Lower Bounds for the Isoperimetric Numbers of Random Regular Graphs

Abstract

The vertex isoperimetric number of a graph G = (V,E ) is the minimum of the ratio |∂ V U |/|U | where U ranges over all nonempty subsets of V with |U |/|V | ≤ u and ∂ V U is the set of all vertices adjacent to U but not in U . The analogously defined edge isoperimetric number-with ∂ V U replaced by ∂ E U , the set of all edges with exactly one endpoint in U -has been studied extensively. Here we study random regular graphs. For the case u = 1/2, we give asymptotically almost sure lower bounds for the vertex isoperimetric number for all d ≥ 3. Moreover, we obtain a lower bound on the asymptotics as d → ∞. We also provide asymptotically almost sure lower bounds on |∂ E U |/|U | in terms of an upper bound on the size of U and analyse the bounds as d → ∞.

Adjacency matrices of random digraphs: Singularity and anti-concentration

“…Given 0 ≤ α, β ≤ 1 we define Ω 0 (α, β) = {M ∈ M n,d : ∃I, J ⊂ [n] such that |I| ≥ αn, |J| ≥ βn, and ∀i ∈ I ∀j ∈ J µ ij = 0}. (20) In other terms, the elements of Ω 0 (α, β) are the matrices in M n,d having a zero submatrix of size at least αn × βn. Theorem 2.6, reformulated below, shows that this set is small whenever α and β are not very small.…”

Modularity of regular and treelike graphs

“…Now consider a random r-regular graph G n,r . For each particular u ∈ (0, 1/2], by Theorem 1.3 in [24] and the discussion in Section 7 of that paper, we see that whp i u (G n,r ) ≥ y/u for each 0 < y < ru(1 − u) withf r (u, y) < 0, wherê…”

“…Now we relate β(G) to edge expansion and the quantity α(G) defined below, so that we can use calculations from [24]. Following the notation of [24], for 0 < u ≤ 1 2 we define the u-edge-expansion i u (G) of an n-vertex graph G by setting where the minimum is over non-empty sets S of at most un vertices (and the value is taken to be ∞ if un < 1). Observe that i 1/2 (G) is the usual edge expansion or isoperimetric number of G. Also, set α(G) = min…”

“…(It is known [24] thatf r (u, y) is strictly concave in y, positive at y = ru(1 − u) and negative for sufficiently small y > 0.) For a random cubic graph G n,3 , by finding appropriate values y, we deduce that whp α(G n,3 ) > 0.196, and so whp q * (G n,3 ) < 0.804.…”

Minimum Power Dominating Sets of Random Cubic Graphs

“…This gives quite reasonable lower bounds on for random d ‐regular graphs. For instance, we have from Kolesnik and Wormald that for a random cubic graph G , a.a.s. Using this and , we see that a.a.s.…”