Now showing 1 - 3 of 3
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    Publication
    Generalized Euclidean distance matrices
    (01-01-2022) ;
    Bapat, R. B.
    ;
    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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    Publication
    On distance matrices of wheel graphs with an odd number of vertices
    (01-01-2022) ;
    Bapat, R. B.
    ;
    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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    Publication
    An inverse formula for the distance matrix of a wheel graph with an even number of vertices
    (01-02-2021) ;
    Bapat, R. B.
    ;
    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.