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Ravindra B. Bapat  One of the best experts on this subject based on the ideXlab platform.

Squared Distance Matrix of a weighted tree
Electronic Journal of Graph Theory and Applications, 2019CoAuthors: Ravindra B. BapatAbstract:Let T be a tree with vertex set {1, …, n } such that each edge is assigned a nonzero weight. The squared Distance Matrix of T , denoted by Δ, is the n × n Matrix with ( i , j ) element d ( i , j ) 2 , where d ( i , j ) is the sum of the weights of the edges on the ( i j ) path. We obtain a formula for the determinant of Δ. A formula for Δ − 1 is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.

Squared Distance Matrix of a weighted tree
arXiv: Combinatorics, 2018CoAuthors: Ravindra B. BapatAbstract:Let $T$ be a tree with vertex set $\{1, \ldots, n\}$ such that each edge is assigned a nonzero weight. The squared Distance Matrix of $T,$ denoted by $\Delta,$ is the $n \times n$ Matrix with $(i,j)$element $d(i,j)^2,$ where $d(i,j)$ is the sum of the weights of the edges on the $(ij)$path. We obtain a formula for the determinant of $\Delta.$ A formula for $\Delta^{1}$ is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.

Moore–Penrose inverse of a Euclidean Distance Matrix
Linear Algebra and its Applications, 2015CoAuthors: Hiroshi Kurata, Ravindra B. BapatAbstract:Abstract We obtain expressions for the Moore–Penrose inverse of a Euclidean Distance Matrix (EDM) that are determined only by the positive semidefinite Matrix associated with the EDM. The results complement formulas for the Moore–Penrose inverse of an EDM given in Balaji and Bapat (2007) [2] . A formula for the inverse of a principal subMatrix of an EDM is also derived, whose expression uses the Schur complement of the Laplacian of the EDM. As an application, we obtain an expression for the terminal Wiener index of a tree.

Distance Matrix of a Tree
Universitext, 2014CoAuthors: Ravindra B. BapatAbstract:The classical Distance between two vertices of a graph, which is defined to be the length of the shortest path, is often intuitively not appealing, and is also not tractable mathematically. The resistance Distance between two vertices is defined to be the effective resistance between the vertices, when the graph is viewed as an electrical network, with a unit resistance along each edge. We begin by giving an equivalent definition in terms of generalized inverse, and prove some basic properties, including the triangle inequality. In the next few sections, interpretaion of the resistance Distance in terms of network flows, random walk on the graph and electrical networks is provided. In the final section some properties of the resistance Matrix are proved. A formula for the inverse of the resistance Matrix is obtained, generalizing the formula for the inverse of the Distance Matrix of a tree.

Product Distance Matrix of a graph and squared Distance Matrix of a tree
Applicable Analysis and Discrete Mathematics, 2013CoAuthors: Ravindra B. Bapat, Sivaramakrishnan SivasubramanianAbstract:Let G be a strongly connected, weighted directed graph. We define a product Distance η(i,j) for pairs i,j of vertices and form the corresponding product Distance Matrix. We obtain a formula for the determinant and the inverse of the product Distance Matrix. The edge orientation Matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained. A formula for the determinant of the (entrywise) squared Distance Matrix of a tree is proved.
Sumit Mohanty  One of the best experts on this subject based on the ideXlab platform.

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse
Special Matrices, 2020CoAuthors: Joyentanuj Das, Sachindranath Jayaraman, Sumit MohantyAbstract: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

Distance Matrix of Weighted Cactoidtype Digraphs
arXiv: Combinatorics, 2020CoAuthors: Joyentanuj Das, Sumit MohantyAbstract:A strongly connected digraph is called a cactoidtype if each of its blocks is a digraph consisting of finitely many oriented cycles sharing a common directed path. In this article, we find the formula for the determinant of the Distance Matrix for weighted cactoidtype digraphs and find its inverse, whenever it exists. We also compute the determinant of the Distance Matrix for a class of unweighted and undirected graphs consisting of finitely many cycles, sharing a common path.

Distance Matrix of Multiblock Graphs: Determinant and Inverse
arXiv: Combinatorics, 2019CoAuthors: Joyentanuj Das, Sumit MohantyAbstract:A connected graph is called a multiblock graph if each of its blocks is a $m$partite graph, whenever $m\geq 2$. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the Distance Matrix for a class of multiblock graphs.

Distance Matrix of a Multiblock Graph: Determinant and Inverse.
arXiv: Combinatorics, 2019CoAuthors: Joyentanuj Das, Sumit MohantyAbstract:A connected graph is called a multiblock graph if each of its blocks is a complete multipartite graph. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the Distance Matrix for a class of multiblock graphs.

