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A. Y. Alfakih

Publications and source records attributed to A. Y. Alfakih.

14 recordsLinked to original sources
arXiv2025

A note on Global Positioning System (GPS) and Euclidean distance matrices

Let $D$ be an $n \times n$ Euclidean distance matrix (EDM) with embedding dimension $r$; and let $d \in R^n$ be a given vector. In this note, we consider the problem of finding a vector $y \in R^n$, that is closest to d in Euclidean norm, such that the augmented matrix $\left[ \begin{array}{cc} 0 & y^T \\ y & D \end{array}\right]$ is itself an EDM of embedding dimension $r$. This problem is motivated by applications in Global Positioning System (GPS). We present a fault detection criterion and three algorithms: one for the case $n=4$, and two for the case $n \geq 5$.

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arXiv2020

On the Colin de Verdiere graph number and penny graphs

The Colin de Verdiere number of graph G, denoted by μ(G), is a spectral invariant of G that is related to some of its topological properties. For example, μ(G) \leq 3 iff G is planar. A penny graph is the contact graph of equal-radii disks with disjoint interiors in the plane. In this note we prove lower bounds on μ(G) when the complement \bar{G} is a penny graph.

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arXiv2019

On Unit Spherical Euclidean Distance Matrices Which Differ in One Entry

A unit spherical Euclidean distance matrix (EDM) D is a matrix whose entries can be realized as the interpoint (squared) Euclidean distances of n points on a unit sphere. In this paper, given such a D and 1 \leq k < l \leq n, we present a characterization of the set of all unit spherical EDMs whose entries agree with those of D except possibly with the entry in the klth and lkth positions. As a result, we show that this set can be discrete, consisting of one or two elements, or it can be continuous. The results are derived using two alternative approaches, the second of which is based on Cayley-Menger matrices.

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arXiv2018

On Representations of Graphs as Two-Distance Sets

Let a \neq b be two positive scalars. A Euclidean representation of a simple graph G in R^r is a mapping of the nodes of G into points in R^r such that the squared Euclidean distance between any two points is a if the corresponding nodes are adjacent and b otherwise. A Euclidean representation is spherical if the points lie on an (r-1)-sphere, and is J-spherical if this sphere has radius 1 and a=2 < b. Let dim_E(G), dim_S(G) and dim_J(G) denote, respectively, the smallest dimension r for which G admits a Euclidean, spherical and J-spherical representation. In this paper, we extend and simplify the results of Roy[18] and Nozaki and shinohara[17] by deriving exact simple formulas for dim_E(G) and dim_S(G) in terms of the eigenvalues of V^TAV, where A is the adjacency matrix of G and V is the matrix whose columns form an orthonormal basis for the orthogonal complement of the vector of all 1's. We also extend and simplify the results of Musin [16] by deriving explicit formulas for determining the J-spherical representation of G and for determining dim_J(G)in terms of the largest eigenvalue of \bar{A}, the adjacency matrix of the complement graph \bar{G}. As a byproduct, we obtain several related results and in particular we answer a question raised by Musin in [16].

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arXiv2018

On yielding and jointly yielding entries of Euclidean distance matrices

An $n \times n$ matrix D is a Euclidean distance matrix (EDM) if there exist $p^1, \ldots, p^n$ in some Euclidean space such that $d_{ij} = || p^i - p^j||^2$ for all $i,j=1,\ldots,n$. Let D be an EDM and let $E^{ij}$ be the $n \times n$ symmetric matrix with 1's in the $ij$th and $ji$th entries and 0's elsewhere. We say that $[l_{ij},u_{ij}]$ is the yielding interval of entry $d_{ij}$ if it holds that $D+t E^{ij}$ is an EDM iff $l_{ij} \leq t \leq u_{ij}$. If the yielding interval of entry $d_{ij}$ has length 0, i.e., if $l_{ij}=u_{ij}$, then $d_{ij}$ is said to be unyielding. Otherwise, if $l_{ij} \neq u_{ij}$, then $d_{ij}$ is said to be yielding. Let $d_{ij}$ and $d_{ik}$ be two unyielding entries of $D$. We say that $d_{ij}$ and $d_{ik}$ are jointly yielding if $D+t_1 E^{ij} + t_2 E^{ik}$ is an EDM for some nonzero scalars $t_1$ and $t_2$. In this paper, we characterize the yielding and the jointly yielding entries of an EDM D in terms of Gale transform of $p^1,\ldots,p^n$. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where $p^1,\ldots,p^n$ are in general position.

