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!
Lexicographic Product
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graph-theory
1 Answers
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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.