Squared distance matrix of a weighted tree

Bapat, R.B.

doi:10.5614/ejgta.2019.7.2.8

Cited by 22 publications

(27 citation statements)

References 8 publications

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“…The difficulties encountered earlier in constructing pseudostationary processes may be attributable to the use of the shortest‐path distance metric. This suggests that the earlier exercises should be attempted using other possible choices of the distance metric, such as Euclidean distance or electrical resistance (Bapat, ).…”

Section: Other Distance Metricsmentioning

confidence: 99%

“Stationary” point processes are uncommon on linear networks

Baddeley

Nair

Rakshit

et al. 2017

Stat

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Statistical methodology for analysing patterns of points on a network of lines, such as road traffic accident locations, often assumes that the underlying point process is "stationary" or "correlation-stationary." However, such processes appear to be rare. In this paper, popular procedures for constructing a point process are adapted to linear networks: many of the resulting models are no longer stationary when distance is measured by the shortest path in the network. This undermines the rationale for popular statistical methods such as the K-function and pair correlation function. Alternative strategies are proposed, such as replacing the shortest-path distance by another metric on the network.

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Section: Other Distance Metricsmentioning

confidence: 99%

“Stationary” point processes are uncommon on linear networks

Baddeley

Nair

Rakshit

et al. 2017

Stat

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“…The Kirchhoff index Kf (G) was defined in [1] as Kf (G) = u<v r uv , where r uv (G) denotes the resistance distance between u and v in G. These novel parameters are in fact intrinsic to the graph theory and has some nice properties and applications in chemistry. For the study of resistance distance and Kirchhoff index, one may be referred to the recent works ( [2], [3], [6]), [13] − [19]) and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Resistance distance and Kirchhoff index in generalized R-vertex and R-edge corona for graphs

Liu

2019

Filomat

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For a graph G, the graph R(G) of a graph G is the graph obtained by adding a new vertex for each edge of G and joining each new vertex to both end vertices of the corresponding edge. Let I(G) be the set of newly added vertices, i.e I(G) = V(R(G))\ V(G). The generalized R-vertex corona of G and Hi for i = 1, 2, ?,n, denoted by R(G) ?? ^n i=1 Hi, is the graph obtained from R(G) and Hi by joining the i-th vertex of V(G) to every vertex in Hi. The generalized R-edge corona of G and Hi for i = 1, 2, ?,m, denoted by R(G)?^m i=1 Hi, is the graph obtained from R(G) and Hi by joining the i-th vertex of I(G) to every vertex in Hi. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of R(G) ?? ^n i=1 Hi and R(G) ? ^m i=1 Hi whenever G and Hi are arbitrary graphs. These results generalize the existing results.

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“…The notion of Wiener index based on the resistance distance has proposed in [9]. The formulae for the determinant and the inverse of the resistance matrix of scalar weighted graph are obtained in [2]. Some inequalities for the eigenvalues of the distance matrix of a tree and the eigenvalues of the resistance matrix of any connected graph are established in [12] and [13], respectively.…”

Section: Introductionmentioning

confidence: 99%

Resistance matrices of graphs with matrix weights

Atik

Bapat

Kannan

2019

Linear Algebra and its Applications

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The resistance matrix of a simple connected graph G is denoted by R, and is defined by R = (r ij ), where r ij is the resistance distance between the vertices i and j of G. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size. We derive a formula for the determinant and the inverse of the resistance matrix. Then, we establish an interlacing inequality for the eigenvalues of resistance and Laplacian matrices. Using this interlacing inequality, we obtain the inertia of the resistance matrix.

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Squared distance matrix of a weighted tree

Cited by 22 publications

References 8 publications

“Stationary” point processes are uncommon on linear networks

“Stationary” point processes are uncommon on linear networks

Resistance distance and Kirchhoff index in generalized R-vertex and R-edge corona for graphs

Resistance matrices of graphs with matrix weights

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