On the Bipartiteness Constant and Expansion of Cayley Graphs
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Let G be a finite, undirected d-regular graph and A(G) its normalized adjacency matrix, with eigenvalues 1 = λ_1(A)≥…≥λ_n ≥ -1. It is a classical fact that λ_n = -1 if and only if G is bipartite. Our main result provides a quantitative separation of λ_n from -1 in the case of Cayley graphs, in terms of their expansion. Denoting h_out by the (outer boundary) vertex expansion of G, we show that if G is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size d) then λ_n ≥ -1 + ch_out^2/d^2 , for c an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on d. This improves upon a recent result by Biswas and Saha who showed λ_n ≥ -1 + h_out^4/(2^9d^8) . We also note that such a result could not be true for general non-bipartite graphs.
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