Bounds on isoperimetric values of trees

Let G=(V,E) be a finite, simple and undirected graph. For S ⊆ V , let δ(S, G) = {(u, v) ∈ E : u ∈ S and v ∈ V − S} be the edge boundary of S. Given an integer i, 1 ≤ i ≤ |V |, let the edge isoperimetric value of G at i be defined as be(i, G) = minS⊆V ;|S|=i |δ(S, G)|. The edge isoperimetric peak of G is defined as be(G) = max1≤j≤|V | be(j, G). The depth of a tree d is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as T 2 d ), cd ≤ be(T 2 d ) ≤ d where c is a suitable constant. For a complete t-ary tree of depth d (denoted as T t d) and d ≥ c′ log t where c′ is a suitable chosen constant, we show that c √ td ≤ be(T t d) ≤ td where c is a suitable constant. Our proof technique can be extended to arbitrary (rooted) trees as follows. Let T = (V, E, r) be a finite, connected and rooted tree the root being the vertex r. Define a weight function w : V → N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index η(T ) be defined as the number of distinct weights in the tree, i.e η(T ) = |{w(u) : u ∈ V }|. For a positive integer k, let `(k) = |{i ∈ N : 1 ≤ i ≤ |V |, be(i, G) ≤ k}|. We show that `(k) ≤ 2 ` 2η+k k ́ . Let bv(G) denote the vertex isoperimetric peak defined in a corresponding way. Since be(G) ≥ bv(G) ≥ be(G) ∆ where ∆ is the maximum degree, we get the following results. We show that cd ≤ bv(T 2 d ) ≤ d and c′ d t ≤ bv(T t d) ≤ d where c and c′ are suitable constants. Similar results can be derived for an arbitrary tree in terms of the weight index η. We observe the following as a consequence of our results : There exists an increasing function f such that if pathwidth(G) ≥ k then there exists a minor G′ of G such that bv(G) ≥ f(k). We also mention an application of our results to a parameter called thinness.