The Star-Extremality of Circulant Graphs

The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. We say a graph G is star extremal if its circular chromatic number is equal to its fractional chromatic number. This paper gives an improvement of a theorem. And we show that several classes of circulant graphs are star extremal.

A Class of Star Extremal Circulant Graphs

The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its circular chromatic number (also known as the star chromatic number). This paper studies the star extremality of the circulant graphs whose generating sets are of the form {±1,±k} .

The Rank of Integral Circulant Graphs

A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulant.The rank of a graph is defined to be the rank of its adjacency matrix.This importance of the rank,due to applications in physics,chemistry and combinatorics.In this paper,using Ramanujan sums,we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.

Resolvability in Circulant Graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V（G） there is a vertex w E W such that d（u, w） ≠ d（v, w）. A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim（G）. For a vertex u of G and a subset S of V（G）, the distance between u and S is the number minses d（u,s）. A k-partition II = {$1,$2,..., Sk} of V（G） is called a resolving partition if for every two distinct vertices u, v E V（G） there is a set Si in H such that d（u, Si） ≠ d（v, Si）. The minimum k for which there is a resolving k-partition of V（G） is called the partition dimension of G, denoted by pd（G）. The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j （rood n） E C, where C C Zn has the property that C = -C and 0 ￠ C. The circulant graph is denoted by Xn,△ where A = ｜C｜. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim（Xn,3） is independent of choice of n by showing that dim（Xn,a） = {3 for all n ≡0 （mod 4）,4 for all n≡2 （mod4）.We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd（Xn,4） is independent of choice of n and show that pd（X5,4） = 5 and pd（Xn,a） ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.

On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

Let F(x)=∑∞n=1 Tsi,s2,...,sk(n)x^n be the generating function for the number,Ts1bs2,...,sk(n) of spanning trees in the circulant graph Cn(s1,S2,...,Sk).We show that F(x)is a rational function with integer coefficients satisfying the property F(x)=F(l/x).A similar result is also true for the circulant graphs C2n(s1,S2,....,Sk,n)of odd valency.We illustrate the obtained results by a series of examples.

Embedding generalized of circulant graphs and Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C（2n ＋ 1; {1, n}） on the projective plane are described, the number of embeddings of C（2n ＋ 1; {1, n}） on the projective plane follows, then the number of embeddings of the generalized Petersen graph P（2n ＋ 1, n） on the projective plane is deduced from that of C（2n ＋ 1; {1, n}）, because C（2n ＋ 1; {1, n}） is a minor of P（2n ＋ 1, n）, their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P（2n, 2） on the projective plane is also obtained.

The k-Diameter of a Kind of Circulant Graph

The diameter of a graph G is the maximal distance between pairs of vertices of G. When a network is modeled as a graph,diameter is a measurement for maximum transmission delay. The k-diameter dk(G) of a graph G, which deals with k internally disjoint paths between pairs of vertices of G, is a extension of the diameter of G. It has widely studied in graph theory and computer science. The circulant graph is a group-theoretic model of a class of symmetric interconnection network. Let Cn(i, n / 2) be a circulant graph of order n whose spanning elements are i and n / 2, where n≥4 and n is even. In this paper, the diameter, 2-diameter and 3-diameter of the Cn(i, n / 2) are all obtained if gcd(n,i)=1, where the symbol gcd(n,i) denotes the maximum common divisor of n and i.

k-Diameter of Circulant Graph with Degree 3

Parameters k-distance and k-diameter are extension of the distance and the diameter in graph theory. In this paper, the k-distance dk （x,y） between the any vertices x and y is first obtained in a connected circulant graph G with order n （n is even） and degree 3 by removing some vertices from the neighbour set of the x. Then, the k-diameters of the connected circulant graphs with order n and degree 3 are given by using the k-diameter dk （x,y）.

A necessary and sufficient condition for a vertex-transitive graph to be star extremal

A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number. We first give a necessary and sufficient condition for a graph G to have circular chromatic number V(G)/α(G) (where V(G) is the vertex number of G and α(G) is its independence number). From this result, we get a necessary and sufficient condition for a vertex-transitive graph to be star extremal as well as a necessary and sufficient condition for a circulant graph to be star extremal. Using these conditions, we obtain several classes of star extremal graphs.

On Network Designs with Coding Error Detection and Correction Application

The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems.Furthermore,combinatorial design theory has several applications like detecting or correcting errors in communication systems.Network(graph)designs(GDs)are introduced as a generalization of the symmetric balanced incomplete block designs(BIBDs)that are utilized directly in the above mentioned application.The networks(graphs)have been represented by vectors whose entries are the labels of the vertices related to the lengths of edges linked to it.Here,a general method is proposed and applied to construct new networks designs.This method of networks representation has simplified the method of constructing the network designs.In this paper,a novel representation of networks is introduced and used as a technique of constructing the group generated network designs of the complete bipartite networks and certain circulants.A technique of constructing the group generated network designs of the circulants is given with group generated graph designs(GDs)of certain circulants.In addition,the GDs are transformed into an incidence matrices,the rows and the columns of these matrices can be both viewed as a binary nonlinear code.A novel coding error detection and correction application is proposed and examined.

Circulant Double Coverings of a Circulant Graph of Valency Five

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory. A covering of G is called circulant if its covering graph is circulant. Recently, the authors [Discrete Math., 277, 73-85 （2004）1 enumerated the isomorphism classes of circulant double coverings of a certain type, called a typical covering, and showed that no double covering of a circulant graph of valency three is circulant. Also, in [Graphs and Combinatorics, 21,386 400 （2005）], the isomorphism classes of circulant double coverings of a circulant graph of valency four are enumerated. In this paper, the isomorphism classes of circulant double coverings of a circulant graph of valency five are enumerated.

INFINITE CIRCULANTS AND THEIR PROPERTIES

In recent years diverse literatures have been published on circulants (cf. [2] and the references cited therein). In this paper we consider the infinite analogues of circulant and random infinite circulant, and their connectivities and hamiltonian properties are discussed. Especially we answer a question of [4] in the case of infinite (undirected) circulants, and some results on random infinite circulants are also obtained.

Enumerating Abelian Typical Cube-Free Fold Coverings of a Circulant Graph

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory.A covering projection p from a Cayley graph Cay(Г,X)onto another Cayley graph Cay(Q,y)is called typical if the function p:Г→Q on the vertex sets is a group epimorphism.A typical covering is called abelian(or circulant,respectively)if its covering graph is a Cayley graph on an abelism(or a cyclic,respectively)group.Recently,the equivalence classes of connected abelian typical prime-fold coverings of a circulant graph are enumerated.As a continuation of this work,we enumerate the equivalence classes of connected abelian typical cube-free fold coverings of a circulant graph.

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action onits ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficientcondition for an automorphism of the graph Γ to be an automorphism of a map with the underlyinggraph Γ is obtained. Using this result, all orientation-preserving automorphisms of maps onsurfaces (orientable and non-orientable) or just orientable surfaces with a given underlyingsemi-regular graph Γ are determined. Formulas for the numbers of non-equivalent embeddings of thiskind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, thenon-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable andgeneral surfaces are enumerated.

A Fault-tolerant Routing Control for Double Loop Networks

This paper divides the vertex set into several disjoined subsets and provides an optimal fault-tolerance routing algorithm based on the vertex set partition. This algorithm is efficient and convergent, in polynomial time, we can get the output if the vertex is given.