Λ∞, Vertex Isoperimetry and Concentration
نویسندگان
چکیده
In an important paper, Alon [2] derived a Cheeger–type inequality [8], by bounding from below the second smallest eigenvalue of the Laplacian of a finite undirected graph by a function of a (vertex) isoperimetric constant. More precisely, let G=(V,E) be a finite, undirected, connected graph, and let λ2(G) denote twice (for reasons explained below) the smallest non-zero eigenvalue of the Laplacian of G. Recall that the Laplacian of G is the matrix D(G)−A(G), where A(G) is a symmetric matrix (indexed by the vertices of G) of order |V | whose i,jth entry is 1 or 0 depending on whether there is an edge or not between the ith and the jth vertex ; and where D(G) is the diagonal matrix whose i, ith element is the degree of the ith vertex. In
منابع مشابه
Isoperimetry and functional inequalities
1.1 Brunn-Minkowski inequality 1.1 Theorem. (Brunn-Minkowski, ’88) If A and B are non-empty compact sets then for all λ ∈ [0, 1] we have vol ((1− λ)A+ λB) ≥ (1− λ)(volA) + λ(volB). (B-M) Note that if either A = ∅ orB = ∅, this inequality does not hold since (1−λ)A+λB = ∅. We can use the homogenity of volume to rewrite Brunn-Minkowski inequality in the form vol (A+B) ≥ (volA) + (volB). (1.1) We ...
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