Equitable partitions for Ramanajun graphs

TL;DR

This paper introduces the concept of equitable partitions to address the Bilu-Linial conjecture, providing solutions for specific regular graphs and methods to construct good signings for graph lifts.

Contribution

It proposes a new approach using equitable partitions to solve the Bilu-Linial conjecture for certain cases and offers algorithms for special graph classes.

Findings

Solved Bilu-Linial conjecture for some cases

Developed methods to find good signings for specific graphs

Provided a way to construct good signings for graph 2-lifts

Abstract

For d-regular graph G, an edge-signing sigma:E(G) \rightarrow {-1,1} is called a good signing if the absolute eigenvalues of adjacency matrix are at most 2 \sqrt{d-1}. Bilu-Linial conjectured that for each regular graph there exists a good signing. In this paper, by using new concept "Equitable Partition", we solve the Bilu-Linial Conjecture for some cases. We show that 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.