
Approximately counting independent sets in bipartite graphs via graph containers
By implementing algorithmic versions of Sapozhenko's graph container met...
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Finding Efficient Domination for S_1,1,5Free Bipartite Graphs in Polynomial Time
A vertex set D in a finite undirected graph G is an efficient dominating...
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An algorithm to evaluate the spectral expansion
Assume that X is a connected (q+1)regular undirected graph of finite or...
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Highgirth nearRamanujan graphs with lossy vertex expansion
Kahale proved that linear sized sets in dregular Ramanujan graphs have ...
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Highgirth nearRamanujan graphs with localized eigenvectors
We show that for every prime d and α∈ (0,1/6), there is an infinite sequ...
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Two (Known) Results About Graphs with No Short Odd Cycles
Consider a graph with n vertices where the shortest odd cycle is of leng...
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Estimating the Cheeger constant using machine learning
In this paper, we use machine learning to show that the Cheeger constant...
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On the Bipartiteness Constant and Expansion of Cayley Graphs
Let G be a finite, undirected dregular 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 nonbipartite 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 nonbipartite graphs.
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