Showing papers on "Vertex (graph theory) published in 1992"

The Subconstituent Algebra of an Association Scheme, (Part I)

Abstract: We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple \Bbb {C}-algebra T e T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y. In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter. We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”. We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur. We close with some conjectures and open problems.

Routing with polynomial communication-space trade-off

Abstract: This paper presents a family of memory-balanced routing schemes that use relatively short paths while storing relatively little routing information. The quality of the routes provided by a scheme is measured in terms of their stretch, namely, the maximum ratio between the length of a route connecting some pair of processors and their distance. The hierarchical schemes$\mathcal{H}_k $ (for every integer $k\geqq 1$) presented in this paper guarantee a stretch factor of $O( k^2 )$ on the length of the routes and require storing at most $O( k \cdot n^{1/k} \cdot \log n\log D )$ bits of routing information per vertex in an n-processor network with diameter D. The schemes are name independent and applicable to general networks with arbitrary edge weights. This improves on previous designs whose stretch bound was exponential in k.The proposed schemes are based on a new efficient solution to a certain graph-theoretic problem concerning sparse graph covers. The new cover technique has already found several other a...

Boundary K-Matrices for the Six Vertex and the n(2n-1) A_{n-1} Vertex Models

Abstract: Boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices $K^{\pm}(\theta)$. For the six vertex model the general solution depending on four arbitrary parameters is found. For the $A_{n-1}$ models all diagonal solutions are found. The associated integrable magnetic Hamiltonians are explicitly derived.

Recursive algorithm for evaluating vertex-type Feynman integrals

Abstract: An algorithm for evaluating vertex-type loop integrals is considered. It is based on applying the integration-by-parts technique. As an example, a class of massless integrals corresponding to triangle diagrams is considered. The presented method can also be applied to loop diagrams with a larger number of external lines as well as to integrals with massive denominators.

Computer-assisted analysis of mixtures (C.A.MAM): statistical algorithms.

Abstract: SUMMARY This paper presents various algorithmic approaches for computing the maximum likelihood estimator of the mixing distribution of a one-parameter family of densities and provides a unifying computeroriented concept for the statistical analysis of unobserved heterogeneity (i.e., observations stemming from different subpopulations) in a univariate sample. The case with unknown number of population subgroups as well as the case with known number of population subgroups, with emphasis on the first, is considered in the computer package C.A.MAN (Computer Assisted Mixture ANalysis). It includes an algorithmic menu with choices of the EM algorithm, the vertex exchange algorithm, a combination of both, as well as the vertex direction method. To ensure reliable convergence, a steplength menu is provided for the three latter methods, each achieving monotonicity for the direction of choice. C.A.MAN has the option to work with restricted support size-that is, the case when the number of components is known a priori. In the latter case, the EM algorithm is used. Applications of mixture modelling in medical problems are discussed.

The extreme eigenvalues and stability of real symmetric interval matrices

Abstract: A novel Khoritonov-like algorithm for computing the minimal and maximal eigenvalues of n*n dimensional symmetric interval matrices is presented. It is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2/sup n-1/ symmetric vertex matrices, whereas its minimal eigenvalue coincides with the minimal of another special set of 2/sup n-1/ symmetric vertex matrices. As immediate corollaries of this algorithm, weak necessary and sufficient conditions for testing the Hurwitz and Schur stability of symmetric interval matrices, where one has to test the stability of 2/sup n-1/ and 2/sup n/ symmetric vertex matrices, respectively, are obtained. >

Guaranteed convergence in a class of Hopfield networks

Abstract: A class of symmetric Hopfield networks with nonpositive synapses and zero threshold is analyzed in detail. It is shown that all stationary points have a one-to-one correspondence with the minimal vertex covers of certain undirected graphs, that the sequential Hopfield algorithm as applied to this class of networks converges in at most 2n steps (n being the number of neurons), and that the parallel Hopfield algorithm either converges in one step or enters a two-cycle in one step. The necessary and sufficient condition on the initial iterate for the parallel algorithm to converge in one step are given. A modified parallel algorithm which is guaranteed to converge in (3n/2) steps ((x) being the integer part of x) for an n-neuron network of this particular class is also given. By way of application, it is shown that this class naturally solves the vertex cover problem. Simulations confirm that the solution provided by this method is better than those provided by other known methods. >

Choice Numbers of Graphs: A Probabilistic Approach

Abstract: The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).

