a b s t r a c tLet G be a graph and S ⊆ V (G). For each vertex u ∈ S and for eachThe smallest cardinality of an exponential dominating set is the exponential domination number, γ e (G). In this paper, we prove: (i) that if G is a connected graph of diameter d, then γ e (G) ≥ (d + 2)/4, and, (ii) that if G is a connected graph of order n, then γ e (G) ≤ 2 5 (n + 2).

The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n‐distance μn(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μn(G) ≤ μk(G) + μn+1−k(G) where 2 ≤ k ≤ n − 1. The range for the average Steiner n‐distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n − 1, it is shown that μn(T) ≤ (n/k)μk(T) and the range for μn(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n‐distance of a tree are outlined. © 1996 John Wiley & Sons, Inc.

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a b s t r a c tIf G is a connected graph with vertex set V , then the degree distance of G, D (G), is defined and d(u, v) denotes the distance between u and v. We prove the asymptotically sharp upper bound D (G) ≤ 1 4 nd(n − d) 2 + O(n 7/2 ) for graphs of order n and diameter d. As a corollary we obtain the bound D (G) ≤ 1 27 n 4 + O(n 7/2 ) for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].

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