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- Peter Dankelmann, David P. Day, David Erwin, Simon Mukwembi, Henda C. Swart
- Discrete Mathematics
- 2009

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 wS(v) = ∑ u∈S 1 2d(u,v)−1 if v 6∈ S, and wS(v) = 2 if v ∈ S. If, for each v ∈ V (G), we have wS(v) ≥ 1, then S is an exponential… (More)

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we establish bounds on the mean eccentricity of a graph. We then examine the change in the average eccentricity when a graph is replaced by a spanning subgraph, in particular the two… (More)

- Peter Dankelmann, Ivan Gutman, Simon Mukwembi, Henda C. Swart
- Discrete Mathematics
- 2009

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 We of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on We in terms of order and size. In particular we prove the asymptotically sharp upper… (More)

- Peter Dankelmann, Neil J. Calkin
- Ars Comb.
- 2004

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)

- Elias Dahlhaus, Peter Dankelmann, Wayne Goddard, Henda C. Swart
- Discrete Applied Mathematics
- 2003

For a graph G with weight function w on the vertices, the total distance of G is the sum over all unordered pairs of vertices x and y of w(x)w(y) times the distance between x and y. A MAD tree of G is a spanning tree with minimum total distance. We develop a linear-time algorithm to find a MAD tree of a distancehereditary graph; that is, those graphs where… (More)

- Peter Dankelmann, Ortrud R. Oellermann
- Discrete Applied Mathematics
- 2003

- Peter Dankelmann, Lutz Volkmann
- Journal of Graph Theory
- 1997

- Peter Dankelmann, Lutz Volkmann
- Ars Comb.
- 1995

- Peter Dankelmann
- Discrete Applied Mathematics
- 1994

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.

- Peter Dankelmann, Ivan Gutman, Simon Mukwembi, Henda C. Swart
- Discrete Applied Mathematics
- 2009

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 degw 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) ≤ 14 nd(n− d) 2 +O(n7/2) for graphs of order n and diameter d. As a corollary we… (More)