Some short notes on oriented line graphs and related matrices
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911351646978048 |
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| author | Antony, Jacob Antony, Cyriac Varughese, Jinitha Joseph, Bloomy |
| author_facet | Antony, Jacob Antony, Cyriac Varughese, Jinitha Joseph, Bloomy |
| contents | Oriented line graph, introduced by Kotani and Sunada (2000), is closely related to Hashimato's non-backtracking matrix (1989). It is known that for regular graphs $G$, the eigenvalues of the adjacency matrix of the oriented line graph $\vec{L}(G)$ of $G$ are the reciprocals of the poles of the Ihara zeta function of $G$. We determine the characteristic polynomial of the $z$-Hermitian adjacency matrix of $\vec{L}(G)$ for each $z\in \mathbb{C}$ and $d$-regular graph $G$ with $d\geq 3$. Special cases of this matrix include the Hermitian adjacency matrix of $\vec{L}(G)$ and the adjacency matrix of the underlying undirected graph of $\vec{L}(G)$. We also exhibit an application to star coloring of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some short notes on oriented line graphs and related matrices Antony, Jacob Antony, Cyriac Varughese, Jinitha Joseph, Bloomy Combinatorics Discrete Mathematics Oriented line graph, introduced by Kotani and Sunada (2000), is closely related to Hashimato's non-backtracking matrix (1989). It is known that for regular graphs $G$, the eigenvalues of the adjacency matrix of the oriented line graph $\vec{L}(G)$ of $G$ are the reciprocals of the poles of the Ihara zeta function of $G$. We determine the characteristic polynomial of the $z$-Hermitian adjacency matrix of $\vec{L}(G)$ for each $z\in \mathbb{C}$ and $d$-regular graph $G$ with $d\geq 3$. Special cases of this matrix include the Hermitian adjacency matrix of $\vec{L}(G)$ and the adjacency matrix of the underlying undirected graph of $\vec{L}(G)$. We also exhibit an application to star coloring of graphs. |
| title | Some short notes on oriented line graphs and related matrices |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2507.13821 |