Alik Abakarov, Yuri Sushkov
This paper is devoted to one theory of hypergraph connectivity and presents the proof of the polynomial algorithm for finding an optimal spanning hyperforest(hypertree) for any given weighed q-uniform hypergraph.
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By using our novel Grassmann formulation we study the phase transition of the spanning-hyperforest model of the k-uniform complete hypergraph for any k>= 2. The case k=2 reduces to the spanning-forest model on the complete graph. Different k are studied at once by using a microcanonical ensemble in which the number of hyperforests is fixed. The low-temperature phase is characterized by the appearance of a giant hyperforest. The phase transition occurs when the number of hyperforests is a fraction (k-1)/k of the total number of vertices. The behaviour at criticality is also studied by means of …
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.
The classical Dodgson identity can be interpreted as a quadratic identity of spanning forest polynomials, where the spanning forests used in each polynomial are defined by how three marked vertices are divided among the component trees. We prove an analogous result with four marked vertices.
It is proved that if $G$ is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of $G$ has no roots in the interval $(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of graphs containing such spanning trees with chromatic roots converging to $t_1$ from above. We employ the Whitney $2$-switch operation to manage the analysis of an infinite class of chromatic polynomials.
On stability of spanning tree degree enumerators
研究了连通图生成树度数枚举多项式的实稳定性，证明其充要条件为该图是距离遗传图。
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger multivariate version of this theorem. We provide a general method by which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph $G$. These polynomials are multivariate generating functions for spanning subgraphs of $G$ with certain weights and constraints imposed, and the theorems specify regions in which these po…
We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing sequence determines an algorithm which can be applied to a single function, or to the set $\mathcal{P}_{G,q}$ of $G$-parking functions. When the latter is chosen, the algorithm uses splitting operations - inspired by the recursive defintion of the Tutte polynomial - to iteratively break $\mathcal{P}_{G,q}$ into disjoint subsets. This results in bijective map…
$h^*$-vectors of graph polytopes using activities of dissecting spanning trees
该研究针对图对称边多面体与半平衡有向图根多面体的生成树剖分，证明其$h^*$-多项式可表为对应生成树活动性的生成函数，提供了计算新方法并揭示了剖分统计量分布的无关性。
Column expansion identities and quadratic spanning forest identities
本文利用行列式列展开恒等式确定二次生成森林恒等式空间的维数，证明2012年的相关猜想，并通过边交换论证给出组合解释，服务于四维量子场论计算。
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.
We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.
On stability of weighted spanning tree degree enumerators
本文将生成树度枚举器的实稳定性结果从连通图推广到加权图，并据此定义了加权距离遗传图类。
This note derives the characteristic polynomial of a graph that represents nonjump moves in a generalized game of checkers. The number of spanning trees is also determined.
Bounded Treewidth and Complete Monotonicity for Scott-Sokal Spanning-Tree Polynomials
本文证明若图G的树宽≤k，则生成树多项式逆幂T_G^{-β}在β>(k-1)/2时完全单调，覆盖了部分3-树等图类。
We concentrate on some recent results of Egawa and Ozeki [J. Graph Theory, 2015 and Combinatorica, 2014], and He et al. [J. Graph Theory, 2002]. We give shorter proofs and polynomial time algorithms as well. We present two new proofs for the sufficient condition for having a spanning tree with prescribed lower bounds on the degrees, achieved recently by Egawa and Ozeki. The first one is a natural proof using induction, and the second one is a simple reduction to the theorem of Lovász. Using an algorithm of Frank we show that the condition of the theorem can be checked in time $O(m\sqrt{n})$, a…
Let $D_2$ denote the $3$-uniform hypergraph with $4$ vertices and $2$ edges. Answering a question of Alon and Shapira, we prove an induced removal lemma for $D_2$ having polynomial bounds. We also prove an Erdős-Hajnal-type result: every induced $D_2$-free hypergraph on $n$ vertices contains a clique or an independent set of size $n^{c}$ for some absolute constant $c > 0$. In the case of both problems, $D_2$ is the only nontrivial $k$-uniform hypergraph with $k\geq 3$ which admits a polynomial bound.
