Some short notes on oriented line graphs and related matrices

The notion of oriented line graphs is introduced by Kotani and Sunada, and they are closely related to Hashimato's non-backtracking matrix. 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 adjacency matrix of the underlying undirected graph of $\vec{L}(G)$ and the skew-symmetric adjacency matrix of $\vec{L}(G)$ for $d$-regular graphs $G$ with $d\geq 3$. We also exhibit a consequence of this result to star coloring of regular graphs.