20337056
On 2-extendable abelian Cayley graphs
Abstract A graph G is 2-extendable if any two independent edges of G are contained in a perfect matching of G. A Cayley graph of even order over an abelian group is 2-extendable if and only if it is not isomorphic to any of the following circulant graphs: 1. (I) Z2n(1, 2n - 1), n ⩾ 3; 2. (II) Z2n(1, 2, 2n - 1, 2n - 2), n ⩾ 3; 3. (III) Z4n(1, 4n - 1, 2n), n ⩾ 2; 4. (IV) Z4n + 2(2,4n,2n + 1), n ⩾ 1; and 5. (V) Z4n +2(1,4n + 1, 2n, 2n + 2), n ⩾ 1.
45692912
Triangle-Free Graphs with Large Degree
20073710
On a class of Hamiltonian laceable 3-regular graphs
Abstract Using the concept of brick-products, Alspach and Zhang showed in Alspach and Zhang (1989) that all cubic Cayley graphs over dihedral groups are Hamiltonian. It is also conjectured that all brick-products C(2n, m, r) are Hamiltonian laceable, in the sense that any two vertices at odd distance apart can be joined by a Hamiltonian path. In this paper, we shall study the Hamiltonian laceability of brick-products C(2n, m, r) with only one cycle (i.e. m = 1). To be more specific, we shall provide a technique with which we can show that when the chord length r is 3, 5, 7 or 9, the corresponding brick-products are Hamiltonian laceable. Let s = gcd((r + 1)/2, n) and t = gcd((r − 1)/2, n). We then show that the brick-product C(2n, 1, r) is Hamiltonian laceable if (i) st is even; (ii) s is odd and rs  r + 1 + 3s (mod 4n); or (iii) t is odd and rt  r − 1 − 3t (mod 4n). In general, when n is sufficiently large, say n ⩾ r2 − r + 1, then the brick-product is also Hamiltonian laceable.
18457324
On the higher-order edge toughness of a graph
For an integer c, 1?c??V(G)??1, we define the cth-order edge toughness of a graph G as tc(G)=min|X|?(G?X)?cX?E(G) & ?(G?X)>c The objective of this paper is to study this generalized concept of edge toughness. Besides giving the of the cth-order edge toughness ?c(G) of a graph G, we prove that `?c(G)?k if and only if G has k edge-disjoint spanning forests with exactly c components?. We also study the `balancity? of a graph G of order p and size q, which is defined as the smallest positive integer c such that ?c(G) = p/(p?c).
28223559
Constructions of point-colour-symmetric graphs
Throuhout this paper, we shall restrict ourselves to finite graphs G which are undirected and line-coloured. For any points x, y in G, we shall write x y to mean that x and y are adjacent and by x w1 y to mean that x y and the colour of the line joining x and y is i. Also, we shall denote the line joining x and y by Xy; the colour of Xy by c(xy); and the set {c(Xy)l x y in Gl by C(G). Let G and H be (undirected, line-coloured) graphs. A one to one mapping ffrom G onto His called an isomorphism if the following two conditions are satisfied:
41020572
Cycle Partitions in Graphs
In this paper, we prove that every graph contains a cycle intersecting all maximum independent sets. Using this, we further prove that every graph with stability number α is spanned by α disjoint cycles. Here, the empty set, the graph of order 1 and the path of order 2 are all considered as degenerate cycles.
2937830
Visualising Semantic Spaces and Author Co-Citation Networks in Digital Libraries
Abstract This paper describes the development and application of visualisation techniques for users to access and explore information in a digital library effectively and intuitively. Salient semantic structures and citation patterns are extracted from several collections of documents, including the ACM SIGCHI Conference Proceedings (1995–1997) and ACM Hypertext Conference Proceedings (1987–1998), using Latent Semantic Indexing and Pathfinder Network Scaling. The unique spatial metaphor leads to a natural combination of search and navigation within the same semantic space in a 3-dimensional virtual world. Author co-citation patterns are visualised through a number of author co-citation maps in attempts to reveal the structure of the hypertext, including an overall co-citation map of 367 authors and three periodical maps. These maps highlight predominant research areas in the field. This approach provides a means for transcending the boundaries of collections of documents and visualising more profound patterns in terms of semantic structures and co-citation networks.
27005423
Binding number and toughness for matching extension
Abstract A graph G of even order is said to be k -extendable if every matching of size k in G can be extended to a 1-factor of G . Plummer (1988) showed that a graph G is k -extendable if tough ( G ) > k , and we here prove that G is also k -extendable if bind ( G ) > max {k, (7k + 13)/12}.
6671949
Identification of critical genes in microarray experiments by a Neuro-Fuzzy approach
Gene expression profiling by microarray technology is usually difficult to interpret into a simpler pattern. One approach to resolve the complexity of gene expression profiles is the application of artificial neural networks (ANNs). A potential difficulty in this strategy, however, is that the non-linear nature of ANN makes it essentially a 'black-box' computation process. Addition of a fuzzy logic approach is useful because it can complement ANN by explicitly specifying membership function during computation. We employed a hybrid approach of neural network and fuzzy logic to further analyze a published microarray study of gene responses to eight bacteria in human macrophages. The original analysis by hierarchical clustering found common gene responses to all bacteria but did not address individual responses. Our method allowed exploration of the gene response of the host to individual bacterium. We implemented a two-layer, feed-forward neural network containing the principle of 'competitive learning' (i.e. 'winner-take-all'). The weights of the trained neural network were fed into a fuzzy logic inference system. A new measurement, called the impact rating (IR) was also introduced to explore the degree of importance of each gene. To assess the reliability of the IR value, a bootstrap re-sampling method was applied to the dataset and a confidence level for each IR was obtained. Our approach has successfully uncovered the unique features of host response to individual bacterium. Further, application of gene ontology (GO) annotation to the genes of high IR values in each response has suggested new biological pathways for individual host-pathogen interactions.
2867815
Matchings and matching extensions in graphs
Abstract Let G be a graph with a perfect matching and k be an integer such that 1 ⩽ k vbV ( G ) vb /2. Then G is said to be k -extendable if every matching of size k in G extends to a perfect matching of G . Plummer (1994) proved that every (2 k + 1)-connected K 1,3 -free graph of even order is k -extendable. In this paper, it was proved that every (2 k + n − 2)-connected K 1, n -free graph of even order is k -extendable. Also, an answer to the problem of characterizing maximal k -extendable graphs posted by Plummer (1994) and Saito (1989/90) is given. Besides, we show that a regular graph of even order belongs to the first class if its any two odd cycles have at least a vertex in common.
45076304
Spatial Semantic Virtual Worlds: The Structural Duality and Beyond