Related papers: Partial Petrial Polynomials of Ribbon Graphs

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,…

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an…

The partial-dual Euler-genus polynomial was defined by Gross, Mansour, and Tucker to analyze how the Euler genus of a ribbon graph changes under partial duality, a generalization of Euler-Poincar\'{e} duality introduced by Chmutov. The…

We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, -petriality, or -triality to the…

The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ of a ribbon graph $G$ is the generating function that enumerates all partial duals of $G$. In this paper, we give a categorification for this polynomial. The key ingredient of…

Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph and asked under what conditions such a polynomial is even-interpolating, odd-interpolating, or both. In this paper, we provide an answer to this open…

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial…

For a ribbon graph $G$, let $\gamma(G)$ denote its Euler genus. Recently, Chen, Gross and Tucker [J. Algebraic Combin. 63 (2026) 13] derived a formula for the maximum partial-dual Euler-genus $\partial\gamma_M(G)$ of a ribbon graph $G$.…

We provide recipe theorems for the Bollob\`as and Riordan polynomial $\mathcal{R}$ defined on classes of ribbon graphs with half-edges introduced in arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial $Q$ on this…

For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon…

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated…

We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $\gamma$-polynomials of their associated graph-associahedra. The partial…

Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes…

In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in three…

Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $\phi(G,x)=\det(xI-A(G))$ and the permanental polynomial $\pi(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For…

We prove that the partial-dual genus polynomial considered as a function on chord diagrams satisfies the four-term relation. Thus it is a weight system from the theory of Vassiliev knot invariants.

We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate…

Recently, Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of…

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…

The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal…