Some short notes on oriented line graphs and related matrices

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.