Tags: #theorem

Statement

Let $G,H$ be two graphs. Suppose the adjacency matrix $A(G)$ has eigenvalues ${\alpha }_1,\dots {\alpha }_m$ and $A(H)$ has ${\beta }_1,\dots ,{\beta }_n$.

Then, $A(G\times H)$ has $m\times n$ eigenvalues given by ${\alpha }_i+{\beta }_j$ for all $i,j$.
and the eigenvectors are given by the tensor product of the eigenvectors.

Proof