Partitioning Into Expanders:Gharan and Trevisan

We also proved a softer version of statement 2 as cheeger’s inequality i.e. Cheeger’s Inequality: 1−λ2 2 ≤ h(G) ≤ √ 2(1− λ2) Note:The same results also hold for a weighted graph which is not d regular.In that case we define define a matrix of degrees D,with Dii=degree of vertex i= ∑ j w(i, j) and Dij = 0∀i 6= j.Then M is given by L=D−1/2AD−1/2.It may be noted that M = A d for a d-regular graph.The above results however hold in general for the generalised definition of M.In this case we define the expansion as w(S, − S) vol(S) , where vol(S) = ∑ u∈S ∑ vinV w(u, v).h(G) = min S:vol(S)≤ vol(S) 2 h(S) In today’s discussion we will generalise this setting in several ways:-