Results 1  10
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15
RiemannRoch and AbelJacobi theory on a finite graph
 Adv. Math
"... Abstract. It is wellknown that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graphtheoretic analogue of the classic ..."
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Cited by 130 (12 self)
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Abstract. It is wellknown that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graphtheoretic analogue of the classical RiemannRoch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the AbelJacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or nonexistence of a winning strategy for a certain chipfiring game played on the vertices of a graph. 1.
A trace on fractal graphs AND THE IHARA ZETA FUNCTION
, 2008
"... Starting with Ihara’s work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and MokhtariSharghi have studied zeta functions for infinite graphs acted u ..."
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Cited by 13 (5 self)
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Starting with Ihara’s work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and MokhtariSharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a selfsimilarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs. The Ihara zeta function, originally associated to certain groups and then combinatorially
IHARA’S ZETA FUNCTION FOR PERIODIC GRAPHS AND ITS APPROXIMATION IN THE AMENABLE CASE
, 2008
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Ihara zeta functions for periodic SIMPLE GRAPHS
, 2008
"... The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs. ..."
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Cited by 11 (4 self)
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The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.
Belief propagation and loop series on planar graphs
, 2008
"... Abstract. We discuss a generic model of Bayesian inference with binary variables defined on edges of a planar graph. The Loop Calculus approach of [1, 2] is used to evaluate the resulting series expansion for the partition function. We show that, for planar graphs, truncating the series at singleco ..."
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Cited by 8 (2 self)
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Abstract. We discuss a generic model of Bayesian inference with binary variables defined on edges of a planar graph. The Loop Calculus approach of [1, 2] is used to evaluate the resulting series expansion for the partition function. We show that, for planar graphs, truncating the series at singleconnected loops reduces, via a map reminiscent of the Fisher transformation [3], to evaluating the partition function of the dimer matching model on an auxiliary planar graph. Thus, the truncated series can be easily resummed, using the Pfaffian formula of Kasteleyn [4]. This allows to identify a big class of computationally tractable planar models reducible to a dimer model via the Belief Propagation (gauge) transformation. The Pfaffian representation can also be extended to the full Loop Series, in which case the expansion becomes a sum of Pfaffian contributions, each associated with dimer matchings on an extension to a subgraph of the original graph. Algorithmic consequences of the Pfaffian representation, as well as relations to quantum and nonplanar models, are discussed.
Quantum graphs where backscattering is prohibited
 J. Phys. A
"... We describe a new class of scattering matrices for quantum graphs in which backscattering is prohibited. We discuss some properties of quantum graphs with these scattering matrices and explain the advantages and interest in their study. We also provide two methods to build the vertex scattering matr ..."
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Cited by 7 (2 self)
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We describe a new class of scattering matrices for quantum graphs in which backscattering is prohibited. We discuss some properties of quantum graphs with these scattering matrices and explain the advantages and interest in their study. We also provide two methods to build the vertex scattering matrices needed for their construction. 1
Zeta Functions of weighted graphs and covering graphs
, 2007
"... We find a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cove ..."
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Cited by 5 (0 self)
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We find a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cover satisfies an approximate Riemann hypothesis while that of the abelian cover does not.
ZETA FUNCTIONS OF GRAPHS WITH Z ACTIONS
, 2006
"... Abstract. Suppose Y is a regular covering of a graph X with covering transformation group π = Z. This paper gives an explicit formula for the L 2 zeta function of Y and computes examples. When π = Z, the L 2 zeta function is an algebraic function. As a consequence it extends to a meromorphic functio ..."
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Cited by 4 (1 self)
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Abstract. Suppose Y is a regular covering of a graph X with covering transformation group π = Z. This paper gives an explicit formula for the L 2 zeta function of Y and computes examples. When π = Z, the L 2 zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta functions of regular graphs, such as the location of singularities and the functional equation. 1.