directed Line graph associated to a graph

Let $X=(V,E)$ be a graph. Then we define a directed graph $(X_L=(V_L,E_L)$ called line graph of $X$ as follows:

1. $V_L=E$
2. $E_L^o=\{(e_1,e_2)\in E\times E; \operatorname{target}(e_1)=\operatorname{source}(e_2). , \bar e_1\neq e_2\}.$

LEMMA 12.5  

$$\det_{C(E)}(1-u\cdot L_{X_L}) = (1-u^2)^{-\chi(X)} \det_{C(V)}(I-u A + u^2(D-I))$$

Reference:

Motoko Kotani and Toshikazu Sunada, Zeta functions of finite graphs, J.Math.Sci.Univ.Tokyo 7(2000) 7-25