Harmonic analysis on metrized graphs

Computation of Polarized Metrized Graph Invariants By Using Discrete Laplacian Matrix

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we give fast algorithms to compute these invariants by expressing them in terms of the discrete Laplacian matrix and its pseudo inverse. Algorithms we give can be used for both symbolic and numerical computations. We present various examples to illustrate the implementation of these algorithms.

可积阿拉克-格林函数在度量图上的计算

Computation of Admissible Arakelov-Green Functions on Metrized Graphs

本文提出一种算法，用于计算度量图上的可积阿拉克-格林函数，扩展了Cinkir的公式，并通过实例展示了算法的实现。

Linear series on metrized complexes of algebraic curves

A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points being in bijection with incident edges. We establish a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical Riemann-Roch theorem and its graph-theoretic and tropical analogues due to Baker-Norine, Gathmann-Kerber, and Mikhalkin-Zharkov. We also establish generalizations of the second author's specialization lemma and its weighted graph analogue due to Caporaso and the first author…

A note on algebraic rank, matroids, and metrized complexes

We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory of metrized complexes to show that equality between the algebraic and combinatorial rank is not a sufficient condition for smoothability of divisors, thus giving a negative answer to a question posed by Caporaso, Melo, and the author.

Admissible Invariants of genus 3 Curves

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus $3$. We find the sharp lower bound for the invariants $φ$, $λ$ and $ε$ for all polarized metrized graphs of genus $3$. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.

Tropical Geometric Compactification of Moduli, I - $M_g$ case -

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are often studied as "tropical curves", are obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic metrics of the Riemann surfaces. This phenomenon can be also seen as an archemidean analogue of the tropicalization of Berkovich analytification of $M_{g}$ (cf., [ACP]…

tau不变量的均匀正性

Uniform positivity of the tau invariant

本文证明度量图tau不变量的正下界与总长度成正比，并构造反例否证通用常数1/108，该下界与亏格、边数及边长分布无关，并应用于雅可比簇高度公式。

Harmonic Index of Total Graphs of Some Graphs

Let G = G(V,E) be a graph with | V | vertices and | E | edges and total graph, T(G) is obtained from G. In This paper we have study the Harmonic index of total graph for standards graphs, bipartite graph of particular type, regular graph, Grid graph Gm,n and Complete Binary tree.

Weak harmonic labeling of graphs and multigraphs

In this article we introduce the notion of weak harmonic labeling of a graph, a generalization of the concept of harmonic labeling defined recently by Benjamini et al. that allows extension to finite graphs and graphs with leaves. We present various families of examples and provide several constructions that extend a given weak harmonic labeling to larger graphs. In particular, we use finite weak models to produce new examples of (strong) harmonic labelings. As a main result, we provide a characterization of weakly labeled graphs in terms of harmonic subsets of the integers and use it to compu…

关于图的调和中心

On harmonic centers of graphs

本文研究图的调和中心及其在图块中的定位，给出使调和中心位于单个块内的充分条件，并证明每个连通图都可作为某些图的调和中心或外围。

Note on the harmonic index of a graph

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{deg(v) + deg(u)}$ of all edges $uv$ of $E (G)$, where $deg (v)$ denotes the degree of a vertex $v$ in $V (G)$. In this note we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561--566] and establish some upper and lower bounds on the harmonic index of $G$.

The minimum harmonic index for bicyclic graphs with given diameter

The harmonic index of a graph $G$, is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ is the degree of the vertex $u$ in $G$. In this paper we find the minimum harmonic index of bicyclic graph of order $n$ and diameter $d$. We also characterized all bicyclic graphs reaching the minimum bound.

特定图的调和特征多项式研究

On the Harmonic characteristic polynomial of specific graphs

本文针对简单图的调和矩阵，研究其谱性质，推导了几类特定图的调和特征多项式与调和能量的显式表达式。

Nowhere-Harmonic Colorings of Graphs

Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.

Lamplighter graphs do not admit harmonic functions of finite energy

We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy.

