scholarly journals Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

2020 ◽  
Vol 8 (1) ◽  
pp. 160-171
Author(s):  
Joyentanuj Das ◽  
Sachindranath Jayaraman ◽  
Sumit Mohanty
Keyword(s):  
Symmetric Matrix ◽  
Nonnegative Matrix ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Positive Semidefinite ◽  
Positive Matrix ◽  
Completely Positive ◽  
Rank One ◽  
Principal Submatrices ◽  
Existing Result

AbstractA real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

scholarly journals Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Hiroshi Kurata ◽  
Ravindra B. Bapat
Keyword(s):  
Euclidean Distance ◽  
Symmetric Matrix ◽  
Distance Matrix ◽  
Positive Semidefinite ◽  
Complete Characterization ◽  
Symmetric Matrices ◽  
Euclidean Distance Matrix ◽  
Positive Semidefinite Matrices ◽  
Special Cases

AbstractBy a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D

scholarly journals The triangle graph $T_6$ is not SPN

2020 ◽  
Vol 36 (36) ◽  
pp. 90-93
Author(s):  
Stephen Drury
Keyword(s):  
Symmetric Matrix ◽  
Nonnegative Matrix ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Real Symmetric Matrix ◽  
Triangle Graph ◽  
Nonnegative Vector ◽  
Copositive Matrix ◽  
Semidefinite Matrix

A real symmetric matrix $A$ is copositive if $x'Ax \geq 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative matrix. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than $4$. A graph $G$ is an SPN graph if every copositive matrix whose graph is $G$ is SPN. We show that the triangle graph $T_6$ is not SPN.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

2013 ◽  
Vol 34 (2) ◽  
pp. 355-368 ◽  
Author(s):  
Naomi Shaked-Monderer ◽  
Immanuel M. Bomze ◽  
Florian Jarre ◽  
Werner Schachinger
Keyword(s):  
Positive Matrix ◽  
Completely Positive ◽  
Completely Positive Matrix

scholarly journals A note on completely positive relaxations of quadratic problems in a multiobjective framework

2021 ◽  
Author(s):  
Gabriele Eichfelder ◽  
Patrick Groetzner
Keyword(s):  
Positive Semidefinite ◽  
Weighted Sum ◽  
Quadratic Problem ◽  
Completely Positive ◽  
Nonconvex Problems ◽  
Completely Positive Matrices ◽  
Convex Problems ◽  
Positive Matrices ◽  
Nondominated Points ◽  
Single Objective

AbstractIn a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.

scholarly journals Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1529 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal Ahmad Ganie ◽  
Yilun Shang
Keyword(s):  
Spectral Radius ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Extremal Graphs ◽  
Special Cases ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Generalized Distance

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

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