7000090
On isomorphisms of finite Cayley graphs--a survey
The isomorphism problem for Cayley graphs has been extensively investigated over the past 30 years. Recently, substantial progress has been made on the study of this problem, many long-standing open problems have been solved, and many new research problems have arisen. The results obtained, and methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the finite simple group classification, through to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.
27555290
Constructing Homogeneous Factorisations of Complete Graphs and Digraphs
Abstract. A homogeneous factorisation of a complete graph Kn is a partition of the edge set that is invariant under a subgroup G of Sn such that G is transitive on the parts of the partition and induces a vertex-transitive automorphism group on the graph corresponding to each part. A product construction is given for such factorisations.
10480954
Isomorphisms of Finite Cayley Digraphs of Bounded Valency
LetGbe a finite group,Sa subset ofG\{1}, and let Cay(G,S) denote the Cayley digraph ofGwith respect toS. If, for all subsetsS,TofG\{1} of size at mostm, Cay(G,S)?Cay(G,T) implies thatS?=Tfor some??Aut(G), thenGis called anm-DCI-group. In this paper, we prove that, form?2, allm-DCI-groups are of the formU×V, where (|U|, |V|)=1,Uis abelian andVbelongs to an explicitly determined list of groups. Moreover Sylow subgroups of such groups satisfy some very restrictive conditions.
23350440
On Isomorphisms of Connected Cayley Graphs, II
A Cayley graph or digraph Cay(G, S) of a finite groupGis called aCI-graphofGif, for anyT?G, Cay(G, S)?Cay(G, T) if and only ifS?=Tfor some??Aut(G). We study the problem of determining which Cayley graphs and digraphs for a given group are CI-graphs. A finite groupGis called aconnected m-DCI-group(orconnected m-CI-group) if all connected Cayley digraphs (or connected Cayley graphs, respectively) ofGof (out)-valency at most m are CI-graphs. For a groupG, letp(G) be the smallest prime divisor of |G|. It was previously shown that all finite groupsGare connected (p(G)?1)-DCI-groups and all finite groupsGare connected 2(p(G)?1)-CI-groups. In this paper, for every primep, we construct infinitely many finite groupsGsuch thatp(G)=pandGis neither a connectedp-DCI-group nor a connected 2p-CI-group, which provides solutions for several open problems in this area.
206571122
On Half-Transitive Metacirculant Graphs of Prime-Power Order
In this paper, infinitely many graphs of large valency which are half-transitive (that is vertex- and edge-transitive but not arc-transitive) are constructed, and a complete classification is given of half-transitive metacirculant graphs of order a p-power and valency less than 2p, where p is a prime. In particular, it is shown that, for any odd prime p, integers n?3 and k?2 such that k divides p?1, there are exactly (pn?2?1)/2+pn?3?1 nonisomorphic connected half-transitive metacirculants of order pn and valency 2k.
16649809
Relative difference sets fixed by inversion and Cayley graphs
Using graph theoretical technique, we present a construction of a (30, 2, 29, 14)-relative difference set fixed by inversion in the smallest finite simple group--the alternating group A5. To our knowledge this is the first example known of relative difference sets in the finite simple groups with a non-trivial forbidden subgroup. A connection is then established between some relative difference sets fixed by inversion and certain antipodal distance-regular Cayley graphs. With the connection, several families of antipodal distance-regular Cayley graphs which are coverings of complete graphs are presented.
17070310
Answering aggregation queries on hierarchical web sites using adaptive sampling
We study how to answer aggregation queries over hierarchical Web sites using adaptive sampling.
15590248
A Class of Discrete Multiresolution Random Fields and Its Application to Image Segmentation
In this paper, a class of Random Field model, defined on a multiresolution array is used in the segmentation of gray level and textured images. The novel feature of one form of the model is that it is able to segment images containing unknown numbers of regions, where there may be significant variation of properties within each region. The estimation algorithms used are stochastic, but because of the multiresolution representation, are fast computationally, requiring only a few iterations per pixel to converge to accurate results, with error rates of 1-2 percent across a range of image structures and textures. The addition of a simple boundary process gives accurate results even at low resolutions, and consequently at very low computational cost.
827639
Segmentation of edge preserving gradient vector flow: an approach toward automatically initializing and splitting of snakes
Active contours or snakes have been extensively utilized in handling image segmentation and classification problems. In traditional active contour models, snake initialization is performed manually by users, and topological changes, such as splitting of the snake, can not be automatically handled. In this paper, we introduce a new method to solve the snake initialization and splitting problem, based on an area segmentation approach: the external force field is segmented first, and then the snake initialization and splitting can be automatically performed by using the segmented external force field. Such initialization and splitting produces multiple snakes, each of which is within the capture range associated to an object and evolved to the object boundary. The external force used in this paper is a gradient vector flow with an edge-preserving property (EPGVF), which can prevent the snakes from passing over weak boundaries. To segment the external force field, we represent it with a graph, and a graph-theory approach can be taken to determine the membership of each pixel. Experimental results establish the effectiveness of the proposed approach.
34090759
Segmentation of external force field for automatic initialization and splitting of snakes
Active contours or snakes have been extensively utilized in handling image segmentation and classification problems. In traditional active contour models, snake initialization is performed manually by users, and topological changes, such as splitting of the snake, cannot be automatically handled. In this paper, we introduce a new method to solve the snake initialization and splitting problem, based on an area segmentation approach: the external force field is segmented first, and then the snake initialization and splitting can be automatically performed by using the segmented external force field. Such initialization and splitting produces multiple snakes, each of which is within the capture range associated to an object and will be evolved to the object boundary. The external force used in this paper is a gradient vector flow with an edge-preserving property (EPGVF), which can prevent the snakes from passing over weak boundaries. To segment the external force field, we represent it with a graph, and a graph-theory approach can be taken to determine the membership of each pixel. Experimental results establish the effectiveness of the proposed approach.
16331610
Calcification Descriptor and Relevance Feedback Learning Algorithms for Content-Based Mammogram Retrieval
In recent years a large number of digital mammograms have been generated in hospitals and breast screening centers. To assist diagnosis through indexing those mammogram databases, we proposed a content-based image retrieval framework along with a novel feature extraction technique for describing the degree of calcification phenomenon revealed in the mammograms and six relevance feedback learning algorithms, which fall in the category of query point movement, for improving system performance. The results show that the proposed system can reach a precision rate of 0.716 after five rounds of relevance feedback have been performed.