suppose that $G$ is a graph with $n$ vertices and $H$ a graph with $m$ vertices,if we consider join of $G$ and $H$ prove that :$$\chi(G \vee H,\lambda)=(-1)^m \chi(G,\lambda)\chi(\overline{H},-\lambda-1)+(-1)^n \chi(H,\lambda)\chi(\overline{G},-\lambda-1)-(-1)^{n+m}\chi(\overline{G},-\lambda-1) \chi(\overline{H},-\lambda-1) $$
I make the adjacency matrix of $G\vee H$ and try deduce characteristic polynomial of $\lambda I-A$ (consider $A$ as adjacency matrix of $G \vee H$ ),but I couldn't arrive something near !any hint or Idea or reference to study will be great thanks!