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 b e ( i , G ) = min S ⊆ V ; | S | = i | δ ( S , G ) | . The edge isoperimetric peak of G is defined as b e ( G ) = max 1 ≤ j ≤ | V | b e ( j , G ) . Let b v ( G ) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t -ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k -ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. pth d of a tree 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 d 2 ), c 1 d ≤ b e ( T d 2 ) ≤ d and c 2 d ≤ b v ( T d 2 ) ≤ d where c 1 , c 2 are constants. For a complete t -ary tree of depth d (denoted as T d t ) and d ≥ c log t where c is a constant, we show that c 1 t d ≤ b e ( T d t ) ≤ t d and c 2 d t ≤ b v ( T d t ) ≤ d where c 1 , c 2 are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t -ary tree. 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 ℓ ( k ) = | { i ∈ N : 1 ≤ i ≤ | V | , b e ( i , G ) ≤ k } | . We show that ℓ ( k ) ≤ 2 2 η + k k .