Related papers: Hypergraph polynomials and the Bernardi process
A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with…
Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding with the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal was to…
The Tutte polynomial is a crucial invariant of matroids. The polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$, introduced by Bernardi et al., is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this…
We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an…
We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte…
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…
The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G,…
Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive…
We prove several theorems concerning Tutte polynomials $T(G,x,y)$ for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the Potts…
Hypergraphs are important objects to model ternary or higher-order relations of objects, and have a number of applications in analysing many complex datasets occurring in practice. In this work we study a new heat diffusion process in…
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…
An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x)…
The Tutte polynomial of a connected graph was originally defined by Tutte as a sum over all spanning trees of monomials depending on a fixed linear order on the set of edges. Tuttle proved that while these monomials do depend on the linear…
The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety.…
A fourientation of a graph $G$ is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph…
The generalized degree polynomial $\mathbf{G}_T(x,y,z)$ of a tree $T$ is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that $\mathbf{G}_T$…
We provide a deterministic polynomial-time algorithm that, for a given $k$-uniform hypergraph $H$ with $n$ vertices and edge density $d$, finds a complete $k$-partite subgraph of $H$ with parts of size at least ${c(d, k)(\log…
The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive…
Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of…
For each uniformity $k \geq 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a root $\lambda$ with $\lvert\lambda\rvert = O\left(\frac{\log…