Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map � : H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues.  conjecture that every d-regular graph has a 2lift such that all new eigenvalues are in the range [−2 √ d − 1,2 √ d − 1] (If true, this is tight , e.g. by the Alon-Boppana bound). Here  show that every graph of maximal degree d has a 2-lift such that all
['Yonatan Bilu', 'Nathan Linial']
I am writing a research paper on a idea Lifts, Discrepancy and Nearly Optimal Spectral Gap
[{"from": "human", "value": "I am writing a research paper on a idea Lifts, Discrepancy and Nearly Optimal Spectral Gap"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map \ufffd : H \u2192 G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n \u201cnew\u201d eigenvalues.  conjecture that every d-regular graph has a 2lift such that all new eigenvalues are in the range [\u22122 \u221a d \u2212 1,2 \u221a d \u2212 1] (If true, this is tight , e.g. by the Alon-Boppana bound). Here  show that every graph of maximal degree d has a 2-lift such that all"}]