922 research outputs found
The equivalence of two graph polynomials and a symmetric function
The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them which also captures Tutte's universal V-function as a specialization. We show that the equivalence remains true for the strong functions thus
answering a question raised by Dominic Welsh
Evaluating a weighted graph polynomial for graphs of bounded tree-width
We show that for any there is a polynomial time algorithm to evaluate the weighted graph polynomial of any graph with tree-width at most at any point. For a graph with vertices, the algorithm requires arithmetical operations, where depends only on
Results on Chromatic Polynomials Inspired by a Correlation Inequality of G.E. Farr
In1993 Graham Farr gave a proof of a correlation inequality involving colourings of graphs. His work eventually led to a conjecture that number of colourings of a graph with certain properties gave a log-concave sequence. We restate Farr's work in terms of the bivariate chromatic polynomial of Dohmen, Poenitz, Tittman and give a simple, self-contained proof of Farr's inequality using a basic combinatorial approach. We attempt to prove Farr's conjecture through methods in stable polynomials and computational verification, ultimately leading to a stronger conjecture
Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
The Equivalence of Two Graph Polynomials and a Symmetric Function
The U-polynomial, the polychromate and the symmetric function generalization
of the Tutte polynomial due to Stanley are known to be equivalent in the sense
that the coefficients of any one of them can be obtained as a function of the
coefficients of any other. The definition of each of these functions suggests a
natural way in which to generalize them which also captures Tutte's universal
V-functions as a specialization. We show that the equivalence remains true for
the extended functions thus answering a question raised by Dominic Welsh.Comment: 17 page
Abstract cluster expansion with applications to statistical mechanical systems
We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions
A weighted graph polynomial from chromatic invariants of knots
Motivated by the work of Chmutov, Duzhin and Lando on Vassiliev invariants, we define a polynomial on weighted graphs which contains as specialisations the weighted chromatic invariants but also contains many other classical invariants including the Tutte and matching polynomials. It also gives the symmetric function generalisation of the chromatic polynomial introduced by Stanley. We study its complexity and prove hardness results for very restricted classes of graphs
The Go polynomials of a graph
Abstract This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This deĂżnition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less "local" in nature than those used to deĂżne more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modiĂżcations similar in spirit to the deletion-contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also brie y record some correlation inequalities
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