270 research outputs found
On the optimality of the Arf invariant formula for graph polynomials
We prove optimality of the Arf invariant formula for the generating function
of even subgraphs, or, equivalently, the Ising partition function, of a graph.Comment: Advances in Mathematics, 201
Permanents, Pfaffian orientations, and even directed circuits
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in
such a way that the permanent of A equals the determinant of the modified
matrix? When does a real square matrix have the property that every real matrix
with the same sign pattern (that is, the corresponding entries either have the
same sign or are both zero) is nonsingular? When is a hypergraph with n
vertices and n hyperedges minimally nonbipartite? When does a bipartite graph
have a "Pfaffian orientation"? Given a digraph, does it have no directed
circuit of even length? Given a digraph, does it have a subdivision with no
even directed circuit?
It is known that all of the above problems are equivalent. We prove a
structural characterization of the feasible instances, which implies a
polynomial-time algorithm to solve all of the above problems. The structural
characterization says, roughly speaking, that a bipartite graph has a Pfaffian
orientation if and only if it can be obtained by piecing together (in a
specified way) planar bipartite graphs and one sporadic nonplanar bipartite
graph.Comment: 47 pages, published versio
Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions
The aim of the paper is to formulate a discrete analogue of the claim made by
Alvarez-Gaume et al., realizing the partition function of the free fermion on a
closed Riemann surface of genus g as a linear combination of 2^{2g} Pfaffians
of Dirac operators. Let G=(V,E) be a finite graph embedded in a closed Riemann
surface X of genus g, x_e the collection of independent variables associated
with each edge e of G (collected in one vector variable x) and S the set of all
2^{2g} Spin-structures on X. We introduce 2^{2g} rotations rot_s and (2|E|
times 2|E|) matrices D(s)(x), s in S, of the transitions between the oriented
edges of G determined by rotations rot_s. We show that the generating function
for the even subsets of edges of G, i.e., the Ising partition function, is a
linear combination of the square roots of 2^{2g} Ihara-Selberg functions
I(D(s)(x)) also called Feynman functions. By a result of Foata--Zeilberger
holds I(D(s)(x))= det(I-D'(s)(x)), where D'(s)(x) is obtained from D(s)(x) by
replacing some entries by 0. Thus each Feynman function is computable in
polynomial time. We suggest that in the case of critical embedding of a
bipartite graph G, the Feynman functions provide suitable discrete analogues
for the Pfaffians of discrete Dirac operators
A Pfaffian formula for monomer-dimer partition functions
We consider the monomer-dimer partition function on arbitrary finite planar
graphs and arbitrary monomer and dimer weights, with the restriction that the
only non-zero monomer weights are those on the boundary. We prove a Pfaffian
formula for the corresponding partition function. As a consequence of this
result, multipoint boundary monomer correlation functions at close packing are
shown to satisfy fermionic statistics. Our proof is based on the celebrated
Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the
authors, and a careful labeling and directing procedure of the vertices and
edges of the graph.Comment: Added referenc
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