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Let G be a graph and S ⊆ V (G). For each vertex u ∈ S and for each v ∈ V (G) − S, we define d(u, v) = d(v, u) to be the length of a shortest path in V (G)−(S−{u}) if such a path exists, and ∞ otherwise. Let v ∈ V (G). We define w S (v) = u∈S 1 2 d(u,v)−1 if v ∈ S, and w S (v) = 2 if v ∈ S. If, for each v ∈ V (G), we have w S (v) ≥ 1, then S is an… (More)

A sharp upper bound on the average distance of a graph depending on the order and the independence number is given. As a corollary we obtain the maximum average distance of a graph with given order and matching number. All extremal graphs are determined.

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index W e of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on W e in terms of order and size. In particular we prove the asymptotically sharp upper… (More)

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in… (More)

If G is a connected graph with vertex set V , then the degree distance of G, D (G), is defined as {u,v}⊆V (deg u + deg v) d(u, v), where deg w is the degree of vertex w, 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… (More)

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph… (More)