Related papers: Partial Petrial Polynomials of Ribbon Graphs

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 $k$.…

In "A survey on the Turaev genus of knots," Champanerkar and Kofman propose several open questions. The first asks whether the polynomial whose coefficients count the number of quasi-trees of the all-A ribbon graph obtained from a diagram…

Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction…

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the…

Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph…

Recently S. Chmutov introduced a generalization of the dual of a ribbon (or embedded) graph and proved a relation between Bollobas and Riordan's ribbon graph polynomial of a ribbon graph and its generalized duals. Here I show that the…

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…

A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a…

A {\it universal labeling} of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ and $u$ in the oriented graph are…

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this…

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…

For every finite simple connected graph $G = (V,E)$, we introduce an invariant, its blowup-polynomial $p_G(\{ n_v : v \in V \})$. This is obtained by dividing the determinant of the distance matrix of its blowup graph $G[{\bf n}]$…

The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections…

In this paper, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the $i$-th (rational) Betti number and Euler characteristic of the real toric variety associated to a graph…

We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$…

We give an excluded minor characterisation of the class of ribbon graphs that admit partial duals of Euler genus at most one.

We prove that the $k$-power of any planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{f(\Delta(G),k)}$ for some graph $H$ with bounded treewidth, some path $P$, and some function $f$. This resolves an open problem of Ossona de…

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…

The permanent of an $n \times n$ matrix $M = (m_{ij})$ is defined as $\mathrm{per}(M) = \sum_{\sigma \in S_n} \prod_{i=1}^n m_{i,\sigma(i)}$, where $S_n$ denotes the symmetric group on $\{1,2,\ldots,n\}$. The permanental polynomial of $M$,…

The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was introduced by Birkhoff in 1912; denoted $P(G,k)$, it equals the number of proper $k$-colorings of graph $G$. Enumerative analogues of the…