619 research outputs found
The Incidence Hopf Algebra of Graphs
This is the published version, also available here: http://dx.doi.org/10.1137/110820075.The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs, and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial
Polynomials associated with graph coloring and orientations
We study colorings and orientations of graphs in two related contexts. Firstly, we generalize Stanley's chromatic symmetric function using the k-balanced colorings of Pretzel to create a new graph invariant. We show that in fact this invariant is a quasisymmetric function which has a positive expansion in the fundamental basis. We also define a graph invariant generalizing the chromatic polynomial for which we prove some theorems analogous to well-known theorems about the chromatic polynomial. Secondly, we examine graphs and graph colorings in the context of the combinatorial Hopf algebras of Aguiar, Bergeron and Sottile. By doing so, we are able to obtain a new formula for the antipode of a Hopf algebra on graphs previously studied by Schmitt. We also obtain new interpretations of evaluations of the Tutte polynomial
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
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 works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
Tutte polynomial of a small-world farey graph
In this paper, we find recursive formulas for the Tutte polynomial of a
family of small-world networks: Farey graphs, which are modular and have an
exponential degree hierarchy. Then, making use of these formulas, we determine
the number of spanning trees, as well as the number of connected spanning
subgraphs. Furthermore, we also derive exact expressions for the chromatic
polynomial and the reliability polynomial of these graphs.Comment: 6 page
Evaluations of topological Tutte polynomials
We find new properties of the topological transition polynomial of embedded
graphs, . We use these properties to explain the striking similarities
between certain evaluations of Bollob\'as and Riordan's ribbon graph
polynomial, , and the topological Penrose polynomial, . The general
framework provided by also leads to several other combinatorial
interpretations these polynomials. In particular, we express , ,
and the Tutte polynomial, , as sums of chromatic polynomials of graphs
derived from ; show that these polynomials count -valuations of medial
graphs; show that counts edge 3-colourings; and reformulate the Four
Colour Theorem in terms of . We conclude with a reduction formula for the
transition polynomial of the tensor product of two embedded graphs, showing
that it leads to additional relations among these polynomials and to further
combinatorial interpretations of and .Comment: V2: major revision, several new results, and improved expositio
The Interlace Polynomial
In this paper, we survey results regarding the interlace polynomial of a
graph, connections to such graph polynomials as the Martin and Tutte
polynomials, and generalizations to the realms of isotropic systems and
delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials,
edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL
Tutte's dichromate for signed graphs
We introduce the ``trivariate Tutte polynomial" of a signed graph as an
invariant of signed graphs up to vertex switching that contains among its
evaluations the number of proper colorings and the number of nowhere-zero
flows. In this, it parallels the Tutte polynomial of a graph, which contains
the chromatic polynomial and flow polynomial as specializations. The number of
nowhere-zero tensions (for signed graphs they are not simply related to proper
colorings as they are for graphs) is given in terms of evaluations of the
trivariate Tutte polynomial at two distinct points. Interestingly, the
bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share
many similar properties with the Tutte polynomial of a graph, does not in
general yield the number of nowhere-zero flows of a signed graph. Therefore the
``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from
the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an
invariant of ordered pairs of matroids on a common ground set -- for a signed
graph, the cycle matroid of its underlying graph and its frame matroid form the
relevant pair of matroids. This invariant is the canonically defined Tutte
polynomial of matroid pairs on a common ground set in the sense of a recent
paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and
Kayibi as a four-variable linking polynomial of a matroid pair on a common
ground set.Comment: 53 pp. 9 figure
On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid
The first author introduced the circuit-cocircuit reversal system of an
oriented matroid, and showed that when the underlying matroid is regular, the
cardinalities of such system and its variations are equal to special
evaluations of the Tutte polynomial (e.g., the total number of
circuit-cocircuit reversal classes equals , the number of bases of
the matroid). By relating these classes to activity classes studied by the
first author and Las Vergnas, we give an alternative proof of the above results
and a proof of the converse statements that these equalities fail whenever the
underlying matroid is not regular. Hence we extend the above results to an
equivalence of matroidal properties, thereby giving a new characterization of
regular matroids.Comment: 7 pages. v2: simplified proof, with new statements concerning other
special evaluations of the Tutte polynomia
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