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应用图论
  • 项目名称：应用图论
  • 项目类别：重点项目
  • 批准号：10831001
  • 申请代码：A011602
  • 项目来源：国家自然科学基金
  • 研究期限：2009-01-01-2012-12-31
  • 项目负责人：张和平
  • 负责人职称：教授
  • 依托单位：兰州大学
  • 批准年度：2008
中文摘要：

图论作为一门新兴的离散数学分支，在理论化学、统计物理、分子生物学、网络设计、运筹学、计算机科学以及经济和社会科学中都得到了广泛的应用，涉及到代数学、拓扑学和组合学等数学工具。图论在相关学科的应用研究也产生了深刻的数学理论，并推动着图论自身的迅速发展。本项目用图论的方法研究化学、分子生物学和统计物理中的理论和实际问题，具体内容有化学图的结构特性及若干拓扑指标（Randic指标， 谱与能量，Wiener指标等），化学分子图的完美匹配与共振问题(包括k-共振、k-圈共振、Clar覆盖多项式，共轭圈多项式等）， 碳纳米管和各种曲面上的富勒烯分子的图理论，DNA序列和蛋白质研究中的图论与拓扑方法，统计物理中的dimer问题和纽结多项式的根的分布等课题，解决其中若干重要的公开问题和猜想，取得具有国际影响的重要成果，为相关学科的进一步研究提供更加有效的数学工具和方法。

中文主题词： 分子拓扑指标；k-共振图；Fullerene图；多面体链环；纽结多项式
英文摘要：

Topological index of molecules；k-resonant graph；Fullerene graph；Polyhedral link；Knot polynomial

英文主题词： Topological index of molecules；k-resonant graph；Fullerene graph；Polyhedral link；Knot polynomial
结论摘要：

本项目围绕有机化学、分子生物学和统计物理中产生的与图论相关的理论问题和实际问题展开了研究工作，完成了预定任务，取得了突出的研究成绩. 在分子的拓扑指标方面，克服拟序比较的缺陷，采用能量计算的Coulson积分公式和参数积分的估算新方法，最终解决了数学化学家Gutman等提出长期未解决的问题确定单圈图类中最大能量图， 在Springer出版一部专著“Graph Energy”；用概率方法得到了树的Randic指标的近似估计，从而证明了Fajtlowicz关于Randic指标和平均距离的猜想对几乎所有的连通图和几乎所有的树都是成立的. 在图的完美匹配与共振方面，研究了曲面嵌入图的匹配扩张，系统地考虑了k-共振系统, 提出了极大共振图的概念, 证明了2-可扩平面二部图是2-共振的，由此完整解决了硼氮富勒烯图的k-共振问题；特别地完整解决了“什么富勒烯图是2-共振”这个困难问题. 在碳纳米管和富勒烯（fullerene）分子的图理论方面，研究了Clar理论在稳定性方面的作用, 对小的fullerene分子，发现从sextet多项式与Clar覆盖多项式的系数之和的最大值都能唯一地从其所有同分异构体中确定出稳定的富勒烯，如C60和C70. 证明了fullerene图的最小匹配强迫数至少为3，反Kekule数恒为4. 证明了开口纳米管等距离嵌入超立方图当且仅当它属三类退化的情况. 在DNA序列和蛋白质方面，从数学上通过多面体链环模型来模拟这些折叠结构，提出了多面体链环的构筑方法，给出了这些多面体链环的对称性、手性、拓扑性质及拓扑指标的计算，特别提出了多面体链环的新Euler公式，在Nova Sciences 出版专著“The Chemistry and Mathematics of DNA Polyhedra”. 在统计物理与纽结多项式方面， 应用排叉链环类证明了琼斯多项式的零点在整个复平面上是稠密的，进一步使用Jaeger链环也证明了Homfly多项式的零点在复平面上是稠密的. 基于平图通过对边进行交错定向tangle替换得到的有向链环的Homfly多项式，建立了其与平图的赋权的Tutte多项式之间的关系. 此外，给出了一些典型图类的支撑树和dimer 构型的计数以及熵的计算. 运用Polya计数定理给出了任意多个苯的Belt及M？bius型的环状苯环数目的计数公

