On 3-restricted edge connectivity of undirected binary Kautz graphs

An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge.     

Efficient calculation of the most reliable pair of link disjoint paths in telecommunication networks

In transmission networks an important routing problem is to find a pair of link disjoint paths which optimises some performance measure. In this paper the problem of obtaining the most reliable pair of link disjoint paths, assuming the reliability of the links are known, is considered. This is a non-linear optimisation problem. It is further introduced the constraint that the length of the paths should not exceed a certain number of links, which makes the efficient resolution of the problem more complex.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

On magic and supermagic circulant graphs

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper, we characterize magic circulant graphs and 3-regular supermagic circulant graphs. We establish some conditions for supermagic circulant graphs.     

Enumeration of 2-regular circulant graphs and directed double networks

We present a formula to enumerate non-isomorphic circulant digraphs of order n with connection sets of cardinality 2. This formula simplifies to C(n,2)=3×2^a−1−4 in the case when n=2^a(a≥3), and when n=p^a(where p is an odd prime and a≥1). The number of non-isomorphic directed double networks are also enumerated.     

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.     

On isomorphisms of circulant digraphs of bounded degree

C.H. Li recently made the following conjecture: Let Γ be a circulant digraph of order n=n₁n₂ and degree m, where gcd(n₁,n₂)=1, n₁ divides 4k, where k is odd and square-free, and every prime divisor of n₂ is greater than m, or, if Γ is a circulant graph, every prime divisor of n₂ is greater than 2m. Then Γ is a CI-digraph of Z_n. In this paper we verify that this conjecture is true.     

On the existence of edge cuts leaving several large components

We characterize graphs of large enough order or large enough minimum degree which contain edge cuts whose deletion results in a graph with a specified number of large components. This generalizes and extends recent results due to Ou [Jianping Ou, Edge cuts leaving components of order at least m, Discrete Math. 305 (2005), 365-371] and Zhang and Yuan [Z. Zhang, J. Yuan, A proof of an inequality concerning k-restricted edge connectivity, Discrete Math. 304 (2005), 128-134].     

On the reduced Euler characteristic of independence complexes of circulant graphs

Let $G$ be the circulant graph $C_n\left(S\right)$ with $S?\{1,\dots ,?\frac{n}{2}?\}$. We study the reduced Euler characteristic $\stackrel{?}{\chi }$ of the independence complex $\Delta \left(G\right)$ for $n=p^k$ with $p$ prime and for $n=2p^k$ with $p$ odd prime, proving that in both cases $\stackrel{?}{\chi }$ does not vanish. We also give an example of circulant graph whose independence complex has $\stackrel{?}{\chi }$ which equals 0, giving a negative answer to R. Hoshino.     

Connectivity of iterated line graphs

Let k≥0 be an integer and L^k(G) be the kth iterated line graph of a graph G. Niepel and Knor proved that if G is a 4-connected graph, then κ(L²(G))≥4δ(G)−6. We show that the connectivity of G can be relaxed. In fact, we prove in this note that if G is an essentially 4-edge-connected and 3-connected graph, then κ(L²(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.     

Star图互连网络的容错性分析

限制连通度和限制容错直径是衡量互连网络可靠性的两个重要参数。当考察这两个参数时，总假设网络中和一台计算机相连接的所有计算机不会同时出现故障。该文证明了Star图互连网络的极小分离集和极小限制分离集的唯一性，然后得到了Star图的限制连通度是2n-4，当n=3,5和n≥7时，它的限制容错直径是|_3(n-1)/2_|+2，对于n =4, 6，限制容错直径是|_3(n-1)/2_|+3，即限制容错直径只比它的容错直径大1。     

REFINED CONNECTIVITY PROPERTIES OF ABELIAN CAYLEY GRAPHS

1.IntroductionAgraphG=(V,E)meansafinitegraphwithoutloopsandmultipleedgeswithvertexsetVandedgesetE,theclassicaledgeconnectivityA(G)ofGistheminimumsizeofasetUofedgessuchthatG--Uisdisconnected,andsuchasetUiscalledaoutsetofG.Notethatintheabovedefinition,absolutelynoconditionsorrestrictionsareimposedeitheronthecomponelltsofG--UoronthesetU.ThusitwouldseemnaturaltogeneralizetheconceptofedgeconnectivitybyintroducingsomeconditionsorrestrictionsonthecomponentsofG--Uand/orthesetU.Asageneralizatio…     

On the diameter of Kronecker graphs

We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter.     

三圈图的Harary指数

图G的Harary指数是指图G中所有顶点对间的距离倒数之和. 三圈图是指边数等于顶点数加2的连通图. 研究了三圈图的Harary数, 给出了所有三圈图中具有极大Harary指数的图的结构以及含有三个圈的三圈图中具有次大Harary指数的图的结构.     

循环图能量的一个上界

本文讨论循环图的能量,得到循环图能量上界的一个估计值.进一步得到整循环图能量的两个计算公式.     

On factor-invariant graphs

We classify trivalent vertex-transitive graphs whose edge sets have a partition into a Hamilton cycle and a 1-factor that is invariant under the action of the full automorphism group.     

The Crossing Number of C(mk;{1, k})

Calculating the crossing number of a given graph is, in general, an elusive problem. Garey and Johnson have proved that the problem of determining the crossing number of an arbitrary graph is NP-complete. The crossing number of a network(graph) is closely related to the minimum layout area required for the implementation of a VLSI circuit for that network. With this important application in mind, it makes most sense to analyze the the crossing number of graphs with good interconnection properties, such as the circulant graphs. In this paper we study the crossing number of the circulant graph C(mk;{1,k}) for m3, k3, give an upper bound of cr(C(mk;{1,k})), and prove that cr(C(3k;{1,k}))=k.Research supported by Chinese Natural Science Foundation     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.