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40
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.
THE NONBACKTRACKING SPECTRUM OF THE UNIVERSAL COVER OF A Graph
, 2007
"... A nonbacktracking walk on a graph, H, is a directed path of directed edges of H such that no edge is the inverse of its preceding edge. Nonbacktracking walks of a given length can be counted using the nonbacktracking adjacency matrix, B, indexed by H’s directed edges and related to Ihara’s Zeta ..."
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Cited by 11 (1 self)
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A nonbacktracking walk on a graph, H, is a directed path of directed edges of H such that no edge is the inverse of its preceding edge. Nonbacktracking walks of a given length can be counted using the nonbacktracking adjacency matrix, B, indexed by H’s directed edges and related to Ihara’s Zeta function. We show how to determine B’s spectrum in the case where H is a tree covering a finite graph. We show that when H is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of B’s spectrum, the corresponding Green function has “periodic decay ratios. ” The existence of such a “ratio system ” can be effectively checked, and is equivalent to being outside the spectrum. We also prove that the spectral radius of the nonbacktracking walk operator on the tree covering a finite graph is exactly √ gr, where gr is the growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras [ST]. Finally, we give experimental evidence that for a fixed, finite graph, H, a random lift of large degree has nonbacktracking new spectrum near that of H’s universal cover. This suggests a new generalization of Alon’s second eigenvalue conjecture.
Uniqueness of Belief Propagation on Signed Graphs
"... While loopy Belief Propagation (LBP) has been utilized in a wide variety of applications with empirical success, it comes with few theoretical guarantees. Especially, if the interactions of random variables in a graphical model are strong, the behaviors of the algorithm can be difficult to analyze d ..."
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Cited by 6 (0 self)
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While loopy Belief Propagation (LBP) has been utilized in a wide variety of applications with empirical success, it comes with few theoretical guarantees. Especially, if the interactions of random variables in a graphical model are strong, the behaviors of the algorithm can be difficult to analyze due to underlying phase transitions. In this paper, we develop a novel approach to the uniqueness problem of the LBP fixed point; our new “necessary and sufficient ” condition is stated in terms of graphs and signs, where the sign denotes the types (attractive/repulsive) of the interaction (i.e., compatibility function) on the edge. In all previous works, uniqueness is guaranteed only in the situations where the strength of the interactions are “sufficiently ” small in certain senses. In contrast, our condition covers arbitrary strong interactions on the specified class of signed graphs. The result of this paper is based on the recent theoretical advance in the LBP algorithm; the connection with the graph zeta function. 1
A homotopical algebra of graphs related to zeta series
 Homology, Homotopy and its Applications 10 (2008), 1–13. MR2506131 (2010f:18010). [C95] Crans, Sjoerd E. Quillen
"... Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equiv ..."
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Cited by 5 (4 self)
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Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two FreydKelly factorization systems based on Folding, Injecting, and Covering graph morphisms. 0. Introduction. In this paper we develop a notion of homotopy within graphs, and demonstrate its relevance to the study of zeta series and spectrum of a finite graph. We will work throughout with a particular category of graphs, described in Section 1 below. Our graphs will be directed and possibly infinite, with loops and multiple arcs allowed.
DEGENERACY IN THE LENGTH SPECTRUM FOR METRIC GRAPHS
"... Abstract. In this note we show that the length spectrum for metric graphs exhibit a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesic (or closed cycles) with the same metric length. Let G = (V, E) be a finite graph with vertices V and ..."
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Abstract. In this note we show that the length spectrum for metric graphs exhibit a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesic (or closed cycles) with the same metric length. Let G = (V, E) be a finite graph with vertices V and edges E. We write E o for the set of oriented edges; for e ∈ E o, ē ∈ E o denotes the same unoriented edge with orientation reversed. (In the physics literature, the vertices are referred to as nodes and the edges as bonds.) The degree of a vertex is the number of outgoing
Bartholdi zeta functions for periodic simple graphs
, 2008
"... The definition of the Bartholdi zeta function is extended to the case of infinite periodic graphs. By means of the analytic determinant for semifinite von Neumann algebras studied by the authors in [7], a determinant formula and functional equations are obtained for this zeta function. ..."
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Cited by 2 (1 self)
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The definition of the Bartholdi zeta function is extended to the case of infinite periodic graphs. By means of the analytic determinant for semifinite von Neumann algebras studied by the authors in [7], a determinant formula and functional equations are obtained for this zeta function.