Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be 2 graphs:

Union of 2 graphs $G_1 \cup G_2=(V_1 \cup V_2,E_1 \cup E_2)$;

Composition of 2 graphs $G_1[G_2]$;

Sum(join) of 2 graphs by $G_1+G_2=(V(G_1)\cup V(G_2),E(G_1) \cup E(G_2) \cup \{\{ u_1,u_2\}|u_1\in V(G_1),u_2 \in V(G_2)\}).$

What can be said about spectrum of $G_1\cup G_2$, $G_1[G_2]$ and $G_1+ G_2$ and replacement and zig-zag product related to $spec(G_1)$ and $spec(G_2)$ ?