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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 $spec(G_2)$ ?

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closed as unclear what you're asking by Wolfgang, Chris Godsil, Jan-Christoph Schlage-Puchta, András Bátkai, Andrej Bauer Aug 16 '16 at 13:24

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  • $\begingroup$ See section 2 (in particular section 2.5) of Cvetkovic, Doob & Sachs, Spectra of graphs. [I only have an old edition, but hopefully this did not change.] They deal with a lot of these as special cases of "NEPS". See also section 2.1 of Cvetkovic, Rowlinson & Simic, An introduction to the theory of graph spectra, 2010. For the zig-zag product, I only know of the spectral gap preservation (see the paper by Reingold, Vadhan et Wigderson from 2002) but there might have been other things published since. $\endgroup$ – ARG Aug 3 '16 at 5:20