Diameters of random circulant graphs

The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n ^1/k , and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.     

小度数循环图的计数

循环图已被用于平行计算，网络等方面．循环图研究的一个基本问题是对互不同构的循环图进行计数．对于给定的一个正整数ｎ，用Ｃ（ｎ，ｋ）表示互不同构的具有几个顶点，度数为ｋ的连通循环图的个数．文中给出了度数为 ４和５的循环图的一般结构，并对ｎ＝ｐａｑｂ（ｐ，ｑ皆为素数，ａ，ｂ＞０），给出了Ｃ（ｎ，４）的计算公式．     

A class of vertex-transitive digraphs,II

(1). We determine the number of non-isomorphic classes of self-complementary circulant digraphs with pq vertices, where p and q are distinct primes. The non-isomorphic classes of these circulant digraphs with pq vertices are enumerated. (2). We also determine the number of non-isomorphic classes of self-complementary, vertex-transitive digraphs with a prime number p vertices, and the number of self-complementary strongly vertex-transitive digraphs with p vertices. The non-isomorphic classes of strongly vertex-transitive digraphs with p vertices are also enumerated.     

Circulant graphs and tessellations on flat tori

Circulant graphs are characterized here as quotient lattices, which are realized as vertices connected by a knot on a k-dimensional flat torus tessellated by hypercubes or hyperparallelotopes. Via this approach we present geometric interpretations for a bound on the diameter of a circulant graph, derive new bounds for the genus of a class of circulant graphs and establish connections with spherical codes and perfect codes in Lee spaces.     

Circulant graphs and tessellations on flat tori

Circulant graphs are characterized here as quotient lattices, which are realized as vertices connected by a knot on a k-dimensional flat torus tessellated by hypercubes or hyperparallelotopes. Via this approach we present geometric interpretations for a bound on the diameter of a circulant graph, derive new bounds for the genus of a class of circulant graphs and establish connections with spherical codes and perfect codes in Lee spaces.     

On Moore bipartite digraphs

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore‐like bound in terms of its diameter k and the maximum out‐degrees (d₁, d₂) of its partite sets of vertices. It has been proved that, when d₁d₂ > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 ≤ k ≤ 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so‐called De Bruijn near‐factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out‐degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k = 4, which means that G = L G′, where G′ is a Moore bipartite digraph of diameter k = 3. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 171–187, 2003     

Circulant Double Coverings of a Circulant Graph of Valency Four

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices. The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com²MaC-KOSEF (R11-1999-054) in Korea.     

On Hamiltonicity of circulant digraphs of outdegree three

This paper deals with Hamiltonicity of connected loopless circulant digraphs of outdegree three with connection set of the form {a,ka,c}, where k is an integer. In particular, we prove that if k=−1 or k=2 such a circulant digraph is Hamiltonian if and only if it is not isomorphic to the circulant digraph on 12 vertices with connection set {3,6,4}.     

Circulant tournaments of prime order are tight

We say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number p≥3 of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory.     

Large graphs with given degree and diameter. II

The following problem arises in the study of interconnection networks: find graphs of given maximum degree and diameter having the maximum number of vertices. Constructions based on a new product of graphs, which enable us to construct graphs of given maximum degree and diameter, having a great number of vertices from smaller ones are given; therefore the best values known before are improved considerably.     

Some Combinatorial Aspects of Constructing Bipartite-Graph Codes

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m, n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.     

Integral circulant graphs

In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2^τ(n)-1 non-isomorphic integral circulant graphs on n vertices, where τ(n) is the number of divisors of n.     

Arc-Disjoint and Edge-Disjoint Hamilton Cycles in Circulants with Two Jumps

In this paper we give new necessary and sufficient conditions for a directed circulant with vertices of outdegree two to have a pair of arc-disjoint Hamilton cycles. These conditions explicitly identify a pair of arc-disjoint Hamilton cycles if such cycles exist. In addition, we give necessary and sufficient conditions for an undirected circulant with vertices of degree four to have a specific pair of edge-disjoint Hamilton cycles.     

