On Spectrum of Neighbourhood Corona Product of Signed Graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913674574168064 |
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| author | Sonar, Bishal Guragain, Satyam Srivastava, Ravi |
| author_facet | Sonar, Bishal Guragain, Satyam Srivastava, Ravi |
| contents | Given two signed graphs $Γ_1$ with nodes $\{u_1,u_2,\cdots,u_n\}$ and $Γ_2$, the neighbourhood corona, $Γ_1*Γ_2$ is the signed graph obtained by taking one copy of $Γ_1$ and $n_1$ copies of $Γ_2$, and joining every neighbour of the $i^{th}$ node with each nodes of the $i^{th}$ copy of $Γ_2$ by a new signed edge. In this paper we will determine the condition for $Γ_1*Γ_2$ to be balanced. We also determine the adjacency spectrum of $Γ_1*Γ_2$ for arbitrary $Γ_1$ and $Γ_2$, and Laplacian and signless Laplacian spectrum of $Γ_1*Γ_2$ for regular $Γ_1$ and arbitrary $Γ_2$, in terms of the corresponding spectrum of $Γ_1$ and $Γ_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12814 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Spectrum of Neighbourhood Corona Product of Signed Graphs Sonar, Bishal Guragain, Satyam Srivastava, Ravi Combinatorics 05C22, 05C50, 05C76 Given two signed graphs $Γ_1$ with nodes $\{u_1,u_2,\cdots,u_n\}$ and $Γ_2$, the neighbourhood corona, $Γ_1*Γ_2$ is the signed graph obtained by taking one copy of $Γ_1$ and $n_1$ copies of $Γ_2$, and joining every neighbour of the $i^{th}$ node with each nodes of the $i^{th}$ copy of $Γ_2$ by a new signed edge. In this paper we will determine the condition for $Γ_1*Γ_2$ to be balanced. We also determine the adjacency spectrum of $Γ_1*Γ_2$ for arbitrary $Γ_1$ and $Γ_2$, and Laplacian and signless Laplacian spectrum of $Γ_1*Γ_2$ for regular $Γ_1$ and arbitrary $Γ_2$, in terms of the corresponding spectrum of $Γ_1$ and $Γ_2$. |
| title | On Spectrum of Neighbourhood Corona Product of Signed Graphs |
| topic | Combinatorics 05C22, 05C50, 05C76 |
| url | https://arxiv.org/abs/2310.12814 |