Mathematical results on harmonic polynomials

Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several properties of this polynomial, and we prove that several properties of graphs can be deduced from their harmonic polynomials. Also, we show that two graphs with the same harmonic polynomial have to be similar.

Harmonic Labeling of Graphs

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite graph $G$ containing at least one edge, there is no harmonic labeling of $ G \times \Z$.

顶点传递图上的 Lipschitz 调和函数

Lipschitz harmonic functions on vertex-transitive graphs

本文证明了每个局部有限的顶点传递图都存在非常值的 Lipschitz 调和函数。

Harmonic Centrality and Centralization of Some Graph Products

Harmonic centrality calculates the importance of a node in a network by adding the inverse of the geodesic distances of this node to all the other nodes. Harmonic centralization, on the other hand, is the graph-level centrality score based on the node-level harmonic centrality. In this paper, we present some results on both the harmonic centrality and harmonic centralization of graphs resulting from some graph products such as Cartesian and direct products of the path $P_2$ with any of the path $P_m$, cycle $C_m$, and fan $F_m$ graphs.

图的加权Tutte-Grothendieck多项式

Weighted Tutte-Grothendieck polynomials of graphs

本文引入图的加权（调和）色多项式与Tutte-Grothendieck多项式概念，推广了配方定理，构造了相关不变量，并探讨了范畴化问题。

Harmonic functions on multiplicative graphs and interpolation polynomials

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by-product, we compute certain Selberg-type integrals.

Harmonic Galois theory for finite graphs

This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched $G$-covers of a fixed base $X$ in terms of homomorphisms from a suitable fundamental group of $X$ together with $G$-inertia structures on $X$. As applications, we show that finite embedding problems for graphs have proper solutions and prove a Grunwald-Wang type result stating that an arbitrary collection of fibers may be realized by a global cover.

Discrete harmonic functions on infinite penny graphs

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincaré inequality hold, which yields the Harnack inequality for positive harmonic functions. Moreover, we prove that the space of polynomial growth harmonic functions, or ancient solutions of the heat equation, with bounded growth rate has finite dimensional property.

Semifinite harmonic functions on the direct product of graded graphs

Indecomposible semifinite harmonic functions on the direct product of graded graphs are classified. As a particular case, the full list of indecomposible traces for the infinite inverse symmetric semigroup is obtained.

Harmonic Centrality in Some Graph Families

One of the more recent measures of centrality in social network analysis is the normalized harmonic centrality. A variant of the closeness centrality, harmonic centrality sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another. It is then normalized by dividing it by m-1, where m is the number of nodes of the graph. In this paper, we present notions regarding the harmonic centrality of some important classes of graphs.

Semifinite harmonic functions on the zigzag graph

We study semifinite harmonic functions on the zigzag graph, which corresponds to Pieri's rule for the fundamental quasisymmetric functions $\{F_λ\}$. The main problem, which we solve here, is to classify the indecomposable semifinite harmonic functions on this graph. We describe the set of classification parameters and an explicit construction that produces a semifinite indecomposable harmonic function out of every point of this set. We also establish a semifinite analog of the Vershik-Kerov ring theorem.

Harmonic functions with finite $p$-energy on lamplighter graphs are constant

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in $\ell^p$ (\ie finite $p$-energy) for any $p\in [1,\infty[$ except constants (and, equivalently, that their reduced $\ell^p$ cohomology is trivial in degree one). Using similar arguments, it is also shown that many direct products of graphs (including all direct products of Cayley graphs) do not admit non-constant harmonic function with gradient in $\ell^p$. The proof relies on a theorem of Thoma…

Harmonic morphisms and hyperelliptic graphs

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connect…

Harmonic evolutions on graphs

We define the harmonic evolution of states of a graph by iterative application of the harmonic operator (Laplacian over $Z_2$). This provides graphs with a new geometric context and leads to a new tool to analyze them. The digraphs of evolutions are analyzed and classified. This construction can also be viewed as a certain topological generalization of cellular automata.

