Inverse of the Squared Distance Matrix of a Complete Multipartite Graph
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914760086257664 |
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| author | Das, Joyentanuj Mohanty, Sumit |
| author_facet | Das, Joyentanuj Mohanty, Sumit |
| contents | Let $G$ be a connected graph on $n$ vertices and $d_{ij}$ be the length of the shortest path between vertices $i$ and $j$ in $G$. We set $d_{ii}=0$ for every vertex $i$ in $G$. The squared distance matrix $Δ(G)$ of $G$ is the $n\times n$ matrix with $(i,j)^{th}$ entry equal to $0$ if $i = j$ and equal to $d_{ij}^2$ if $i \neq j$. For a given complete $t$-partite graph $K_{n_1,n_2,\cdots,n_t}$ on $n=\sum_{i=1}^t n_i$ vertices, under some condition we find the inverse $Δ(K_{n_1,n_2,\cdots,n_t})^{-1}$ as a rank-one perturbation of a symmetric Laplacian-like matrix $\mathcal{L}$ with $\textup{rank} (\mathcal{L})=n-1$. We also investigate the inertia of $\mathcal{L}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01069 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inverse of the Squared Distance Matrix of a Complete Multipartite Graph Das, Joyentanuj Mohanty, Sumit Combinatorics 05C12, 05C50 Let $G$ be a connected graph on $n$ vertices and $d_{ij}$ be the length of the shortest path between vertices $i$ and $j$ in $G$. We set $d_{ii}=0$ for every vertex $i$ in $G$. The squared distance matrix $Δ(G)$ of $G$ is the $n\times n$ matrix with $(i,j)^{th}$ entry equal to $0$ if $i = j$ and equal to $d_{ij}^2$ if $i \neq j$. For a given complete $t$-partite graph $K_{n_1,n_2,\cdots,n_t}$ on $n=\sum_{i=1}^t n_i$ vertices, under some condition we find the inverse $Δ(K_{n_1,n_2,\cdots,n_t})^{-1}$ as a rank-one perturbation of a symmetric Laplacian-like matrix $\mathcal{L}$ with $\textup{rank} (\mathcal{L})=n-1$. We also investigate the inertia of $\mathcal{L}$. |
| title | Inverse of the Squared Distance Matrix of a Complete Multipartite Graph |
| topic | Combinatorics 05C12, 05C50 |
| url | https://arxiv.org/abs/2311.01069 |