A framework for spanning forest identities
本文建立生成森林恒等式的框架，研究其通用条件与空间维数，识别全 minors 矩阵树定理给出的映射，证明其秩由Narayana数计数，助力相关领域发展。
For any graph G with n edges, the spanning subgraphs and the orientations of G are both counted by the evaluation T_G(2,2)=2^n of its Tutte polynomial. We define a bijection $Φ$ between spanning subgraphs and orientations and explore its enumerative consequences regarding the Tutte polynomial. The bijection $Φ$ is closely related to a recent characterization of the Tutte polynomial relying on a combinatorial embedding of the graph G, that is, on a choice of cyclic order of the edges around each vertex. Among other results, we obtain a combinatorial interpretation for each of the evaluations T_…
Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F form an increasing sequence. Hallam and Sagan showed that the generating function ISF(G,t) for increasing spanning forests of G has all nonpositive integral roots. Furthermore they proved that, up to a change of sign, this polynomial equals the chromatic polynomial of G precisely when 1,...,n is a perfect elimination order for G. We give new, purely combinatorial proofs of these results which permit us to generalize them in sever…
Consider the problem of determining whether there exists a spanning hypertree in a given k-uniform hypergraph. This problem is trivially in P for k=2, and is NP-complete for k>= 4, whereas for k=3, there exists a polynomial-time algorithm based on Lovasz' theory of polymatroid matching. Here we give a completely different, randomized polynomial-time algorithm in the case k=3. The main ingredients are a Pfaffian formula by Vaintrob and one of the authors (G.M.) for a polynomial that enumerates spanning hypertrees with some signs, and a lemma on the number of roots of polynomials over a finite f…
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property …
In this paper, we investigate the polynomial integrand of an integral formula that yields the expected length of the minimal spanning tree of a graph whose edges are uniformly distributed over the interval [0, 1]. In particular, we derive a general formula for the coefficients of the polynomial and apply it to express the first few coefficients in terms of the structure of the underlying graph; e.g. number of vertices, edges and cycles.
Let $T_G(x,y)$ be the Tutte polynomial of a graph $G$. In this paper we show that if $(G_n)_n$ is a sequence of $d$-regular graphs with girth $g(G_n)\to \infty$, then for $x\geq 1$ and $0\leq y\leq 1$ we have $$\lim_{n\to \infty}T_{G_n}(x,y)^{1/v(G_n)}=t_d(x,y),$$ where $$t_d(x,y)=\left\{\begin{array}{lc} (d-1)\left(\frac{(d-1)^2}{(d-1)^2-x}\right)^{d/2-1}&\ \ \mbox{if}\ x\leq d-1,\\ x\left(1+\frac{1}{x-1}\right)^{d/2-1} &\ \ \mbox{if}\ x> d-1. \end{array}\right.$$ independently of $y$ if $0\leq y\leq 1$. If $(G_n)_n$ is a sequence of random $d$-regular graphs, then the same statement holds tr…
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynom…
Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi …
In the present paper we find a simple algorithm for counting Jacobian group of the generalized Petersen graph GP(n,k). Also, we obtain a closed formula for the number of spanning trees of this graph in terms of Chebyshev polynomials.
