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 2d−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.

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