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Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Θ(√n) distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by... 

Let G be a simple connected graph of order n. Let (Formula presented.) be the Laplacian eigenvalues of G. In this paper, we show that if X and Y are two subsets of vertices of G such that (Formula presented.) and the set of all edges between X and Y decomposed into r disjoint perfect matchings, then, (Formula presented.) where (Formula presented.). Also, we determine a relation between the Laplacian eigenvalues and matchings in a bipartite graph by showing that if (Formula presented.) is a bipartite graph, (Formula presented.) and (Formula presented.), then G has a matching that saturates U. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group  

Let G be a graph and M be a matrix associated with G whose characteristic polynomial is M(G,x)=∑i=0nαi(G)xn-i. We say that the spectrum of G is determined by non-constant coefficients (simply M-SDNC), if for any graph H with ai(H)=ai(G), 0≤i≤n-1, then Spec(G)=Spec(H) (if M is the adjacency matrix or the Laplacian matrix of G, then G is called an A-SDNC graph or L-SDNC graph). In this paper, we study some properties of graphs which are A-SDNC or L-SDNC. Among other results, we prove that the path of order at least five is L-SDNC and moreover stars of order at least five are both A-SDNC and L-SDNC. Furthermore, we construct infinitely many trees which are not A-SDNC graphs. More precisely, we... 

Let G be a graph and L(G) be the Laplacian matrix of G. In this paper, we explicitly determine the multiplicity of Laplacian eigenvalue 2 for any unicyclic graph containing a perfect matching  

In this paper, we present sufficient conditions to address a larger class of digraphs, including balanced ones, whose members' Laplacian (L) makes L 1L + LTL1 to be positive semi-definite, where L1 is the Laplacian associated with a fully connected equally-edged weighted graphs. This property can be later utilized to introduce an appropriate energy function for stability analysis of flocking algorithms in a larger class of networks with switching directed information flow. Also, some of their properties are investigated in the line of matrix theory and graph theory  

This brief proposes a joint graph signal recovery and topology learning algorithm using a Variational Bayes (VB) framework in the case of non-Gaussian measurement noise. It is assumed that the graph signal is Gaussian Markov Random Field (GMRF) and the graph weights are considered statistical with the Gaussian prior. Moreover, the non-Gaussian noise is modeled using two distributions: Mixture of Gaussian (MoG), and Laplace. All the unknowns of the problem which are graph signal, Laplacian matrix, and the (Hyper)parameters are estimated by a VB framework. All the posteriors are calculated in closed forms and the iterative VB algorithm is devised to solve the problem. The efficiency of the...