Zeta functions and complexities of a semiregular bipartite graph and its line graph

We treat zeta functions and complexities of semiregular bipartite graphs. Furthermore, we give formulas for zeta function and the complexity of a line graph of a semiregular bipartite graph. As a corollary, we present the complexity of a line graph of a complete bipartite graph.     

The Ihara zeta function of the complement of a semiregular bipartite graph

We study the Ihara zeta function of the complement of a semiregular bipartite graph. A factorization formula for the Ihara zeta function is derived via which the number of spanning trees is computed. For a class of complements of semiregular bipartite graphs, it is shown that they have the same Ihara zeta function if and only if they are cospectral.     

A New Proof of Bartholdi's Theorem

We give a new proof of Bartholdi's theorem for the Bartholdi zeta function of a graph. Supported by Grant-in-Aid for Science Research (C)     

On the Ihara Zeta Function of Cones Over Regular Graphs

In this paper we derive a formula of the Ihara zeta function of a cone over a regular graph that involves the spectrum of the adjacency matrix of the cone. We show that the Ihara zeta function and the spectrum of the adjacency matrix of the cone determine each other and we characterize those cones that satisfy the graph theory Riemann hypothesis.     

用超广义线图构造整谱图

线图在图的谱理论研究中起着重要的作用.在本文中,通过研究超广义线图成为整谱图的充分条件,获得了一种全新的构造新的整 谱图的方法,运用这种方法,可以构造出无穷多个新的整谱图.     

图论中若干基本概念的历史注记

考察了图与子图,树,匹配,欧拉图与哈密尔顿图,可平面图,以及与图的连通性和图的着色有关的若干图论基本概念的历史背景.     

Some properties of graphs determined by edge zeta functions

In 1989, Hashimoto introduced an edge zeta function of a finite graph, which is a generalization of the Ihara zeta function. The edge zeta function is the reciprocal of a polynomial in twice as many indeterminants as edges in the graph and can be computed via a determinant expression. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal. Finally, we present a new example illustrating that the Ihara zeta function cannot necessarily do the same.     

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.     

The Zagreb coindices of graph operations

Recently introduced Zagreb coindices are a generalization of classical Zagreb indices of chemical graph theory. We explore here their basic mathematical properties and present explicit formulae for these new graph invariants under several graph operations.     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

Generalized characteristic polynomials of graph bundles

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.     

Dynamic Scheduling on a Single Batch Processing Machine with Split Compatibility Graphs

We consider the problem of minimizing the makespan on a batch processing machine, in which jobs are not all compatible. Only compatible jobs can be included into the same batch. This relation of compatibility is represented by a split graph. Jobs have release dates. The capacity of the batch processing machine is finite or infinite. The processing time of a batch is given by the processing time of the longest job in the batch. We establish the NP-hardness of the general problem and present polynomial algorithms for several special cases. Relating scheduling theory and graph theory appears to be an interesting and important concept.     

Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219-235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.     

CYCLE SPACES OF GRAPHS ON THE SPHERE AND THE PROJECTIVE PLANE

Cycle base theory of a graph has been well studied in abstract mathematical field such matroid theory as Whitney and Tutte did and found many applications in pratical uses such as electric circuit theory and structure analysis, etc. In this paper graph embedding theory is used to investigate cycle base structures of a 2-(edge)-connected graph on the sphere and the projective plane and it is shown that short cycles do generate the cycle spaces in the case of ““““small face-embeddings““““. As applications the authors find the exact formulae for the minimum lengthes of cycle bases of some types of graphs and present several known results. Infinite examples shows that the conditions in their main results are best possible and there are many 3-connected planar graphs whose minimum cycle bases can not be determined by the planar formulae but may be located by re-embedding them into the projective plane.     

关于联图P_1VP_n的k-强优美性

本文研究了联图P_1VP_n的k-强优美性问题.利用K-强优美图的定义,获得了联图P_1VP_n是k-强优美图的必要条件,还得到了当n:2k-1时联图P_1VP_n是k-强优美图,亦是k-优美图,及当n≥3时联图P_1VP_n是2-强优美图,也是2-优美图的结果,推广了联图P_1VP_n是优美图的结果.     

The Inducibility of Graphs on Four Vertices

We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for a handful of small graphs and a specific set of complete multipartite graphs. Answering questions of Brown–Sidorenko and Exoo, we determine the inducibility of K_{1, 1, 2} and the paw graph. The proof is obtained using semidefinite programming techniques based on a modern language of extremal graph theory, which we describe in full detail in an accessible setting.     

可定向曲面上具有较短不可收缩圈图嵌入的唯一性

拓扑图论中的一个基本问题就是要决定图在一个(可定向)曲面上的嵌入之数目(既嵌入的柔性问题).H.Whitney的经典结果表明:一个3-连通图至多有一个平面嵌入;C.Thomassen的LEW-嵌入(大边宽度)理论将这一结果推广到一般的可定向曲面.本文给出了几个关于一般可定向曲面上嵌入图的唯一性定理.结果表明:一些具有大的面迹的可定向嵌入仍然具有唯一性.这在本质上推广了C.Thomassen在LEW-嵌入方面的工作.     

完全多部图的区间图完全化问题

The interval graph completion problem of a graph G includes two class problems: the profile problem and the pathwidth problem, denoted as P(G) and PW(G) respectively, where the profile problem is to find an interval supergraph with the smallest possible number of edges; the pathwidth problem is to find an interval supergraph with the smallest possible cliquesize. These two class problems have important applications to numerical algebra, VLSI-layout and algorithm graph theory respectively; And they are known to be NP-complete for general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the exact solutions of the profile and the pathwidth of the complete multipartite Graph Kn1,n2,…,nr(r≥2) are determined.     

图的倍图与补倍图

计算机科学数据库的关系中遇到了可归为倍图或补倍图的参数和哈密顿圈的问题．对简单图C,如果V（D（G））：V（G）∪V（G′）E（D（G））=E（C）∪E（C″）U{vivj′｜vi∈V（G）,Vj′∈V（G′）且vivj∈E（G））那么,称D（C）是C的倍图,如果V（D（G））=V（C）∪V（G′）,E（D（C））：E（C）∪E（G′）∪{vivj′}vi∈V（G）,vj′∈V（G’）and vivj∈（G））,称D（C）是G的补倍图,这里G′是G的拷贝．本文研究了D（G）和D的色数,边色数,欧拉性,哈密顿性和提出了D（G） 的边色数是D（G）的最大度等公开问题．     

应用图论分析与最优化理论来数据挖掘大规模水牛普里昂蛋白结构数据

图论、最优化理论显然在蛋白质结构的研究中大有用场. 首先, 调查/回顾了研究蛋白质结构的所有图论模型. 其后, 建立了一个图论模型: 让蛋白质的侧链来作为图的顶点, 应用图论的诸如团、 $k$-团、 社群、 枢纽、聚类等概念来建立图的边. 然后, 应用数学最优化的现代摩登数据挖掘算法/方法来分析水牛普里昂蛋白结构的大数据. 成功与令人耳目一新的数值结果将展示给朋友们.