Equitable partition for some Ramanujan graphs

Abstract

For a [Formula: see text]-regular graph [Formula: see text], an edge-signing [Formula: see text] is called a good signing if the absolute eigenvalues of its adjacency matrix are at most [Formula: see text]. 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 [Formula: see text], then we can find a good signing for a 2-lift of [Formula: see text].