Graph Polynomials and Their Applications II: Interrelations and Interpretations

  title={Graph Polynomials and Their Applications II: Interrelations and Interpretations},
  author={Joanna A. Ellis-Monaghan and Criel Merino},
  booktitle={Structural Analysis of Complex Networks},
This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore how an… 

Graph polynomials and their representations

This work presents the edge elimination polynomial and introduces several graph polynomials equivalent to it, and connects a recursive deVnition to the counts of colorings and to the counting of (spanning) subgraphs.

Interactions with Graph Polynomials

Chapter 4 explores twisted duality as a tool for extracting both combinatorial and topological information from topological graph polynomials. We begin with the topological transition polynomial of

Computing Tutte Polynomials

The implementation of a program that exploits isomorphisms in the computation tree to extend the range of graphs for which it is feasible to compute their Tutte polynomials is described and the utility of the program is demonstrated by finding counterexamples to a conjecture of Welsh on the location of the real flow roots of a graph.

Edge-Selection Heuristics for Computing Tutte Polynomials

This paper presents and discusses two edge-selection heuristics which (respectively) give good performance on sparse and dense graphs and develops the most efficient algorithm to-date for computing the Tutte polynomial of a graph.

Twisted duality and polynomials of embedded graphs

We consider two operations on the edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two

On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

  • J. White
  • Mathematics
    Electron. J. Comb.
  • 2011
It is proved that specializations of these new polynomials recover polynmials which enumerate hyperedge coverings, matchings, transversals, and section hypergraphs, all weighted according to certain statistics.

Graph Polynomials: From Recursive Definitions to Subset Expansion Formulas

This article presents a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials, and proves that every recursive definition of a graphs polynomial can be converted into a subset expansion formula.

The zero forcing polynomial of a graph

Exact and approximate counting of graph objects: independent sets, eulerian tours, and more

Counting problems are studied in a variety of areas. For example, enumerative combinatorics, statistics, statistical physics, and artificial intelligence. In this dissertation, we investigate



The interlace polynomial of a graph

Identities for circuit partition polynomials, with applications to the Tutte polynomial

Some Applications of the Proper and Adjacency Polynomials in the Theory of Graph Spectra

An upper bound for the cardinality of k (u) is derived, showing that j k (U)j decreases at least as O( 2 ), and the cases in which the bound is attained are characterized.

Graph polynomials derived from Tutte-Martin polynomials

A Multivariate Interlace Polynomial and its Computation for Graphs of Bounded Clique-Width

We define a multivariate polynomial that generalizes in a unified way the two-variable interlace polynomial defined by Arratia, Bollobas and Sorkin on the one hand, and a one-variable variant of it

A Tutte Polynomial for Coloured Graphs

We define a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sufficient conditions on

A Two-Variable Interlace Polynomial

A new graph polynomial in two variables, “interlace”, which can be computed in two very different ways and considers a few properties and specializations of the two-variable interlacePolynomial.

On the interlace polynomials

  • L. Traldi
  • Mathematics
    J. Comb. Theory, Ser. B
  • 2013

Twisted duality and polynomials of embedded graphs

We consider two operations on the edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two