Using real numbers as vertex invariants for third-generation topological indexes

Abstract: First-generation topological indexes (TI’s) were integer numbers obtained by simple (“bookkeeping”) operations from local vertex invariants (LOVI’s), which were integer numbers. Second-generation TI’s were real numbers obtained via sophisticated (‘structural”) operations from integer LOVI’s. Third-generation TI’s are real numbers based on real-number LOVI’s. In successive generations, there is an increasing correlational abiiity and a decreasing degeneracy of TI’s. Four types of newly developed real-number LOVI’s are reviewed: (i) Information-based LOVI’s obtained from topological distances to all other graph vertexes; (ii) Solutions of linear equation systems obtained from triplets consisting of a matrix (adjacency or distance matrix) and two column vectors; (iii) LOVI’s based on eigenvalues and eigenvectors of the two above matrices; (iv) Regressive distance sums and regressive vertex degrees, which are the corresponding LOVI’s (distance sums or vertex degrees) augmented slightly by all other vertexes, whose contributions decrease with increasing distance. When the LOVI’s are based on topological distances, it is easy to include information on the presence and location of multiple bonds and/or heteroatoms. All LOVI’s are validated by intramolecular comparison within various alkanes, and all TI’s are validated both by intermolecular comparison within series of isomeric alkanes and by correlations with physicochemical properties.

A note on bipartite subgraphs of triangle‐free graphs

Abstract: Let G be a triangle‐free graph on n points with m edges and vertex degrees d1, d2,…, dn. Let k be the maximum number of edges in a bipartite subgraph of G. In this note we show that k ⩾ m/2 + Σ **image** √di. It follows as a corollary that k ⩾ m/2 + cm3/4. © 1992 Wiley Periodicals, Inc.

Weber's problem with attraction and repulsion

Abstract: . Weber's problem consists of locating a facility in the plane in such a way that the weighted sum of Euclidean distances to n given points be minimum. The case where some weights are positive and some are negative is shown to be a d.-c. program (i.e., a global optimization problem with both the objective function and constraint functions written as differences of convex functions), reducible to a problem of concave minimization over a convex set. The reduced problem is then solved by outer-approximation and vertex enumeration. Moreover, locational constraints can be taken into account by combining the previous algorithm with an enumerative procedure on the set of feasible regions. Finally, the algorithm is extended to solve the case where obnoxiousness of the facility is assumed to have exponential decay. Computational experience with n up to 1000 is described.

Uninodal 4‐connected 3D nets. I. Nets without 3‐ or 4‐rings

Abstract: A description is given of 4-connected nets with one kind of vertex in which the shortest rings containing each pair of edges are N-rings (N > 4). Eleven uniform nets (66) are identified; seven of these are believed to be new. A further thirteen nets with one type of vertex and without 3- or 4-rings are described; nine of these are also believed to be new

On the irregularity strength of the m x n grid

Abstract: Given a graph G with weighting w: E(G) Z+, the Strength of G(w) is the maximum weight on any edge. The sum of a vertex in G(w) is the sum of the weights of all its incident edges. The network G(w) is irregular if the vertex sums are distinct. The irregularity strength of G is the minimum strength of the graph under all irregular weightings. In this paper we determine the irregularity strength of the m × n grid for certain m and n. In particular, for every positive integer d we find the irregularity strength for all but a finite number of m × n grids where n - m = d. In addition, we present a general lower bound for the irregularity strength of graphs. © 1992 John Wiley & Sons, Inc.

IEEE Trans. Reliab.

Abstract: A stochastic network is considered the underlying deterministic graph of which is assumed to be undirected and connected. This graph is split into two edge—disjoint subgraphs by a given separating vertex set. Reliability analysis is carried out for the corresponding stochastic subgraphs and the results are combined to obtain a formula of the K—terminal reliability of the stochastic network. Special cases are discussed in detail and numerical examples are presented.

