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.

翻译：Kotani与Sunada (2000) 提出的有向线图与Hashimato (1989) 的非回溯矩阵密切相关。已知对于正则图$G$，其有向线图$\vec{L}(G)$的邻接矩阵特征值即为$G$的伊原ζ函数极点的倒数。本文针对任意$z\in \mathbb{C}$及度数$d\geq 3$的正则图$G$，确定了$\vec{L}(G)$的$z$-埃尔米特邻接矩阵的特征多项式。该矩阵的特殊情形包含$\vec{L}(G)$的埃尔米特邻接矩阵及其基础无向图的邻接矩阵。文中还展示了该结论在图星着色问题中的应用。