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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 Bollobas and Riordans 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 $R(G)$.
We introduce and study the notion of the $G$-Tutte polynomial for a list $mathcal{A}$ of elements in a finitely generated abelian group $Gamma$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite
In [A polynomial invariant of graphs on orientable surfaces, Proc. Lond. Math. Soc., III Ser. 83, No. 3, 513-531 (2001)] and [A polynomial of graphs on surfaces, Math. Ann. 323, 81-96 (2002)], Bollobas and Riordan generalized the classical Tutte poly
A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobas-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-va
Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points (a,b) where (a-1)(b-1) equals 2 or 4. These evaluations are expressed in terms of eulerian su
A cycle $C$ in a graph $G$ is called a Tutte cycle if, after deleting $C$ from $G$, each component has at most three neighbors on $C$. Tutte cycles play an important role in the study of Hamiltonicity of planar graphs. Thomas and Yu and independently