A Cheeger type inequality in finite Cayley sum graphs

2019/07/17 by Arindam Biswas, Biswas, Arindam, Jyoti Prakash Saha +1
Mathematics · #05C25 #05C50 #05C75 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:05C25 #msc:05C50 #msc:05C75

paper · pdf · doi:10.48550/arxiv.1907.07710

Grant number added

arxiv created 2019/07/22 · arxiv updated 2019/07/23

Abstract

Let G be a finite group and S be a symmetric generating set of G with |S| = d. We show that if the undirected Cayley sum graph C_Σ(G,S) is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from -1. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval (-1+\frach(G)⁴η, 1-\frach(G)²2d²], where h(G) denotes the (vertex) Cheeger constant of the d-regular graph C_Σ(G,S) and η= 2⁹d⁸. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph C(G,S).

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