Equitable partition for some Ramanujan graphs.

For a d -regular graph G , an edge-signing σ : E (G) → { − 1 , 1 } is called a good signing if the absolute eigenvalues of its adjacency matrix are at most 2 d − 1 . Bilu and Linial conjectured that for each regular graph there exists a good signing. In this paper, by using the concept of "Equitable Partition", we prove the conjecture for some cases. We show how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph G , then we can find a good signing for a 2-lift of G. [ABSTRACT FROM AUTHOR]