24184742
On Distance Two Labellings of Graphs
17515399
Sizes of Graphs with Fixed Orders and Spans for Circular-Distance-Two Labelings
A k-circular-distance-two labeling (or k-c-labeling) of a simple graph G is a vertex-labeling, using the labels 0, 1, 2, · · · , k − 1, such that the “circular difference” (mod k) of the labels for adjacent vertices is at least two, and for vertices of distance-two apart is at least one. The σ-number, σ(G), of a graph G is the minimum k of a k-c-labeling of G. For any given positive integers n and k, let G(n, k) denote the set of graphs G on n vertices and σ(G) = k. We determine the maximum size (number of edges) and the minimum size of a graph G ∈ G(n, k). Furthermore, we prove that for any value p between the maximum and the minimum size, there exists a graph G ∈ G(n, k) of size p. These results are analogues of the ones by Georges and Mauro [4] on distance-two labelings.
45408024
Star Extremal Circulant Graphs
A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number (also known as the star chromatic number). We prove that members of a certain family of circulant graphs are star extremal. The result generalizes some known theorems of Sidorenko [Discrete Math., 91 (1991), pp. 215--217] and Gao and Zhu [Discrete Math., 152 (1996), pp. 147--156]. We show relations between circulant graphs and distance graphs and discuss their star extremality. Furthermore, we give counterexamples to two conjectures of Collins [SIAM J. Discrete Math., 11 (1998), pp. 330--339] on asymptotic independence ratios of circulant graphs.
16699960
Circular chromatic number for iterated Mycielski graphs
Abstract For a graph G , let M( G ) denote the Mycielski graph of G . The t th iterated Mycielski graph of G , M t ( G ), is defined recursively by M 0 ( G )= G , and M t ( G )=M(M t −1 ( G )) for t ⩾1. Let χ c ( G ) denote the circular chromatic number of G . We prove two main results: (1) If G has a universal vertex x , then χ c (M( G ))= χ (M( G )) if χ c (G−x)>χ(G)− 1 2 and G is not a star, otherwise χ c ( M (G))=χ( M (G))− 1 2 ; and (2) χ c (M t ( K m ))= χ (M t ( K m )) if m ⩾2 t −1 +2 t −2 and t ⩾2.
5450384
Gradient-based methods for simultaneous blind separation of mixed source signals
This paper presents gradient-based methods for simultaneous blind separation of arbitrarily mixed source signals. We consider the regular case where the mixing matrix has full column rank as well as ill-conditioned cases. Two cost functions based on fourth-order cumulants are introduced to simultaneously separate all separable single sources and all inseparable mixtures. By minimizing the cost functions, two gradient-based methods are developed. Our algorithms derived from gradient-based methods are guaranteed to converge. Finally, simulation results show the effectiveness of our methods.
38948507
Analysis and synthesis of a class of neural networks with sparse interconnections
The authors present results for the analysis and synthesis of a class of neural networks with sparse or partial interconnecting structure. The, design procedure guarantees to synthesize neural networks with arbitrarily predetermined sparse interconnection structures and to store any given set of desired bipolar patterns as memories. It is shown that a sufficient condition for the existence of a design for neural networks with sparse interconnection is self-feedback for every neuron in the network. Several specific examples are included to demonstrate the applicability of the methodology.<<ETX>>
1708763
Multilevel Distance Labelings for Paths and Cycles
For a graph $G$, let $\diam(G)$ denote the diameter of $G$. For any two vertices $u$ and $v$ in $G$, let $d(u, v)$ denote the distance between $u$ and $v$. A multilevel distance labeling (or distance labeling) for $G$ is a function $f$ that assigns to each vertex of $G$ a nonnegative integer such that for any vertices $u$ and $v$, $|f(u)-f(v)| \geq \diam(G) - d_G(u, v) +1$. The span of $f$ is the largest number in $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the minimum span of a distance labeling for $G$. In this paper, we completely determine the radio numbers for paths and cycles.
36089150
Hamiltonicity and circular distance two labellings
Abstract A k -circular distance two labelling (or k -c-labelling) of a graph G is a vertex-labelling such that the circular difference ( mod k ) of the labels is at least two for adjacent vertices, and at least one for vertices at distance two. Given G , denote σ(G) the minimum k for which there exists a k -c-labelling of G . Suppose G has n vertices, we prove σ(G)⩽n if G c is Hamiltonian; and σ(G)=n+p v (G c ) otherwise, where p v (G) is the path covering number of G . We give exact values of σ(G) for some families of graphs such that G c is Hamiltonian, and discuss injective k -c-labellings especially for joins and unions of graphs.
12746246
JRV: an interactive tool for data mining visualization
In this paper, we demonstrate JRV, a new data mining visualization tool for the knowledge discovery process where the user and computer can cooperate with each other. First, the computer can be instructed by the user interactively to compute values of several evaluation functions. Then, the user can take advantage of domain knowledge and assess the intermediate results obtained. Furthermore, by providing effective and efficient data visualization, the pattern recognition capacities of users can be greatly improved. Instead of being limited to two attributes at a given time in independence diagrams, this novel tool will allow simultaneous analyses of multiple attribute dependencies using four different drawing panels. Also, by utilizing the existing techniques of data visualization, we design a general model which can handle both categorical and numerical attributes in an intuitive way. With this model, we can identify patterns of interests efficiently. Through actual examples, we show that it might help users to find novel attribute relationships. This work is supported by NIH grant #RO1-CA98932-01.