Eigenvalues of subgraphs of the cube. (May 2018)

• Record Type: Journal Article
• Authors: Bollobás, Béla; Lee, Jonathan; Letzter, Shoham
• Is Part Of: European journal of combinatorics. Volume 70(2018)
• Page Start: 125
• Page End: 148
• Publication Date: 2018-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2017.12.007
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube Q d of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius o ( d ) have largest eigenvalue that is within 1 + o ( 1 ) of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when d is sufficiently large. Our proofs rely on the method of compressions.