相关论文: Chromatic polynomials and bialgebras of graphs

We introduce two coproducts on mixed graphs (that is to say graphs with both oriented and unoriented edges), the first one by separation of the vertices into two parts, and the second one given by contraction and extractions of subgraphs.…

Using the formalism of species and twisted objects, we introduce two structures of cointeracting bialgebras on hypergraphs, induced by two notions of induced sub-hypergraphs. We study the associated unique morphisms of cointeracting…

Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three…

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 $k$.…

Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and…

We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a double bialgebra in the sense of L. Foissy, who recently proved there is then a unique…

We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie…

Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic…

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying…

We prove that the generating function for the symmetric chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev--Petviashvili integrable hierarchy of mathematical physics.…

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very…

A topological index of a graph $G$ is a real number which is preserved under isomorphism. Extensive studies on certain polynomials related to these topological indices have also been done recently. In a similar way, chromatic versions of…

We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. For various joins of all-positive or all-negative signed complete…

The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the…

Given a graph $G$ of order $n$, the $\sigma$-$polynomial$ of $G$ is the generating function $\sigma(G,x) = \sum a_{i}x^{i}$ where $a_{i}$ is the number of partitions of the vertex set of $G$ into $i$ nonempty independent sets. Such…

This paper discusses ways to categorify chromatic, dichromatic and Penrose polynomials, including categorifications of integer evaluations of chromatic polynomials. We show that with an appropriate choice of variables the coefficients of…

We study "holomorphic quadratic differentials" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $\chi$ of a graph $G$, at…

The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $\lambda$ colors such that adjacent vertices either receive distinct colors from a set of $\lambda$…

For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known…

In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously…