On the Squared Distance Matrix of a Starlike Block Graph
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909805686292480 |
|---|---|
| author | Das, Joyentanuj Mohanty, Sumit |
| author_facet | Das, Joyentanuj Mohanty, Sumit |
| contents | Let $D(G)$ be the distance matrix of a simple connected graph $G$. The Hadamard product $D(G)~\circ~ D(G)$ is called the squared distance matrix of $G$, and is denoted by $Δ(G)$. A simple connected graph is called a starlike block graph if it has a central cut vertex, and each of its blocks is a complete graph. Let $ \mathcal{S}(n_1, n_2, \ldots, n_b)$ be the starlike block graph with blocks $K_{n_1+1}, K_{n_2+1}, \ldots, K_{n_b+1} $ on $n=1 + \sum_{i=1}^b n_i$ vertices. In this article, we compute the determinant of $Δ( \mathcal{S}(n_1, n_2, \ldots, n_b))$ and find its inverse as a rank-one perturbation of a positive semidefinite Laplacian-like matrix $\mathcal{L}$ with rank $n-1$. We also investigate the inertia of $Δ( \mathcal{S}(n_1, n_2, \ldots, n_b))$. Furthermore, for a fixed value of $ n $ and $ b $, we determine the extremal graphs that uniquely attain the maximum and minimum spectral radius of the squared distance matrix for starlike block graphs on $ n $ vertices and $ b $ blocks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Squared Distance Matrix of a Starlike Block Graph Das, Joyentanuj Mohanty, Sumit Combinatorics 05C12, 05C50 Let $D(G)$ be the distance matrix of a simple connected graph $G$. The Hadamard product $D(G)~\circ~ D(G)$ is called the squared distance matrix of $G$, and is denoted by $Δ(G)$. A simple connected graph is called a starlike block graph if it has a central cut vertex, and each of its blocks is a complete graph. Let $ \mathcal{S}(n_1, n_2, \ldots, n_b)$ be the starlike block graph with blocks $K_{n_1+1}, K_{n_2+1}, \ldots, K_{n_b+1} $ on $n=1 + \sum_{i=1}^b n_i$ vertices. In this article, we compute the determinant of $Δ( \mathcal{S}(n_1, n_2, \ldots, n_b))$ and find its inverse as a rank-one perturbation of a positive semidefinite Laplacian-like matrix $\mathcal{L}$ with rank $n-1$. We also investigate the inertia of $Δ( \mathcal{S}(n_1, n_2, \ldots, n_b))$. Furthermore, for a fixed value of $ n $ and $ b $, we determine the extremal graphs that uniquely attain the maximum and minimum spectral radius of the squared distance matrix for starlike block graphs on $ n $ vertices and $ b $ blocks. |
| title | On the Squared Distance Matrix of a Starlike Block Graph |
| topic | Combinatorics 05C12, 05C50 |
| url | https://arxiv.org/abs/2509.10854 |