On Structural Properties and Adjacency Spectrum of Coprime Graph of Integers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909695929745408 |
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| author | Banerjee, Subarsha |
| author_facet | Banerjee, Subarsha |
| contents | Let $TCG_n$ denote the coprime graph having vertex set $\{1,2,\ldots,n\}$ with any two vertices $i,j$ being adjacent if and only if $\gcd(i,j)=1$. In this article, we first study some structural properties of $TCG_n$. We study the vertex connectivity and crossing number of the coprime graph of integers. We discover a lower constraint on the multiplicity of $-1$, which appears as an eigenvalue in the adjacency matrix of $TCG_n$. We demonstrate our findings with a variety of cases. We also show that the adjacency matrix of $TCG_n$ is singular, i.e. has determinant $0$. Furthermore, we give a lower bound on the multiplicity of $0$, which appears as an eigenvalue in the adjacency matrix of $TCG_n$. Finally, we establish that the greatest eigenvalue of the adjacency matrix of $TCG_n$ is always above $2.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Structural Properties and Adjacency Spectrum of Coprime Graph of Integers Banerjee, Subarsha Combinatorics Group Theory 05C25, 05C50 Let $TCG_n$ denote the coprime graph having vertex set $\{1,2,\ldots,n\}$ with any two vertices $i,j$ being adjacent if and only if $\gcd(i,j)=1$. In this article, we first study some structural properties of $TCG_n$. We study the vertex connectivity and crossing number of the coprime graph of integers. We discover a lower constraint on the multiplicity of $-1$, which appears as an eigenvalue in the adjacency matrix of $TCG_n$. We demonstrate our findings with a variety of cases. We also show that the adjacency matrix of $TCG_n$ is singular, i.e. has determinant $0$. Furthermore, we give a lower bound on the multiplicity of $0$, which appears as an eigenvalue in the adjacency matrix of $TCG_n$. Finally, we establish that the greatest eigenvalue of the adjacency matrix of $TCG_n$ is always above $2.$ |
| title | On Structural Properties and Adjacency Spectrum of Coprime Graph of Integers |
| topic | Combinatorics Group Theory 05C25, 05C50 |
| url | https://arxiv.org/abs/2506.10583 |