Related papers: Tutte Short Exact Sequences of Graphs
A rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere $\widehat{\mathbb{C}}$ is called critically fixed if each critical point of $f$ is fixed under $f$. In this article, we study the properties of a…
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,…
From any directed graph $E$ one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in $E$. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for…
Two landmark results in combinatorial random matrix theory, due to Koml\'os and Costello-Tao-Vu, show that discrete random matrices and symmetric discrete random matrices are typically nonsingular. In particular, in the language of graph…
The perfect matching complex of a simple graph $G$ is a simplicial complex having facets (maximal faces) as the perfect matchings of $G$. This article discusses the perfect matching complex of polygonal line tilings and the $\left(2 \times…
The q-state Potts model is a fundamental framework in statistical physics and graph theory, with its partition function encoding rich information about spin configurations. The multivariate Tutte polynomial (known as the partition function…
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…
To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path…
In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate…
The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly…
Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$. We obtain that $G$ satisfies the Tutte's condition \[ o(G-S)\le \sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$}, \] with respect…
A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge harms the equimatchability, called edge-critical equimatchable graphs…
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 present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality…
We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with…
Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, \textsc{$T$-Cycle}: given an undirected $n$-vertex graph $G$ and a set of $k$ vertices…
Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general…
In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach…
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 study the space ${\mathcal X}^{d}(G)$ of pictures of a graph $G$ in complex projective $d$-space. The main result is that the homology groups (with integer coefficients) of ${\mathcal X}^{d}(G)$ are completely determined by the Tutte…