about these polynomials, including their recursive relations, are both more efficiently and more generally obtained. In addition, we obtain several interrelationships and results for polynomials on multimatroids and delta-matroids that correspond to new interrelationships and results for the corresponding graphs polynomials. As a tool we prove the equivalence of tight 3-matroids and delta-matroids closed under the operations of twist and loop complementation, called vf-safe delta-matroids. This result is of independent interest and related to the equivalence between tight 2-matroids and even delta-matroids observed by Bouchet.Comment: 35 pages, 3 figure