互连网络的向量图模型

n-超立方体,环网,k元n超立方体,Star网络,煎饼(pancake)网络,冒泡排序(bubble sort)网络,对换树的Cayley图,De Bruijn图,Kautz图,Consecutive-d有向图,循环图以及有向环图等已被广泛的应用做处理机或通信互连网络.这些网络的性能通常通过它们的度,直径,连通度,hamiltonian性,容错度以及路由选择算法等来度量.在本文中,首先,我们提出了有向向量图和向量图的概念;其次,我们开发了有向向量图模型和向量图模型来更好地设计,分析,改良互连网络;我们进一步证明了上述各类著名互连网络都可表示为有向向量图模型或向量图模型;更重要的是该模型能够使我们设计出了新的互连网络---双星网络和三角形网络.     

New results on the energy of integral circulant graphs

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph ICG_n(D) has the vertex set Z_n = {0, 1, 2, … , n − 1} and vertices a and b are adjacent if gcd(a − b, n) ∈ D, where D ⊆ {d : d∣n, 1 ? d < n}. These graphs are highly symmetric, have integral spectra and some remarkable properties connecting chemical graph theory and number theory. The energy of a graph was first defined by Gutman, as the sum of the absolute values of the eigenvalues of the adjacency matrix. Recently, there was a vast research for the pairs and families of non-cospectral graphs having equal energies. Following Bapat and Pati [R.B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc. 1 (2004) 129-132], we characterize the energy of integral circulant graph modulo 4. Furthermore, we establish some general closed form expressions for the energy of integral circulant graphs and generalize some results from Ili? [A. Ili?, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009), 1881-1889]. We close the paper by proposing some open problems and characterizing extremal graphs with minimal energy among integral circulant graphs with n vertices, provided n is even.     

The Kirchhoff indexes of some composite networks

In electric circuit theory, it is of great interest to compute the effective resistance between any pair of vertices of a network, as well as the Kirchhoff index. During the past decade these parameters have been applied in organic chemistry as natural structural indexes different from the usual ones in order to achieve an improvement in the discrimination between different molecules that have similar structural behaviours. This new application has started an important and fruitful line of research which has carried the computation of the Kirchhoff index into some symmetrical networks such as distance-regular grahs or circulant graphs. Moreover, a wide range of generalized Kirchhoff indexes for some networks have been introduced. The objective of the present work is to obtain the Kirchhoff index for composite networks such as corona or cluster networks.     

有关循环图C(n;{1,k})的独立数的一些结果

令G=(V(G),E(G))是一个简单有限无向图.如果V(G)的子集S中任意两个顶点均不相邻,则S是图G的一个独立集.顶点独立集大小的最大值,称为图G的独立数,记作α(G).本文研究了循环图C(n;{1,k})的独立数问题,并给出了当k=2,3,4,5时的准确值.     

Scale Free Properties of Random <Emphasis Type="Italic">k</Emphasis>-Trees

Scale free graphs have attracted attention as their non-uniform structure that can be used as a model for many social networks including the WWW and the Internet. In this paper, we propose a simple random model for generating scale free k-trees. For any fixed integer k, a k-tree consists of a generalized tree parameterized by k, and is one of the basic notions in the area of graph minors. Our model is quite simple and natural; it first picks a maximal clique of size k + 1 uniformly at random, it then picks k vertices in the clique uniformly at random, and adds a new vertex incident to the k vertices. That is, the model only makes uniform random choices twice per vertex. Then (asymptotically) the distribution of vertex degree in the resultant k-tree follows a power law with exponent 2 + 1/k, the k-tree has a large clustering coefficient, and the diameter is small. Moreover, our experimental results indicate that the resultant k-trees have extremely small diameter, proportional to o(log n), where n is the number of vertices in the k-tree, and the o(1) term is a function of k.     

Circulant discrete dynamical systems with threshold functions of at most three variables

We propose a method for finding sources of discrete dynamical systems of the circulant type with a q-valued arbitrary function at vertices. We find all sources, all fixed points, and some cycles, as well as lengths of some maximal chains outside cycles for the systems with Boolean threshold functions of at most three variables at the vertices.