Chain Lengths in Certain Random Directed Graphs

Abstract: We study the random directed graph with vertex set {1, …, n} in which the directed edges (i, j) occur independently with probability cn/n for i

New algorithms for the intersection problem of submodular systems

Abstract: We consider the problem of finding a maximum common subbase of two submodular systems onE with |E|=n. First, we present a new algorithm by finding the shortest augmenting paths, which begins with a pair of subbases of the given submodular systems and is convenient for adopting the preflow-push approach of Goldberg and Tarjan. Secondly, by using the basic ideas of the preflow-push method, we devise a faster algorithm for the intersection problem, which requiresO(n3) push andO(n2) relabeling operations in total by the largest-label implementation with a specific order on the arc list of each vertex in the auxiliary graph.

Cogrowth of Regular Graphs

Abstract: Let & be a ?-regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Lapla- cian on 9 is zero. Grigorchuk's criterion for amenability of finitely generated groups follows. In this note, we shall relate the first eigenvalue of the Laplacian on a connected regular graph to the size of the kernel of the universal covering map. The main results have been proven in (C, G, P). The proof presented here appears simpler; it depends on the explicit formula for minimal positive solutions of AF + eF = -I. Let f be a connected simple graph with constant vertex degree d > 3, F be the universal covering tree of "§, and 6 the covering map (i.e., 6 is a vertex surjection of T on ^ that preserves adjacency and vertex degree). We let T and & denote the vertex sets of the corresponding graphs. Note that F has constant vertex degree d. Since F is connected, F may be considered a metric space with the usual graph metric a (S(x, y) is the length of the shortest path connecting x and y). For x £ T and zz > 0, let (x) = 6~x(6(x)) and Sn(x) = {y: S(x, y) = n} . For x, y £ T, note that

Tournament-like oriented graphs

Abstract: A local tournament is an oriented graph in which the inset as well as the outset of each vertex induces a tournament. Local tournaments possess many properties of tournaments and have interesting structure. In 1982, Skrien proved (in different terminology), using a deep structural characterization of proper circular arc graphs by Tucker, that a connected graph is local-tournament-orientable if and only if it is a proper circular arc graph. In Chapter 2, we shall give a simple O($m\Delta$) algorithm to decide if a graph can be oriented as a local tournament, and hence whether or not it is a proper circular arc graph. We analyze relationships among local tournaments, local transitive tournaments, and proper circular arc graphs. We obtain theorems to describe all possible local-tournament orientations of a proper circular arc graph. In Chapter 3, we shall present an O($m\Delta$) algorithm to recognize comparability graphs and to calculate transitive orientations. Our method can be applied to recognize proper circular arc graphs and to find local-transitive-tournament orientations, and can also be applied to recognize proper interval graphs and to find acyclic local-tournament orientations. We shall give a simple proof of Skrien's theorem, which does not depend on Tucker's result. In Chapter 4, we shall present two O(m + n) time algorithms. One is for recognizing proper interval graphs and for finding an associated interval family. The other is for recognizing proper circular arc graphs and for finding an associated circular arc family. In Chapter 5, we shall obtain two additional O(m + n) time algorithms for proper circular arc graphs by using the auxiliary local-tournament orientations. One is for finding maximum cliques, and the other is for determining c-colourability. In Chapter 6, we shall introduce a new class of oriented graphs, namely, in-tournaments, which contains the class of local tournaments. We shall show that some of the basic and very nice properties of tournaments extend not only to local tournaments, but also to this more general class of digraphs. Our results imply a polynomial time algorithm for finding hamiltonian paths and cycles in the class of in-tournaments. We shall also investigate the class of graphs which are orientable as in-tournaments. Finally, in Chapter 7, we shall introduce another class of oriented graphs, i.e., those of Moon type. We shall find a close relationship between the class of oriented graphs of Moon type and the class of local tournaments. In fact, oriented graphs of Moon type can be characterized in terms of local transitive tournaments.

On super-edge-connected digraphs and bipartite digraphs

Abstract: A maximally edge-connected digraph is called super-λ if every minimum edge disconnecting set is trivial, i.e., it consists of the edges adjacent to or from a given vertex. In this paper sufficient conditions for a digraph to be super-λ are presented in terms of parameters such as diameter and minimum degree. Similar results are also given for bipartite digraphs. As a corollary, some characterizations of super-λ graphs and bipartite graphs are obtained. © 1929 John Wiley & Sons, Inc.

Uninodal 4‐connected 3D nets. II. Nets with 3‐rings

Abstract: A description is given of nineteen 4-connected nets with one kind of vertex and containing 3-rings. Many of them are believed to be new.