成果综合统计
成果类型
数量
  • 期刊论文
  • 会议论文
  • 专利
  • 获奖
  • 著作
  • 159
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  • 1
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期刊论文
Dimer statistics on the Klein bottle
Dimer statistics of honeycomb lattices on Klein bottle, Mobius strip and cylinder
The Laplacian energy of random graphs
On the similarity/dissimilarity of DNA sequences based on 4D graphical representation
On the minimal energy of trees with a given number of vertices of degree two
A Note on Path-Matchings
The graph of perfect matching polytope and an extreme problem
A novel characterization of n-extendable bipartite graphs
Smallest generalized cuts and diameter-increasing sets of Johnson graphs
DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO TRIANGULAR AND HONEYCOMB LATTICES
On plane graphs with link component number equal to the nullity
The atom-bond connectivity index of chemical unicyclic graphs
Small Randic Index Ordering of Trees with k Pendant Vertices
Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links
Trees with a given order and matching number that have maximum general Randic index
PI index of toroidal polyhexes
Randic Index, Diameter and Average Distance
Direct Sum of Distributive Lattices on the Perfect Matchings of a Plane Bipartite Graph
Complete solution to a conjecture on Randic index
All Connected Graphs with Maximum Degree at Most 3 whose Energies are Equal to the Number of Vertice
Enumeration of perfect matchings of a type of quadratic lattice on the torus
On Triregular Graphs whose Energy Exceeds the Number of Vertices
Maximally resonant polygonal systems
Embedding on alphabet overlap digraphs
The Minimal Estrada Index of Trees with Two Maximum Degree Vertices
Extreme Atom-Bond Connectivity Index of Graphs
On a kind of reliability analysis of networks
l(1)-Embeddability of Hexagonal and Quadrilateral Mobius graphs
Models of virus capsids containing 72 pentameric capsomeres
The HOMFLY Polynomial for a Family of Polyhedral Links
On the restricted matching extension of graphs in surfaces
On forcing matching number of boron-nitrogen fullerene graphs
The architecture and the Jones polynomial of polyhedral links
Character of Trees with Extreme Balaban Index
Connectivity and minimal distance spectral radius of graphs
Bounding the sum of powers of the Laplacian eigenvalues of graphs
Dimers Belonging to Three Orientations on Plane Honeycomb Lattices
New Nordhaus-Gaddum-type results for the Kirchhoff index
A New Euler's Formula for DNA Polyhedra
Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model
2-extendability and k-resonance of non-bipartite Klein-bottle polyhexes
The optimal strong radius and optimal strong diameter of the Cartesian product graphs
Removable Edges in Cycles of a k-Connected Graph
On the location of zeros of the Homfly polynomial
Embeddability of open-ended carbon nanotubes in hypercubes
Note on the energy of regular graphs
The Homfly Polynomial for Even Polyhedral Links
Forcing matching numbers of fullerene graphs
Note on Conjugated Unicyclic Graphs with Minimal Energy
On a Relation Between Randic Index and Chromatic number
Topological transformation of dual polyhedral links
BOUNDS ON THE PATH-MATCHING NUMBER
Various energies of random graphs
Maximal resonance of cubic bipartite polyhedral graphs
A general sufficient condition for a graph G with lambda (m) (G) a (c) 1/2 zeta (m) (G)
2-resonance of plane bipartite graphs and its applications to boron-nitrogen fullerenes
Note on Bipartite Unicyclic Graphs of Given Bipartition with Minimal Energy
Counting the Cyclocized Polyphenacenes
Super connectivity of Kronecker products of graphs
Contractible edges of k-connected graphs for k=4, 5
A kind of conditional fault tolerance of (n, k)-star graphs
REMOVABLE EARS OF 1-EXTENDABLE GRAPHS
ORIENTED STATE MODEL OF THE JONES POLYNOMIAL AND ITS CONNECTION TO THE DICHROMATIC POLYNOMIAL