Semifinite harmonic functions on branching graphs

We study semifinite harmonic functions on arbitrary branching graphs. We give a detailed exposition of an algebraic method which allows one to classify semifinite indecomposable harmonic functions on some multiplicative branching graphs. This method was proposed by A. Wassermann in terms of operator algebras, while we rephrase, clarify, and simplify the main arguments, working only with combinatorial objects. This work was inspired by the theory of traceable factor representations of the infinite symmetric group $S(\infty)$.

Harmonic functions of polynomial growth on infinite penny graphs

For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above spaces.

Combinatorial Iterated Integrals and the Harmonic Volume of Graphs

Let $Γ$ be a connected bridgeless metric graph, and fix a point $v$ of $Γ$. We define combinatorial iterated integrals on $Γ$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of $Γ$ at $v$ and the tensor algebra on the first homology of $Γ$, $\int\colon \mathbf{Z}π_1(Γ,v) \times T\mathrm{H}_1(Γ,\mathbf{R}) \to \mathbf{R}$. We show that this pairing on the two-step unipotent quotient of the group a…

Genus Bounds for Harmonic Group Actions on Finite Graphs

This paper develops graph analogues of the genus bounds for the maximal size of an automorphism group of a compact Riemann surface of genus $g\ge 2$. Inspired by the work of M. Baker and S. Norine on harmonic morphisms between finite graphs, we motivate and define the notion of a harmonic group action. Denoting by M(g) the maximal size of such a harmonic group action on a graph of genus $g\ge 2$, we prove that $4(g-1)\le M(g)\le 6(g-1)$, and these bounds are sharp in the sense that both are attained for infinitely many values of g. Moreover, we show that the values $4(g-1)$ and $6(g-1)$ are th…

Semifinite harmonic functions on the Gnedin-Kingman graph

We study the Gnedin-Kingman graph, which corresponds to Pieri's rule for the monomial basis $\{M_λ\}$ in the algebra $\mathrm{QSym}$ of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin-Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik-Kerov ring theorem.

Layering $\partial$-Graphs and Networks

We consider the inverse problem for countable, locally finite electrical networks with edge weights in an arbitrary field. The electrical inverse problem seeks to determine the weights of the edges knowing only the potential and current data of harmonic functions on a set of boundary nodes. Motivated by the results of Curtis-Ingerman-Morrow and de-Verdiere-Gitler-Vertigan and others, we formalize the idea of using layer-stripping and harmonic continuation to solve the inverse problem. Our strategy is to iteratively recover "vulnerable" edges near the boundary, then remove them by deletion or c…

Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition

Brighton [Bri13] proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. $CD(0,\infty).$ In this paper, we provide a new proof of this result by using the reverse Poincaré inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the $CD(0,\infty)$ condition.

伪紧型度量图上的因子与调和态射

Divisors and harmonic morphisms on metric graphs of pseudocompact type

本文证明伪紧型度量图中因子d-多边形性等价于存在到树的度为d的调和态射，并给出构造性证明及正秩因子提升结果，最后研究路径情形的Brill-Noether理论。

2-同构图的调和向量之间的幺模双射

A unimodular bijection between harmonic vectors of 2-isomorphic graphs

本文证明2-同构连通图的Laplacian矩阵通过幺模矩阵U相合，且U建立了图上特定调和向量空间之间的双射，将顶点对的调和向量对应到另一图的对应顶点对。

Harmonic analysis invariants for infinite graphs via operators and algorithms

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, Fourier dua…

Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths

We introduce and characterize central probability distributions on Littelmann paths. Next we establish a law of large numbers and a central limit theorem for the generalized Pitmann transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we show there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and explain how it can be computed.

Maximal harmonic group actions on finite graphs

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group $Γ=\left<x,y \ | \ x^2=y^3=1\right>$ of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the $(2,3,7)$-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establi…

通过调和函数研究两端图

A study of 2-ended graphs via harmonic functions

本文利用拟等距于实数轴的常返图上本质唯一的 Lipschitz 调和函数，证明两端顶点传递图的连接常数下界，并构造非 Cayley 的三次两端顶点传递图。

Characterisations of algebraic properties of groups in terms of harmonic functions

We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite …