Shivani Goel

Indian Institute of Science Bangalore

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Indian Institute of Science Bangalore

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Indian Institute of Technology Madras

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Shivani Goel has a total of 10 co-authors. Here are the top co-authors by number of shared publications:

Ravindra Bhalchandra Bapat
Indian Statistical Institute
6 shared publications
Ramamurthy Balaji
DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY MADRAS, CHENNAI, INDIA
5 shared publications
R. Balaji
Department of Mathematics, IIT Madras, Chennai, India
3 shared publications
R. B. Bapat
2 shared publications
Ali Azimi
University of Neyshabur
1 shared publication
Anita Arora
1 shared publication

Latest Publications

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- 21 June 2023
- Published by arXiv in arXiv
Distance matrix of enhanced power graphs of finite groups
by Anita Arora, Hiranya Kishore Dey, Shivani Goel
The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups $\mathrm{El}(p^n)$ and the non-cyclic abelian groups $\mathrm{El}(p^n)\times\mathrm{El}(q^m)$ and $\mathrm{El}(p^n)\times \mathbb{Z}_m$, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group $\mathrm{El}(p^n)\times \mathrm{El}(q^m)$, we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.
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- 21 June 2023
- Published by arXiv in arXiv
Open Access
Distance matrix of enhanced power graphs of finite groups
by Anita Arora, Hiranya Kishore Dey, Shivani Goel
The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups $\mathrm{El}(p^n)$ and the non-cyclic abelian groups $\mathrm{El}(p^n)\times\mathrm{El}(q^m)$ and $\mathrm{El}(p^n)\times \mathbb{Z}_m$, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group $\mathrm{El}(p^n)\times \mathrm{El}(q^m)$, we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.
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Cited by 3
- 1 July 2022
- Published by Taylor & Francis in Linear and Multilinear Algebra
On resistance matrices of weighted balanced digraphs
by Balaji Ramamurthy, Ravindra B. Bapat, Shivani Goel
Let G be a connected graph with $V (G) = {1, \dots, n}$ . Then the resistance distance between any two vertices i and j is given by $r_{i j} := l_{i i}^{†} + l_{j j}^{†} - 2 l_{i j}^{†}$ , where $l_{i j}^{†}$ is the $(i, j)$ th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix $R := [r_{i j}]$ , there is an elegant formula to compute the inverse of R. This says that $R^{- 1} = - \frac{1}{2} L + \frac{1}{τ^{'} R τ} τ τ^{'},$ where $τ := (τ_{1}, \dots, τ_{n})^{'} and τ_{i} := 2 - \sum_{j \sim i} r_{i j} i = 1, \dots, n .$ A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.
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- 1 July 2022
- Published by Taylor & Francis in Linear and Multilinear Algebra
On resistance matrices of weighted balanced digraphs
by Balaji Ramamurthy, Ravindra B. Bapat, Shivani Goel
Let G be a connected graph with $V (G) = {1, \dots, n}$ . Then the resistance distance between any two vertices i and j is given by $r_{i j} := l_{i i}^{†} + l_{j j}^{†} - 2 l_{i j}^{†}$ , where $l_{i j}^{†}$ is the $(i, j)$ th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix $R := [r_{i j}]$ , there is an elegant formula to compute the inverse of R. This says that $R^{- 1} = - \frac{1}{2} L + \frac{1}{τ^{'} R τ} τ τ^{'},$ where $τ := (τ_{1}, \dots, τ_{n})^{'} and τ_{i} := 2 - \sum_{j \sim i} r_{i j} i = 1, \dots, n .$ A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.
Cited by 3
Open Access
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Cited by 5
- 19 March 2022
- Published by Walter de Gruyter GmbH in Special Matrices
Steiner distance matrix of caterpillar graphs
by Ali Azimi, Ravindra B. Bapat, Shivani Goel
In this article, we show that the rank of the 2-Steiner distance matrix of a caterpillar graph having N vertices and p pendant veritices is 2N - p - 1.
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- 19 March 2022
- Published by Walter de Gruyter GmbH in Special Matrices
Open Access
Steiner distance matrix of caterpillar graphs
by Ali Azimi, Ravindra B. Bapat, Shivani Goel
In this article, we show that the rank of the 2-Steiner distance matrix of a caterpillar graph having N vertices and p pendant veritices is 2N - p - 1.
Cited by 5
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Cited by 11
- 6 September 2021
- Published by Taylor & Francis in Linear and Multilinear Algebra
Generalized Euclidean distance matrices
by R. Balaji, R.B. Bapat, Shivani Goel
Euclidean distance matrices ( $EDM$ ) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ( $GEDM$ s) that include $EDM$ s. Each $GEDM$ is an entry-wise nonnegative matrix. A $GEDM$ is not symmetric unless it is an $EDM$ . 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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- 6 September 2021
- Published by Taylor & Francis in Linear and Multilinear Algebra
Generalized Euclidean distance matrices
by R. Balaji, R.B. Bapat, Shivani Goel
Euclidean distance matrices ( $EDM$ ) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ( $GEDM$ s) that include $EDM$ s. Each $GEDM$ is an entry-wise nonnegative matrix. A $GEDM$ is not symmetric unless it is an $EDM$ . 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.
Cited by 11
Open Access
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- 19 August 2021
- Published by arXiv in arXiv
Generalized Euclidean distance matrices
by R. Balaji, R. B. Bapat, Shivani Goel
Euclidean distance matrices (EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices (GDMs) that include EDMs. Each GDM is an entry-wise nonnegative matrix. A GDM is not symmetric unless it is an EDM. 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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- 19 August 2021
- Published by arXiv in arXiv
Open Access
Generalized Euclidean distance matrices
by R. Balaji, R. B. Bapat, Shivani Goel
Euclidean distance matrices (EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices (GDMs) that include EDMs. Each GDM is an entry-wise nonnegative matrix. A GDM is not symmetric unless it is an EDM. 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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Cited by 5
- 1 July 2021
- Published by Elsevier in Linear Algebra and its Applications
On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices
by Shivani Goel
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- 1 July 2021
- Published by Elsevier in Linear Algebra and its Applications
On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices
by Shivani Goel
Cited by 5
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Cited by 18
- 1 February 2021
- Published by Elsevier in Linear Algebra and its Applications
An inverse formula for the distance matrix of a wheel graph with an even number of vertices
by R. Balaji, R.B. Bapat, Shivani Goel
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- 1 February 2021
- Published by Elsevier in Linear Algebra and its Applications
An inverse formula for the distance matrix of a wheel graph with an even number of vertices
by R. Balaji, R.B. Bapat, Shivani Goel
Cited by 18
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Cited by 7
- 29 October 2020
- Published by Taylor & Francis in Linear and Multilinear Algebra
On distance matrices of wheel graphs with an odd number of vertices
by R. Balaji, R. B. Bapat, Shivani Goel
Let $W_{n}$ denote the wheel graph having n-vertices. If i and j are any two vertices of $W_{n}$ , define $d_{i j} := \{\begin{cases} 0 & if i = j \\ 1 & if i and j are adjacent \\ 2 & else . \end{cases}$ Let D be the $n \times n$ matrix with $(i, j) t h$ entry equal to $d_{i j}$ . The matrix D is called the distance matrix of $W_{n}$ . Suppose $n \geq 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an $n \times n$ matrix $\tilde{L}$ and a rank one matrix $w w^{'}$ such that $D^{†} = - \frac{1}{2} \tilde{L} + \frac{4}{n - 1} w w^{'} .$ Here, $\tilde{L}$ is positive semidefinite, $r a n k (\tilde{L}) = n - 2$ and all row sums are equal to zero.
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- 29 October 2020
- Published by Taylor & Francis in Linear and Multilinear Algebra
On distance matrices of wheel graphs with an odd number of vertices
by R. Balaji, R. B. Bapat, Shivani Goel
Let $W_{n}$ denote the wheel graph having n-vertices. If i and j are any two vertices of $W_{n}$ , define $d_{i j} := \{\begin{cases} 0 & if i = j \\ 1 & if i and j are adjacent \\ 2 & else . \end{cases}$ Let D be the $n \times n$ matrix with $(i, j) t h$ entry equal to $d_{i j}$ . The matrix D is called the distance matrix of $W_{n}$ . Suppose $n \geq 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an $n \times n$ matrix $\tilde{L}$ and a rank one matrix $w w^{'}$ such that $D^{†} = - \frac{1}{2} \tilde{L} + \frac{4}{n - 1} w w^{'} .$ Here, $\tilde{L}$ is positive semidefinite, $r a n k (\tilde{L}) = n - 2$ and all row sums are equal to zero.
Cited by 7
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Cited by 3
- 1 October 2020
- Published by Institute of Mathematics, Czech Academy of Sciences in Applications of Mathematics
DISTANCE MATRICES PERTURBED BY LAPLACIANS
by Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel
Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s. Let D-ij denote the sum of all the weights lying in the path connecting the verticesiand j of T. We now say that D-ij is the distance between i andj. Define D := [D-ij], where D-ii is thesxsnull matrix and for i not equal j, D-ij is the distance betweeniandj. Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of orders. If i andjare adjacent, then define L-ij colon equals - W-ij(-1), where W-ij is the weight of the edge (i, j). Define L-ii = Sigma W-n(i not equal j,j=1)ij(-1). The Laplacian of G is now thens x ns block matrix L := [L-ij]. In this paper, we first note that D-1-L is always nonsingular and then we prove that D and its perturbation (D-1-L)(-1) have many interesting properties in common.
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- 1 October 2020
- Published by Institute of Mathematics, Czech Academy of Sciences in Applications of Mathematics
DISTANCE MATRICES PERTURBED BY LAPLACIANS
by Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel
Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s. Let D-ij denote the sum of all the weights lying in the path connecting the verticesiand j of T. We now say that D-ij is the distance between i andj. Define D := [D-ij], where D-ii is thesxsnull matrix and for i not equal j, D-ij is the distance betweeniandj. Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of orders. If i andjare adjacent, then define L-ij colon equals - W-ij(-1), where W-ij is the weight of the edge (i, j). Define L-ii = Sigma W-n(i not equal j,j=1)ij(-1). The Laplacian of G is now thens x ns block matrix L := [L-ij]. In this paper, we first note that D-1-L is always nonsingular and then we prove that D and its perturbation (D-1-L)(-1) have many interesting properties in common.
Cited by 3
Open Access
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Cited by 7
- 7 April 2020
- Published by Taylor & Francis in Linear and Multilinear Algebra
Resistance matrices of balanced directed graphs
by R. Balaji, R.B. Bapat, Shivani Goel
Let G be a strongly connected and balanced directed graph. We define the resistance $r_{i j}$ 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 $R := [r_{i j}]$ . 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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- 7 April 2020
- Published by Taylor & Francis in Linear and Multilinear Algebra
Open Access
Resistance matrices of balanced directed graphs
by R. Balaji, R.B. Bapat, Shivani Goel
Let G be a strongly connected and balanced directed graph. We define the resistance $r_{i j}$ 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 $R := [r_{i j}]$ . 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.
Cited by 7

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