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse
arXiv: Combinatorics, 2019CoAuthors: Joyentanuj Das, Sachindranath Jayaraman, Sumit MohantyAbstract:A real symmetric Matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative Matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative Matrix realization of $G$ 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. Similar to trees, we obtain a relation for the inverse of the Distance Matrix of a class of completely positive graphs involving the Laplacian Matrix, a rank one Matrix and a Matrix $\mathcal{R}$. We also determine the eigenvalues of some principal submatrices of Matrix $\mathcal{R}$.
Nenad Trinajstić  One of the best experts on this subject based on the ideXlab platform.

Maximum Eigenvalues of the Reciprocal Distance Matrix and the Reverse
2008CoAuthors: Bo Zhou, Nenad TrinajstićAbstract:We report some properties of the maximum eigenvalues of the reciprocal Distance Matrix and the reverse Wiener Matrix of a connected graph, in particular, various lower and upper bounds, and the NordhausGaddumtype results for them. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 108: 858  864, 2008

ResistanceDistance Matrix: A computational algorithm and its application
International Journal of Quantum Chemistry, 2002CoAuthors: Darko Babić, Sonja Nikolić, Douglas J. Klein, István Lukovits, Nenad TrinajstićAbstract:The Distance Matrix D, the resistanceDistance Matrix Ω, the related quotient matrices D/Ω and Ω/D and the corresponding Distancerelated and resistanceDistancerelated descriptors: the Wiener index W, the Balaban indices J and JΩ, the Kirchhoff index Kf, the Wienersum index WS, and Kirchhoffsum index KfS are presented. A simple algorithm for computing the resistanceDistance Matrix is outlined. The Distancerelated and the resistanceDistancerelated indices are used to study cyclicity in four classes of polycyclic graphs: fivevertex graphs containing a fivecycle and Schlegel graphs representing platonic solids, buckminsterfullerene isomers and C70 isomers. Among the considered indices only the Kirchhoff index correctly ranks according to their cyclicity, the Schlegel graphs for platonic solids, C60 isomers, and C70 isomers. The Kirchhoff index further produces the reverse order of fivevertex graphs containing a fivecycle (which could be simply altered to the correct order by adding a minus sign to the Kirchhoff indices for these graphs). © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

The Distance Matrix in chemistry
Journal of Mathematical Chemistry, 1992CoAuthors: Zlatko Mihalić, Sonja Nikolić, Darko Veljan, Dragan Amić, Dejan Plavšić, Nenad TrinajstićAbstract:The graphtheoretical (topological) Distance Matrix and the geometric (topographic) Distance Matrix and their invariants (polynomials, spectra, determinants and Wiener numbers) are presented. Methods of computing these quantities are discussed. The uses of the Distance Matrix in both forms and the related invariants in chemistry are surveyed. Special attention is paid to the 2D and 3D Wiener numbers, defined respectively as one half of the sum of entries in the topological Distance Matrix and the topographic Distance Matrix. These numbers appear to be very valuable molecular descriptors in the structure property correlations.

On the geometricDistance Matrix and the corresponding structural invariants of molecular systems
Chemical Physics Letters, 1991CoAuthors: Sonja Nikolić, Nenad Trinajstić, Zlatko Mihalić, Stuart CarterAbstract:Abstract A recently introduced concept of the geometricDistance polynomial by Balasubramanian is used for differentiating conformational isomers. Related geometrydependent invariants (the spectrum of the geometricDistance Matrix and the threedimensional Wiener number) are also used for the same purpose. The geometricDistance Matrix is computed by means of a molecularmechanics method, and its characteristic polynomial by a modified Le Verrier—Fadeev—Frame—Balasubramanian method.
Piotr Pokarowski  One of the best experts on this subject based on the ideXlab platform.

