An inverse formula for the distance matrix of a wheel graph with an even number of vertices

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. I...

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Bibliographic Details
Published in:Linear algebra and its applications 2021-02, Vol.610, p.274-292
Main Authors: Balaji, R., Bapat, R.B., Goel, Shivani
Format: Article
Language:English
Subjects:
Apexes
Circulant matrices
Distance matrices
Eigenvalues
Graph theory
Laplacian matrices
Linear algebra
Mathematical analysis
Matrix methods
Shortest-path problems
Wheel graphs
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Summary: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 thatD−1=−12L˜+4n−1wwT. An interlacing property between the eigenvalues of D and L˜ is also proved.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2020.10.003