Positively curved graphs

Abstract

This paper consists of two halves. In the first half of the paper, we consider real-valued functions f whose domain is the vertex set of a graph G and that are Lipschitz with respect to the graph distance. By placing a uniform distribution on the vertex set, we treat as a random variable. We investigate the link between the isoperimetric function of G and the functions f that have maximum variance or meet the bound established by the subgaussian inequality. We present several results Question 1.2 (Alon et al. [2], p. 416). Let X be a variance-optimal function over G. Is it true for d sufficiently large and r, ℓ in appropriate ranges that |B rOur initial suspicion was that the question would have a positive answer, based on the following intuition. Constructions of functions that maximize variance whose expectations are 0 will evenly spread values toward ∞ and −∞ as possible. Because X is Lipschitz we have that B r (S ℓ ) ⊆ S ℓ+r . The hope is that V(G d ) − S ℓ + r, X = S −(ℓ + r), −X is relatively large.A positive answer to the question would have significance. Let X 1 , X 2 be variance-optimal over G 1 , G 2 , respectively. Let H = G 1 ◻ G 2 , and define variable Y over H as Y(u 1 , u 2 ) = X 1 (u 1 ) + X 2 (u 2 ). Bobkov YANCEY | 541 X .Once again we are able to use our statements describing optimal functions to characterize the optimal functions of specific examples.Theorem 2.11. Conjecture 1.5 is true.All of our discussion so far generalizes in the obvious way to arbitrary finite metric spaces. An example is the symmetric group S d on d elements equipped with the Hamming distance, which is d H (π, π′) = |{i : π(i) ≠ π′(i)}|. Bobkov et al [9] bounded σ S d . YANCEY | 543

Remarks on the Spectral Approach to Finding Short Paths

“…Our result connects the distance between two vertices with the length of a path formed by locally-optimal choices over a spectral function, although we use a different spectral formulation than (2). In particular, we replace the Fiedler vector of the Laplacian with the "spread" of a graph, which we will define in Section 3 (its connection to the Fiedler vector of a Laplacian is described in Section 1.3 of [5]).…”

“…Chung and Yau [3] established an upper bound on L D when the graph satisfies graph curvature conditions. There are many different variations of graph curvature; we survey eight variations of positive curvature on Page 545 of [5] and five variations of negative curvature are discussed at various places in [6]; neither survey includes either of the two curvature variations in [3]. The central definition of curvature in [3] is simply called "curvature;" let κ denote the curvature of G when it is well-defined.…”