Related papers: Evaluations of topological Tutte polynomials
Let $T(G;X,Y)$ be the Tutte polynomial for graphs. We study the sequence $t_{a,b}(n) = T(K_n;a,b)$ where $a,b$ are non-negative integers, and show that for every $\mu \in \N$ the sequence $t_{a,b}(n)$ is ultimately periodic modulo $\mu$…
For a ribbon graph $G$ we consider an alternating link $L_G$ in the 3-manifold $G\times I$ represented as the product of the oriented surface $G$ and the unit interval $I$. We show that the Kauffman bracket $[L_G]$ is an evaluation of the…
The $2$-decomposition for ribbon graphs was introduced in [Annals of Combinatorics 15 (2011), pp 675-706]. We extend this result to half-edged ribbon graphs and to rank $D$-weakly colored graphs [SIGMA 12 (2016), 030], generalizing…
This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial…
We study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ \chi_G^\lambda(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } \lambda_v c(v) }, \] where $\lambda \in…
The Tutte polynomial is an important invariant of graphs and matroids. Chen and Guo \emph{[Adv. in Appl. Math. 166 (2025) 102868.]} proved that for a $(k+1)$-edge connected graph $G$ and for any $i$ with $0\leq i <\frac{3(k+1)}{2}$,…
The Tutte polynomial is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, $T_G(X,Y)\; =\; {T}_{G^*}(Y,X)$ where $G^*$ denotes the…
In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW…
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the…
We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. The $G$-parking critical module and the toppling critical module are canonical modules (with suitable twists)…
Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},\tau+1)| \le (\tau-1)^{n-5}$, where $\tau=(1+\sqrt{5} \,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study…
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the…
The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…
Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane…
For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old…
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the…
The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to…
Let $P(G,q)$ be the chromatic polynomial for coloring the $n$-vertex graph $G$ with $q$ colors, and define $W=\lim_{n \to \infty}P(G,q)^{1/n}$. Besides their mathematical interest, these functions are important in statistical physics. We…
We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this…