Jun-Ming Xu

Topological Structure and Analysis of Interconnection Networks

This book provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number. Audience: The book is suitable for those readers who are working on or intend to start research in design analysis of the topological structure of interconnection networks, particularly undergraduates and postgraduates specializing in computer science and applied mathematics.

Theory and Application of Graphs

In the spectrum of mathematics, graph theory which studies a mathe matical structure on a set of elements with a binary relation, as a recognized discipline, is a relative newcomer. In recent three decades the exciting and rapidly growing area of the subject abounds with new mathematical devel opments and significant applications to real-world problems. More and more colleges and universities have made it a required course for the senior or the beginning postgraduate students who are majoring in mathematics, computer science, electronics, scientific management and others. This book provides an introduction to graph theory for these students. The richness of theory and the wideness of applications make it impossi ble to include all topics in graph theory in a textbook for one semester. All materials presented in this book, however, I believe, are the most classical, fundamental, interesting and important. The method we deal with the mate rials is to particularly lay stress on digraphs, regarding undirected graphs as their special cases. My own experience from teaching out of the subject more than ten years at University of Science and Technology of China (USTC) shows that this treatment makes hardly the course di:fficult, but much more accords with the essence and the development trend of the subject.

On restricted edge-connectivity of graphs

This paper considers the concept of restricted edge-connectivity, and relates that to the edge-degree of a connected graph. The author gives some necessary conditions for a graph whose restricted edge-connectivity is smaller than its minimum edge-degree, then uses these conditions to show some large classes of graphs, such as all connected edge-transitive graphs except a star, and all connected vertex-transitive graphs with odd order or without triangles, have equality of the restricted edge-connectivity and the minimun edge-degree.

Cycles in folded hypercubes

This work investigates important properties related to cycles of embedding into the folded hypercube FQ n for n ≥ 2. The authors observe that FQn is bipartite if and only if n is odd, and show that the minimum length of odd cycles is n + 1i fn is even. The authors further show that every edge of FQn lies on a cycle of every even length from 4 to 2 n ;i fn is even, every edge of FQn also lies on a cycle of every odd length from n + 1t o 2 n − 1.

Panconnectivity of locally twisted cubes

The locally twisted cube LTQn which is a newly introduced interconnection network for parallel computing is a variant of the hypercube Qn. Yang et al. [X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Applied Mathematics Letters 17 (2004) 919–925] proved that LTQn is Hamiltonian connected and contains a cycle of length from 4 to 2 n for n ≥ 3. In this work, we improve this result by showing that for any two different vertices u and v in LTQn (n ≥ 3), there exists a uv-path of length l with d(u ,v )+ 2 ≤ l ≤ 2 n − 1 except for a shortest uv-path. c

On the complexity of the bondage and reinforcement problems

Let G=(V,E) be a graph. A subset D@?V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with larger domination (resp. total domination) number of G. The reinforcement (resp. total reinforcement) number of G is the smallest number of edges whose addition to G results in a graph with smaller domination (resp. total domination) number. This paper shows that the decision problems for the bondage, total bondage, reinforcement and total reinforcement numbers are all NP-hard.

A combinatorial problem related to distributed loop networks

The problem under consideration arises from studies on local networks and multimodule memory organizations. The ring network has been one of the popular network topologies used in the design and implementation of local area networks and other configurations. We consider here a generalization of the ring network by adding two fixed-step links to each node. The resulting networks have low diameter, easy routing, and switching structure and therefore are suitable for implementation in the design of reliable networks. Let N denote the number of nodes in the network. For a given N, we are concerned with the problem of determining the best topologies to minimize the diameter (and, hence, the transmission delay) of the network. We obtain new classes of values of N for which topologies can be found that achieve the lower bound lb = [(√2N - 1 - 1)/2] for the minimum diameter. We also show that for some infinite classes of N this lower bound lb is not achievable.

The bondage numbers and efficient dominations of vertex-transitive graphs

The bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number. A dominating set D is called an efficient dominating set of G if |N^-[v]@?D|=1 for every vertex v@?V(G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the relation between the bondage number and the efficient domination. As applications, we determine the bondage number for some circulant graphs and tori by characterizing the existence of efficient dominating sets in these graphs.

Edge fault tolerance analysis of a class of interconnection networks

Fault tolerant measures have played an important role in the reliability of an interconnection network. Edge connectivity, restricted-edge-connectivity, extra-edge-connectivity and super-edge-connectivity of many well-known interconnection networks have been explored. In this paper, we study the 2-extra-edge connectivity of a special class of graphs G(G0, G1; M) proposed by Chen et al. [Appl. Math. Comput. 140 (2003) 245– 254]. Then by showing that several well-known interconnection networks such as hyper

The bondage numbers of extended de Bruijn and Kautz digraphs

In this paper, we consider the bondage number b(G) for a digraph G, which is defined as the minimum number of edges whose removal results in a new digraph with larger domination number. This parameter measures to some extent the robustness of an interconnection network with respect to link failures. By constructing a family of minimum dominating sets, we compute the bondage numbers of the extended de Bruijn digraph and the extended Kautz digraph. As special cases, we obtain for the de Bruijn digraph B(d, n) and the Kautz digraph K(d, n) that b(B(d, n)) = d if n is odd and d =< b(B(d, n)) < 2d if n is even, and b(K(d, n)) = d + 1.