Now showing 1 - 6 of 6
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    Resistance matrices of balanced directed graphs
    (01-01-2022) ;
    Bapat, R. B.
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    Goel, Shivani
    Let G be a strongly connected and balanced directed graph. We define the resistance (Formula presented.) between any two vertices i and j of G by using the Moore–Penrose inverse of the Laplacian matrix of G and define the resistance matrix by (Formula presented.). This generalizes the resistance in the undirected case. In this paper, we show that R is a non-negative matrix and obtain an expression to compute the inverse, determinant and cofactor sums of R.
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    Resistance distance in directed cactus graphs
    (01-01-2020) ;
    Bapat, R. B.
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    Goel, Shivani
    Let G = (V, E) be a strongly connected and balanced digraph with vertex set V = {1, …, n}. The classical distance dij between any two vertices i and j in G is the minimum length of all the directed paths joining i and j. The resistance distance (or, simply the resistance) between any two vertices i and j in V is defined by rij:= lii† + l†jj − 2l†ij, where l†pq is the (p, q)th entry of the Moore-Penrose inverse of L which is the Laplacian matrix of G. In practice, the resistance rij is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between i and j is always less than or equal to the classical distance, i.e., rij ≤ dij . However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.
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    Generalized Euclidean distance matrices
    (01-01-2022) ;
    Bapat, R. B.
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    Goel, Shivani
    Euclidean distance matrices ((Formula presented.)) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ((Formula presented.) s) that include (Formula presented.) s. Each (Formula presented.) is an entry-wise nonnegative matrix. A (Formula presented.) is not symmetric unless it is an (Formula presented.). By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore–Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices.
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    On distance matrices of wheel graphs with an odd number of vertices
    (01-01-2022) ;
    Bapat, R. B.
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    Goel, Shivani
    Let (Formula presented.) denote the wheel graph having n-vertices. If i and j are any two vertices of (Formula presented.), define (Formula presented.) Let D be the (Formula presented.) matrix with (Formula presented.) entry equal to (Formula presented.). The matrix D is called the distance matrix of (Formula presented.). Suppose (Formula presented.) is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an (Formula presented.) matrix (Formula presented.) and a rank one matrix (Formula presented.) such that (Formula presented.) Here, (Formula presented.) is positive semidefinite, (Formula presented.) and all row sums are equal to zero.
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    An inverse formula for the distance matrix of a wheel graph with an even number of vertices
    (01-02-2021) ;
    Bapat, R. B.
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    Goel, Shivani
    Let n≥4 be an even integer and Wn be the wheel graph with n vertices. The distance dij between any two distinct vertices i and j of Wn is the length of the shortest path connecting i and j. Let D be the n×n symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to dij. In this paper, we find a positive semidefinite matrix L˜ such that rank(L˜)=n−1, all row sums of L˜ equal to zero, and a rank one matrix wwT such that [Formula presented] An interlacing property between the eigenvalues of D and L˜ is also proved.
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    Simple expressions for the long walk distance
    (01-01-2013)
    Chebotarev, Pavel
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    Bapat, R. B.
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    The walk distances in graphs are defined as the result of appropriate transformations of the ∑k=0∞(tA)k proximity measures, where A is the weighted adjacency matrix of a connected weighted graph and t is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter t approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix L=ρI-A, where ρ is the Perron root of A. © 2013 Elsevier Inc. All rights reserved.