Distance Matrixbased approach to protein structure prediction
Journal of Structural and Functional Genomics, 2009CoAuthors: Andrzej Kloczkowski, Andrzej Kolinski, Robert L. Jernigan, Guang Song, Zhijun Wu, Lei Yang, Piotr PokarowskiAbstract:Much structural information is encoded in the internal Distances; a Distance Matrixbased approach can be used to predict protein structure and dynamics, and for structural refinement. Our approach is based on the square Distance Matrix D = [ r _ij ^2 ] containing all square Distances between residues in proteins. This Distance Matrix contains more information than the contact Matrix C , that has elements of either 0 or 1 depending on whether the Distance r _ij is greater or less than a cutoff value r _cutoff. We have performed spectral decomposition of the Distance matrices $$ {\mathbf{D}} = \sum {\lambda_{k} {\mathbf{v}}_{k} {\mathbf{v}}_{k}^{T} } $$ , in terms of eigenvalues $$ \lambda_{k} $$ and the corresponding eigenvectors $$ {\mathbf{v}}_{k} $$ and found that it contains at most five nonzero terms. A dominant eigenvector is proportional to r ^2—the square Distance of points from the center of mass, with the next three being the principal components of the system of points. By predicting r ^2 from the sequence we can approximate a Distance Matrix of a protein with an expected RMSD value of about 7.3 Å, and by combining it with the prediction of the first principal component we can improve this approximation to 4.0 Å. We can also explain the role of hydrophobic interactions for the protein structure, because r is highly correlated with the hydrophobic profile of the sequence. Moreover, r is highly correlated with several sequence profiles which are useful in protein structure prediction, such as contact number, the residuewise contact order (RWCO) or mean square fluctuations (i.e. crystallographic temperature factors). We have also shown that the next three components are related to spatial directionality of the secondary structure elements, and they may be also predicted from the sequence, improving overall structure prediction. We have also shown that the large number of available HIV1 protease structures provides a remarkable sampling of conformations, which can be viewed as direct structural information about the dynamics. After structure matching, we apply principal component analysis (PCA) to obtain the important apparent motions for both bound and unbound structures. There are significant similarities between the first few key motions and the first few lowfrequency normal modes calculated from a static representative structure with an elastic network model (ENM) that is based on the contact Matrix C (related to D ), strongly suggesting that the variations among the observed structures and the corresponding conformational changes are facilitated by the lowfrequency, global motions intrinsic to the structure. Similarities are also found when the approach is applied to an NMR ensemble, as well as to atomic molecular dynamics (MD) trajectories. Thus, a sufficiently large number of experimental structures can directly provide important information about protein dynamics, but ENM can also provide a similar sampling of conformations. Finally, we use Distance constraints from databases of known protein structures for structure refinement. We use the distributions of Distances of various types in known protein structures to obtain the most probable ranges or the meanforce potentials for the Distances. We then impose these constraints on structures to be refined or include the meanforce potentials directly in the energy minimization so that more plausible structural models can be built. This approach has been successfully used by us in 2006 in the CASPR structure refinement ( http://predictioncenter.org/caspR ).
R B Bapat  One of the best experts on this subject based on the ideXlab platform.

moore penrose inverse of a euclidean Distance Matrix
Linear Algebra and its Applications, 2015CoAuthors: Hiroshi Kurata, R B BapatAbstract:Abstract We obtain expressions for the Moore–Penrose inverse of a Euclidean Distance Matrix (EDM) that are determined only by the positive semidefinite Matrix associated with the EDM. The results complement formulas for the Moore–Penrose inverse of an EDM given in Balaji and Bapat (2007) [2] . A formula for the inverse of a principal subMatrix of an EDM is also derived, whose expression uses the Schur complement of the Laplacian of the EDM. As an application, we obtain an expression for the terminal Wiener index of a tree.

inverse of the Distance Matrix of a block graph
Linear & Multilinear Algebra, 2011CoAuthors: R B Bapat, Sivaramakrishnan SivasubramanianAbstract:A connected graph G, whose 2connected blocks are all cliques (of possibly varying sizes) is called a block graph. Let D be its Distance Matrix. By a theorem of Graham, Hoffman and Hosoya, we have det(D) ≠ 0. We give a formula for both the determinant and the inverse, D −1 of D.

a q analogue of the Distance Matrix of a tree
Linear Algebra and its Applications, 2006CoAuthors: R B Bapat, S PatiAbstract:We consider a qanalogue of the Distance Matrix (called the qDistance Matrix) of an unweighted tree and give formulae for the inverse and the determinant, which generalize the existing formulae for the Distance Matrix. We obtain the Smith normal form of the qDistance Matrix of a tree. The relationship of the qDistance Matrix with the Laplacian Matrix leads to qanalogue of the Laplacian Matrix of a tree, some of whose properties are also given. We study another Matrix related to the Distance Matrix (the exponential Distance Matrix) and show its relationship with the qLaplacian and the qDistance Matrix. A formula for the determinant of the qDistance Matrix of a weighted tree is also given.

Distance Matrix and laplacian of a tree with attached graphs
Linear Algebra and its Applications, 2005CoAuthors: R B BapatAbstract:A tree with attached graphs is a tree, together with graphs defined on its partite sets. We introduce the notion of incidence Matrix, Laplacian and Distance Matrix for a tree with attached graphs. Formulas are obtained for the minors of the incidence Matrix and the Laplacian, and for the inverse and the determinant of the Distance Matrix. The case when the attached graphs themselves are trees is studied more closely. Several known results, including the Matrix Tree theorem, are special cases when the tree is a star. The case when the attached graphs are paths is also of interest since it is related to the transportation problem.