Related papers: A most general edge elimination graph polynomial

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of…

In this note we study a certain graph polynomial arising from a special recursion. This recursion is a member of a family of four recursions where the other three recursions belong to the chromatic polynomial, the modified matching…

As a variant of the Ulam's vertex reconstruction conjecture and the Harary's edge reconstruction conjecture, Cvetkovi\'c and Schwenk posed independently the following problem: Can the characteristic polynomial of a simple graph $G$ with…

The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing 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…

Let $G$ be a graph and let $m_{ij}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The {\em $M$-polynomial} of $G$ is introduced with $\displaystyle{M(G;x,y) = \sum_{i\le j} m_{ij}(G)x^iy^j}$.…

Let $G=(V,E)$ be a simple connected graph. A connected edge cover of $G$ is a subset $S\subseteq E$ such that every vertex of $G$ is incident with at least one edge in $S$ and the subgraph induced by $S$ is connected. The connected edge…

T. K\'{a}lm\'{a}n (A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1/x,1)$ and…

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollob\'as-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted…

The chromatic number $\chi$ of a graph is bounded from below by its clique number $\omega,$ but it can be arbitrary large. Perfect graphs are defined by $\chi=\omega$ for all induced subgraphs. An interesting relaxation are $\chi$-bounded…

For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model…

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\"obner basis can be computed by…

Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is $2^gT(3,1/2)$. We provide a short deletion-contraction…

A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its…

We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

The line graph of a graph $G$ is the graph $L(G)$ whose vertex set is the edge set of $G$ and there is an edge between $e,f\in E(G)$ if $e$ and $f$ share an endpoint in $G$. A graph is called line graph if it is a line graph of some graph.…

The symmetric edge polytope of a simple graph is a lattice polytope defined as the convex hull of a subset of the type A roots corresponding to the edges of the graph. In this article we prove a sharp lower bound for the number of edges of…

Tutte's dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In the…

Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for…