A lower bound on the number of removable ears of 1-extendable graphs
The complexity of Platonic and Archimedean polyhedral links
Transfer-matrix Calculation of the Clar Covering Polynomial of Hexagonal Systems
The Architecture and Jones Polynomials of Cycle-crossover Polyhedral Links
Some results on Laplacian spectral radius of graphs with cut vertices
Hypoenergetic and Strongly Hypoenergetic k-Cyclic Graphs
A combination of Clar number and Kekul, count as an indicator of relative stability of fullerene iso
Character of Graphs with Extremal Balaban Index
ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS
The architecture of polyhedral links and their HOMFLY polynomials
On super 2-restricted and 3-restricted edge-connected vertex transitive graphs
Topological Analysis of Enzymatic Actions on DNA Polyhedral Links
A proof of a conjecture on the Randic index of graphs with given girth
Degree conditions for graphs to be lambda(3)-optimal and super-lambda(3)
Characterizations for P(>= 2)-factor and P(>= 3)-factor covered graphs
Extended Goldberg polyhedral links with odd tangles
Johnson图的哈密尔顿连通性、泛圈性和删边保持直径不变性
The Laplacian spectral radius of unicyclic graphs with k pendent vertices
Connectivity of the Mycielskian of a digraph
(n,m)-Graphs with maximum zeroth-order general Randic index for alpha in (-1,0)
A novel molecular design of polyhedral links and their chiral analysis
All hypoenergetic graphs with maximum degree at most 3
Hosoya Polynomials of TUC4C8® Nanotubes
The maximum Randic index of chemical trees with k pendants
Bicyclic graphs with extremal Kirchhoff index
Some graphs determined by their spectra
Extremal fullerene graphs with the maximum Clar number
Enumerating spanning trees of graphs with an involution
ON k-RESONANT FULLERENE GRAPHS
Algorithms and extremal problem on Wiener polarity index
Complete solution to a conjecture on the Randic index of triangle-free graphs
Optimal strong (k, d)-orientation of complete k-partite graphs
Lower and upper orientable strong diameters of graphs satisfying the Ore condition
Minimal 2-matching-covered graphs
Kirchhoff index of composite graphs
2-extendability of toroidal polyhexes and Klein-bottle polyhexes
奇图的匹配可扩性
On f-fault tolerant arc-forwarding and optical indices of all-optical folded hypercubes
Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes
The atom-bond connectivity index of chemical bicyclic graphs
On Vertex-Distinguishing Proper Edge Colorings of Graphs Satisfying the Ore Condition
Note on unicyclic graphs with given number of pendent vertices and minimal energy
Extended Goldberg polyhedral links with even tangles
化学单圈图的原子键连通性指数(英文)
On the Distance Spectra and Energies of Cartesian Product of Some Graphs
Some rules on resistance distance with applications
GRAPHICAL CALCULI FOR THE DUBROVNIK POLYNOMIAL WITH APPLICATIONS
The Braid Index of Complicated DNA Polyhedral Links
On the Anti-Kekule Number of Fullerenes
Super lambda(3)-optimality of regular graphs
Enumerating the total colorings of a polyhedron and application to polyhedral links
On the characterizing properties of the permanental polynomials of graphs
Super-connectivity of Kronecker products of split graphs, powers of cycles, powers of paths and comp
The number of Kekule structures of polyominos on the torus
Dimer coverings on random multiple chains of planar honeycomb lattices
Hosoya polynomials of the capped zig-zag nanotubes
Wiener Index in Random Polyphenyl Chains
Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph
On the restricted matching extension of graphs on the torus and the Klein bottle
Computing the permanental polynomials of bipartite graphs by Pfaffian orientation
THE HOMFLY AND DICHROMATIC POLYNOMIALS
Global Forcing Number of Some Chemical Graphs
Hexagonal resonance of (3,6)-fullerenes
若干与统计物理相关的组合问题
基于4D图形表示的DNA序列相似性分析
极大3-限制性边连通图的若干充分条件
关于Randic指数及图的直径
获奖
图的匹配理论与图能量的研究
著作
Graph Energy
DNA多面体的数学与化学
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