渺位四角系统完美匹配数的计算

四角系统的完美匹配有很强的统计物理背景．本文给出了渺位四角系统完美匹配数的一个计算方法

Pfaffian orientation and enumeration of perfect matchings for some Cartesian products of graphs

The importance of Pfaffian orientations stems from the fact that if a graph G is Pfaffian, then the number of perfect matchings of G (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovasz pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph G and a cycle C(n) with n vertices (or a path P(n) with n vertices), we define C(n) (or P(n)) x G as the Cartesian product of graphs C(n) (or P(n)) and G. In the present paper, we construct Pfaffian orientations of graphs C(4) x G, P(4) x G and P(3) x G, where G is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of (G) over right arrow to enumerate their perfect matchings by Pfaffian approach, where (G) over right arrow is an arbitrary orientation of G

若干图类的常返构型

研究图上Abelian沙堆模型的常返构型问题,证明了图的常返构型与子图的常返构型的关系.进一步地,将其应用到具体图类,得到单圈图、lollypop图、dumbbell图的常返构型

Dimers on two types of lattices on the Klein bottle

NFSC [11171279]The problem of enumerating close-packed dimers, or perfect matchings, on two types of lattices (the so-called 8.8.4 and 8.8.6 lattices) embedded on the Klein bottle is considered, and we obtain the explicit expression of the number of close-packed dimers and entropy. Our results imply that 8.8.4 lattices have the same entropy under three different boundary conditions (cylindrical, toroidal and Klein bottle) and 8.8.6 lattices have the same property

Enumeration of perfect matchings of a type of quadratic lattice on the torus

NSFC [10831001]A quadrilateral cylinder of length m and breadth n is the Cartesian product of a m-cycle(with m vertices) and a n-path(with n vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as x(1), x(2), ... , x(m) and y(1), y(2), ... , y(m), respectively, where x(i) corresponds to y(i) (i = 1, 2, ..., m). We denote by Q(m,n,r), the graph obtained from quadrilateral cylinder of length m and breadth n by adding edges x(i)y(i+r) (r is a integer, 0 <= r < m and i + r is modulo m). Kasteleyn had derived explicit expressions of the number of perfect matchings for Q(m,n,0) [P.W. Kasteleyn, The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961), 1209-1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressions of the number of perfect matchings for Q(m,n,r) by enumerating Pfaffians

Avalanche Sizes of the Abelian Sandpile Model on Unicyclic Graphs

研究图上AbElIAn沙堆模型问题.首先给出关于沙堆模型常返构型的极大雪崩序列,然后刻画了一些雪崩性质.基于这些性质,我们确定了单圈图的基本雪崩中每个顶点的TOPPlIngS数及它的雪崩多项式,推广了r.COrI的结果.We first give a definition of a maximal avalanche sequence for a configuration of the Abelian sandpile model and characterize some avalanche properties.Applying these properties to unicyclic graphs, we determine their number of topplings on each vertex in principal avalanches and avalanche polynomials of the Abelian sandpile model, which generalizes R.Cori's results on cycles.supportedbyNSFC(11171279;11101358); theNSFofFujian(2012D140); theScientificResearchFundofFujianProvincialEducationDepartment(JA12208;JA12266); theScientificResearchFoundationofGuangxiEducationCommittee(200103YB069); theDoctorFoundationofMinnanNormalU

The Edge Surviving Rate of a Class of Planar Graphs for the Firefighter Problem

设g是一个有n个点M条边的连通图.假设火在图g的一条边uV的两个端点燃起,消防员保护若干个没有着火的顶点,火接着蔓延到其他未保护且没有着火的邻点,火和消防员交替地在图g上移动.设Sn(g,uV;(k1,k2))表示当火在边uV的两个端点燃起时,消防员采取第一步保护k1个点,后面每步保护k2个点的策略所能救下的最大顶点数.定义图g的边存活率ρ(g,E;(k1,k2))=∑uV∈E(g)Sn(g,uV;(k1,k2))/nM,即当火随机地在图g的一条边的两个端点燃起时,消防员最多能救下的顶点数的平均率.本文证明了如果g是一个至少有3个点且最小度至少为3的不含4-圈连通平面图,那么ρ(g,E;(4,2))>7/705.Let Gbe a connected graph with n vertices and m edges.Suppose that a fire breaks out at an edge uv of G(two adjacent vertices).At each time interval,the firefighter protects vertices not yet on fire.At the end of each time interval,the fire spreads to all the unprotected vertices that are associated with a neighbour on fire.Then the firefighter and the fire alternately move on the graph.Let sn(G;uv;(k1;k2))denote the maximum number of vertices in Gthat the firefighter can save when a fire breaks out at two adjacent vertices u and v,protecting k1 vertices at the first step and k2 at subsequent steps.The surviving rate of Gdenotesρ(G,e;(k1,k2))=∑uv∈E(G)sn(G,uv;(k1,k2))/nm,which is the average proportion of saved vertices when a fire breaks out at any edge uv.In this paper,we prove that,if Gis a planar graph without 4-cycles andδ(G)≥3,thenρ(G,e;(4,2))>7/705.国家自然科学基金(11171279;11471273); 国家留学基金委项目(201406310108

最大Randi指标的k悬挂点化学树的性质(英文)

化学分子图G的Randi指标为R(G)=∑_(u,v)(d_G(u)d_G(v))~(-(1/2).其中uv是G的边,d_G(u)表示G的顶点u的度.本文刻画了具有最大Randi指标的k悬挂点化学树的一些性质

The Pfaffian property of Cartesian products of graphs

NSFC [10831001, 11171279]Suppose that G=(V,E) is a graph with even vertices. An even cycle C is a nice cycle of G if G-V(C) has a perfect matching. An orientation of G is a Pfaffian orientation if each nice cycle C has an odd number of edges directed in either direction of the cycle. Let P (n) and C (n) denote the path and the cycle on n vertices, respectively. In this paper, we characterize the Pfaffian property of Cartesian products GxP (2n) and GxC (2n) for any graph G in terms of forbidden subgraphs of G. This extends the results in (Yan and Zhang in Discrete Appl Math 154:145-157, 2006)