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 polynomials of the adjacency matrix of the underlying undirected graph of $\vec{L}(G)$ and the skew-symmetric adjacency matrix (and Hermitian adjacency matrix) of $\vec{L}(G)$ for $d$-regular graphs $G$ with $d\geq 3$. A locally bijective (resp. injective) homomorphism from a graph $G$ to a graph $H$ is a mapping $\psi\colon V(G)\to V(H)$ such that for every vertex $v$ of $G$, the restriction of $\psi$ to the neighborhood $N_G(v)$ is a bijection (resp. injection) from $N_G(v)$ to $N_H(\psi(v))$ (Fiala and Kratochv\'il, 2008). An out-neighborhood bijective (resp. injective) homomorphism from a directed graph $\vec{G}$ to a directed graph $\vec{H}$ is a mapping $\psi\colon V(\vec{G})\to V(\vec{H})$ such that for every vertex $v$ of $\vec{G}$, the restriction of $\psi$ to the out-neighborhood $N_{\vec{G}}^+(v)$ is a bijection (resp. injection) from $N_{\vec{G}}^+(v)$ to $N_{\vec{H}}^+(\psi(v))$ (Antony and Shalu, 2025). We prove that the existence of a locally bijective (resp. injective) homomorphism from a graph $G$ of minimum degree at least 3 to a graph $H$ is equivalent to the existence of an out-neighborhood bijective (resp. injective) homomorphism from $\vec{L}(G)$ to $\vec{L}(H)$. We also prove some results on the coloring variants distance-two coloring and star coloring.