With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The problem has been proved to be NP-hard and fixed-parameter polynomial algorithms have been obtained for some special classes of graphs. In this paper, we concentrate on the optimality characterizations for typical classes of graphs. We determine the exact optimality representations for Petersen graph, the complete k-partite graphs, split graphs, generalized con…
It has been previously shown by the authors that a directed graph on a linearly ordered set of edges (ordered graph) with adjacent unique source and sink (bipolar digraph) has a unique fully optimal spanning tree, that satisfies a simple criterion on fundamental cycle/cocycle directions. This result yields, for any ordered graph, a canonical bijection between bipolar orientations and spanning trees with internal activity 1 and external activity 0 in the sense of the Tutte polynomial. This bijection can be extended to all orientations and all spanning trees, yielding the active bijection, prese…
Hypermaps with hyperedges of length at most $3$
本文研究了超映射中超边长度不超过3的超树计数问题，通过发展涉及六种广义环和桥的删除-收缩公式，并证明了特殊代入到Whitney多项式中的结果，进而推广了可靠性多项式和随机聚类模型到超映射中，最终明确计数了平面图中每个顶点度数不超过3的超树。
We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane.…
Topological indices on self-similar graphs generated by groups
本文研究了自相似图的拓扑指数，推导了树自动群的Schreier图的直径、匹配数和Tutte多项式公式，并给出了Wiener和Szeged指数的精确值。
In this paper, we study the graph polynomial that counts spanning rooted forests f_g of a given graph. This polynomial has a remarkable reciprocity property. We give a new bijective proof for this theorem which has Prufer coding as a special case.
Combinatorial interpretations of Tutte polynomials at the point $(2,-1)$
本文引入偶左生成森林和奇G-可划分排列，证明Tutte多项式在(2,-1)处的值等于这两种组合结构的计数，并特别对完全图K_n给出与交错排列的等价性。
Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures
本文利用 Artin-Ihara L-函数在 u=1 处的特殊值定义 Ihara 多项式，并用其马勒测度及 p-进马勒测度刻画 Z-覆盖的有限中间图生成树数目的渐近增长和 p-进赋值，统一并推广了循环图、I-图及 Fibonacci 数赋值等已有结果。
Recently Guth and Katz \cite{GK2} invented, as a step in their nearly complete solution of Erdős's distinct distances problem, a new method for partitioning finite point sets in $\R^d$, based on the Stone--Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi--Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generaliza…
In this paper, we develop a new method to produce explicit formulas for the number $f_{G}(n)$ of rooted spanning forests in the circulant graphs $ G=C_{n}(s_1,s_2,\ldots,s_k)$ and $ G=C_{2n}(s_1,s_2,\ldots,s_k,n).$ These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form $f_{G}(n)=p\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the parity of $n$. Finally, we find an asymptotic formula for $f_{G}(n)$ through the Mahler measure of the associated …
We investigate the problem of simultaneously dominating all spanning trees of a given graph. We prove that on 2-connected graphs, a subset of the vertices dominates all spanning trees of the graph if and only if it is a vertex cover. Using this fact we present an exact algorithm that finds a simultaneous dominating set of minimum size using an oracle for finding a minimum vertex cover. The algorithm can be implemented to run in polynomial time on several graph classes, such as bipartite or chordal graphs. We prove that there is no polynomial time algorithm that finds a minimum simultaneous dom…
A spanning tree model for chromatic homology
本文提出了色复形的生成树模型，利用该模型在$\mathcal{A}_m$代数上证明了关于色同调同调跨度及挠存在性的两个猜想。
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a spectral sequence that converges to its homology. For alternating links, all differentials on the spanning tree complex are zero and the reduced Khovanov homology is determined by the Jones polynomial and signature. We prove some analogous theorems for (unreduced) Khovanov homolo…
Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's conjecture. However, we can evaluate f(G,q) explicitly for certain graphs G, such as the complete graph. We also point out the connection between Kontsevich's conjecture and such topics as the Matrix-Tree Theorem and orthogonal geometry.
Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fresán, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we cons…
Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph $G$ and the dimer model defined on a decorated version $\GD$ of this graph \cite{Fisher}. In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristi…
On the complexity of finding a spanning even tree in a graph
本文研究了判定图是否存在生成偶树的问题，证明该问题在平面图上为NP完全，并为分裂图、余图、余二部图、单位区间图和块图等受限图类提供了多项式时间算法。
Minimal $L^p$-congestion spanning trees on weighted graphs
本文将生成树拥塞推广为 L^p 拥塞，给出若干图族的显式估计，并提出针对一般加权图和加权平面图的多项式时间近似算法。
Sampling Balanced Forests of Grids in Polynomial Time
该研究解决了Charikar等人的猜想，提出多项式时间算法用于采样平衡网格图划分，相关结果可应用于选区划分等领域。
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a sin…
Choi et. al (2011) introduced a minimum spanning tree (MST)-based method called CLGrouping, for constructing tree-structured probabilistic graphical models, a statistical framework that is commonly used for inferring phylogenetic trees. While CLGrouping works correctly if there is a unique MST, we observe an indeterminacy in the method in the case that there are multiple MSTs. In this work we remove this indeterminacy by introducing so-called vertex-ranked MSTs. We note that the effectiveness of CLGrouping is inversely related to the number of leaves in the MST. This motivates the problem of f…
Let $(X,τ)$ be a Polish space with Borel probability measure $μ,$ and $G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a Borel one-ended spanning tree generically. If $G$ is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result $μ$-a.e. (resp., everywhere). Our results generalize recent work of Timár, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even degree graphs and measurable and Baire mea…
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably defined matrix. This result can be interpreted topologically as an expression for the lowest order term of the Alexander-Conway polynomial of an algebraically split link. We also prove some algebraic properties of our Pfaffian-tree polynomial.
Two Proofs of the Hamiltonian Cycle Identity
本文采用将哈密顿环多项式视为所有长度为n的生成环列表的视角，给出两种原创证明以建立其与行列式、积和式多项式的关联恒等式，并推导了哈密顿路径多项式的类似恒等式。
Counting spanning quasi-trees of ribbon graphs: determinants and #P-completeness
研究连通带状图拟树计数问题，证明其在多项式时间图灵归约下是#P完全的，还通过带框Cohn - Lempel等式等将计数与交错多项式求值联系起来，同时给出可定向带状图等情形的行列式证明。
We introduce two graph polynomials and discuss their properties. One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function. The other is a polynomial of one variable that is obtained by the specialization of the first one. It is shown that these polynomials satisfy deletion-contraction relations and are new examples of the V-function, which was introduced by Tutte (1947, Proc. Cambridge Philos. Soc. 43, 26-40). For these polynomials, we discuss the interpretations of special values and then obtai…
Spanning Trees with Many Leaves in Graphs of Minimum Degree at Least 7
本文提出一个多项式时间算法，在最小度至少为7的连通图中构造具有至少0.5455n个叶子的生成树，并给出对所有δ≥7的改进下界。
Homaloidal Polynomials and Gaussian Models of Maximum Likelihood Degree One
本文借助homaloidal多项式刻画最大似然度为一的高斯模型，证明弦图的生成树生成函数为homaloidal，而$n$环对应第$n$个欧拉数，并给出模型方程、构造族与矩阵规模上界。
Many recent approximation algorithms for different variants of the traveling salesman problem (asymmetric TSP, graph TSP, s-t-path TSP) exploit the well-known fact that a solution of the natural linear programming relaxation can be written as convex combination of spanning trees. The main argument then is that randomly sampling a tree from such a distribution and then completing the tree to a tour at minimum cost yields a better approximation guarantee than simply taking a minimum cost spanning tree (as in Christofides' algorithm). We argue that an additional step can help: reassembling the sp…
(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of its combinatorial Laplacian matrix. This is a 165 year old result in graph theory and the proof is conceptually simple. However, the elegance of this result is it connects many apparently unrelated concepts in linear algebra and graph theory. Our motivation behind this work was to make the proof accessible to anyone with beginner\slash intermediate grasp of linear algebra. Therefore in this paper…
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