Lexicographic Product I found out that the lexicographic product of a complete graph of order m and a cycle of order n is singular iff n is divisible by 4. But I am having a hard time on how to prove it using eigenvalues. Can someone help me? Thank you!

The eigenvalues of the second graph (if regular) are also the eigenvalues of the product (lexicographic) graph. Further, if the second graph is $C_n$, then its eigenvalues are given by $2\cos\left(\frac{2\pi j}{n}\right),$ for $j=0, 1, \ldots, n-1$. Now if $n=4k$, for some $k$, then choosing $j=k$, we get $0$ as an eigenvalue of $C_n$. Hence the result.  
