Graph rigidity via Euclidean distance matrices

Let G=(V,E,ω) be an incomplete graph with node set V, edge set E, and nonnegative weights ω_ij's on the edges. Let each edge (v_i,v_j) be viewed as a rigid bar, of length ω_ij, which can rotate freely around its end nodes. A realization of a graph G is an assignment of coordinates, in some Euclidean space, to each node of G. In this paper, we consider the problem of determining whether or not a given realization of a graph G is rigid. We show that each realization of G can be epresented as a point in a compact convex set ; and that a generic realization of G is rigid if and only if its corresponding point is a vertex of Ω, i.e., an extreme point with full-dimensional normal cone.     

Extremality of stochastic matrices in nonusual cones

The set of doubly stochastic matrices can be regarded as a set in a cone. Thus, one can perturb the cone in different ways, in such a way that it is important to characterize the new extremals. In addition, we study some special families of cones obtaining the corresponding extremals.This paper has been supported by a grant from CONICET, Buenos Aires, Argentina.     

Recursive Construction of Optimal Self-Concordant Barriers for Homogeneous Cones

We give a recursive formula for optimal dual barrier functions on homogeneous cones. This is done in a way similar to the primal construction of Güler and Tunçel (Math. Program. 81(1):55–76, 1998) by means of the dual Siegel cone construction of Rothaus (Bull. Am. Math. Soc. 64:85–86, 1958). We use invariance of the primal barrier function with respect to a transitive subgroup of automorphisms and the properties of the duality mapping, which is a bijection between the primal and the dual cones. We give simple direct proofs of self-concordance of the primal optimal barrier and provide an alternative expression for the dual universal barrier function.     

Two theorems on Euclidean distance matrices and Gale transform

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ(E−C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones

where is the elliptope (set of correlation matrices) and is the (closed convex) cone of EDMs.

The characterization is given using the Gale transform of the points generating D. We also show that given points , for any scalars λ₁,λ₂,…,λ_n such that

∑_j=1ⁿλ_jp^j=0, ∑_j=1ⁿλ_j=0,

we have

∑_j=1ⁿλ_jpⁱ−p^j2= forall i=1,…,n,

for some scalar independent of i.     

Developments on spectral characterizations of graphs

In [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.     

Recognizing Helly Edge-Path-Tree graphs and their clique graphs

We present a unifying procedure for recognizing intersection graphs of Helly families of paths in a tree and their clique graphs. The Helly property makes it possible to look at these recognition problems as variants of the Graph Realization Problem, namely, the problem of recognizing Edge-Path-Tree matrices. Our result heavily relies on the notion of pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8-22] and on the observation that Helly Edge-Path-Tree matrices form a self-dual class of Helly matrices. Coupled to the notion of reduction presented in the paper, these facts are also exploited to reprove and slightly refine some known results for Edge-Path-Tree graphs.     

On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B₁,B₂ of a certain ten element matroid. Given that B₁,B₂ are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.     

The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W _n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W _n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.     

On the integrality of an extreme solution to pluperfect graph and balanced systems

Let A be a nonnegative integer matrix, and let e denote the vector all of whose components are equal to 1. The pluperfect graph theorem states that if for all integer vectors b the optimal objective value of the linear program minse′xvbAx ? b, x ? 0 s is integer, then those linear programs possess optimal integer solutions. We strengthen this theorem and show that any lexicomaximal optimal solution to the above linear program (under any arbitrary ordering of the variables) is integral and an extreme point of sxvbAx ? b, x ? 0 s. We note that this extremality property of integer solutions is also shared by covering as well as packing problems defined by a balanced matrix A.     

Editorial Policy

Consider the parametric linear complementarity problem w=Mz+q+p, w0, z0, w ^T z=0, where p0, 0q0, and 0. We show that a necessary condition for every complementary map z() to be isotone for every nonzero q0 and every p is that M be either a P-matrix or a -matrix. The Cottle necessary and sufficient conditions for strong and uniform isotonicity for P-matrices are restated, with slight modifications, for -matrices.     

Solvability Theory and Projection Methods for a Class of Singular Variational Inequalities: Elastostatic Unilateral Contact Applications

The mathematical modeling of engineering structures containing members capable of transmitting only certain type of stresses or subjected to noninterpenetration conditions along their boundaries leads generally to variational inequalities of the form , where C is a closed convex set of (kinematically admissible set), (loading strain vector), and (stiffness matrix). If rigid body displacements and rotations cannot be excluded from these applications, then the resulting matrix M is singular and serious mathematical difficulties occur. The aim of this paper is to discuss the existence and the numerical computation of the solutions of problem (P) for the class of cocoercive matrices. Our theoretical results are applied to two concrete engineering problems: the unilateral cantilever problem and the elastic stamp problem.     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

On Balanced Graphs

Berge defined a hypergraph to be balanced if its incidence matrix is balanced. We consider this concept applied to graphs, and call a graph to be balanced when its clique matrix is balanced. Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs. Two of them are characterized by 0–1 matrices and can be recognized in polynomial time. Furthermore, we propose polynomial time combinatorial algorithms for the problems of stable set, clique-independent set and clique-transversal for one of these subclasses of balanced graphs. Finally, we analyse the behavior of balanced graphs and these four subclasses under the clique graph operator. Received: April, 2004