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arXiv2015

Universal Rigidity of Bar Frameworks via the Geometry of Spectrahedra

A bar framework (G,p) in dimension r is a graph G whose vertices are points p^1,...,p^n in R^r and whose edges are line segments between pairs of these points. Two frameworks (G,p) and (G,q) are equivalent if each edge of (G,p) has the same (Euclidean) length as the corresponding edge of (G,q). A pair of non-adjacent vertices i and j of (G,p)is universally linked if ||p^i-p^j||=||q^i-q^j|| in every framework (G,q) that is equivalent to (G,p). Framework (G,p) is universally rigid iff every pair of non-adjacent vertices of (G,p) is universally linked. In this paper, we present a unified treatment of the universal rigidity problem based on the geometry of spectrahedra. A spectrahedron is the intersection of the positive semidefinite cone with an affine space. This treatment makes it possible to tie together some known, yet scattered, results and to derive new ones. Among the new results presented in this paper are: (i) A sufficient condition for a given pair of non-adjacent vertices of (G,p) to be universally linked. (ii) A new, weaker, sufficient condition for a framework (G,p) to be universally rigid thus strengthening the existing known condition. An interpretation of this new condition in terms of the Strong Arnold Property and transversal intersection is also presented.

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arXiv2014

Graph connectivity and universal rigidity of bar frameworks

Let $G$ be a graph on $n$ nodes. In this note, we prove that if $G$ is $(r+1)$-vertex connected, $1 \leq r \leq n-2$, then there exists a configuration $p$ in general position in $R^r$ such that the bar framework $(G,p)$ is universally rigid. The proof is constructive and is based on a theorem by Lovasz et al concerning orthogonal representations and connectivity of graphs [12,13].

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arXiv2014

On Farkas Lemma and Dimensional Rigidity of Bar Frameworks

We present a new semidefinite Farkas lemma involving a side constraint on the rank. This lemma is then used to present a new proof of a recent characterization, by Connelly and Gortler, of dimensional rigidity of bar frameworks.

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arXiv2013

On affine motions and universal rigidity of tensegrity frameworks

Recently, Alfakih and Ye [Lin. Algebra Appl. 438:31--36, 2013] proved that if an $r$-dimensional bar framework $(G,p)$ on $n \geq r+2$ nodes in general position in $\R^r$ admits a positive semidefinite stress matrix with rank $n-r-1$, then $(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration $p$, each point and its neighbors in $G$ affinely span $\R^r$.

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arXiv2012

A Remark on the Manhattan Distance Matrix of a Rectangular Grid

Consider the Quadratic Assignment Problem (QAP): given two matrices A and D, minimize {trace AXDX^T: X is a permutation matrix}. New lower bounds were obtained recently (Mittelmann and peng [8]) for the QAP where D is either the Manhattan distance matrix of a rectangular grid, or the Hamming distance of a hypercube. In this note, we show that the results in [8,11] extend to the case where D is a spherical Euclidean distance matrix, which includes the Manhattan distance matrix and the Hamming distance matrix as special cases.

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arXiv2012

On stress matrices of chordal bar frameworks in general position

A bar framework in R^r, denoted by G(p), is a simple connected graph G whose vertices are points p^1,...,p^n in R^r that affinely span R^r, and whose edges are line segments between pairs of these points. In this paper, we use stress matrices to characterize the universal and global rigidities of chordal bar frameworks in general position in R^r, i.e., bar frameworks where graph G is chordal and the points p^1,...,p^n are in general position in R^r. We also prove that if a chordal bar framework in R^r admits a stress matrix of rank n-r-1 with generic rank profile, then it admits a positive semidefinite stress matrix of rank n-r-1.

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arXiv2012

Universal rigidity of bar frameworks in general position: a Euclidean distance matrix approach

A configuration p in r-dimensional Euclidean space is a finite collection of labeled points p^1,p^2,...,p^n in R^r that affinely span R^r. Each configuration p defines a Euclidean distance matrix D_p = (d_ij) = (||p^i-p^j||^2), where ||.|| denotes the Euclidean norm. A fundamental problem in distance geometry is to find out whether or not, a given proper subset of the entries of D_p suffices to uniquely determine the entire matrix D_p. This problem is known as the universal rigidity problem of bar frameworks. In this chapter, we present a unified approach for the universal rigidity of bar frameworks, based on Euclidean distance matrices (EDMs), or equivalently, on projected Gram matrices. This approach makes the universal rigidity problem amenable to semi-definite programming methodology. Using this approach, we survey some recently obtained results and their proofs, emphasizing the case where the points p^1,...,p^n are in general position.

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arXiv2010

On affine motions and bar frameworks in general position

A configuration p in r-dimensional Euclidean space is a finite collection of points (p^1,...,p^n) that affinely span R^r. A bar framework, denoted by G(p), in R^r is a simple graph G on n vertices together with a configuration p in R^r. A given bar framework G(p) is said to be universally rigid if there does not exist another configuration q in any Euclidean space, not obtained from p by a rigid motion, such that ||q^i-q^j||=||p^i-p^j|| for each edge (i,j) of G. It is known that if configuration p is generic and bar framework G(p) in R^r admits a positive semidefinite stress matrix S of rank n-r-1, then G(p) is universally rigid. Connelly asked whether the same result holds true if the genericity assumption of p is replaced by the weather assumption of general position. We answer this question in the affirmative in this paper.

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