Peter Dankelmann

Average distance, minimum degree, and spanning trees

A graph G is perfectly orderable, if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph, if it contains no P4, 2K2, and C4 as induced subgraph. A theorem of Chvatal, Hoang, Mahadev, and de Werra states that a graph is perfectly orderable, if it is the union of two threshold graphs. In this article, we investigate possible generalizations of the above theorem. Hoang has conjectured that, if G is the union of two graphs G1 and G2, then G is perfectly orderable whenever G1 and G2 are both P4-free and 2K2-free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter-example to this conjecture, and we prove a special case of it: if G1 and G2 are two edge-disjoint graphs that are P4-free and 2K2-free, then the union of G1 and G2 is perfectly orderable.

The average Steiner distance of a graph

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.

Maximum sizes of graphs with given domination parameters

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

SOME CENTRALITY RESULTS NEW AND OLD

We prove a number of results on betweenness and closeness centrality and centralization. In particular, we prove the much used normalization expression for closeness centrality first given by Freeman (1979), correcting an error in the justification given in his paper. We explore the relationship between betweenness and the cutting number and use these results to prove and correct some centrality and centralization formulae first proposed by Borgatti and Everett (1997).

On the average Steiner distance of graphs with prescribed properties

The average n-distance of a connected graph G, p,,(G), is the average of the Steiner distances of all n-sets of vertices of G. In this paper, we give bounds on pn for two-connected graphs and for k-chromatic graphs. Moreover, we show that pn(G) does not depend on the n-diameter of G. Let G = (V. E) be a connected graph of order p. The average distance of G, p(G), is defined to be the average of all distances between pairs of vertices in G, i.e.

MAD Trees and distance-hereditary graphs

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 distance-hereditary graph; that is, those graphs where distances are preserved in every connected induced subgraph.

Degree sequence conditions for maximally edge-connected graphs depending on the clique number

Abstract In this paper we give degree sequence conditions for the equality of edge-connectivity and minimum degree of a graph with given clique number.

Bounds on the average connectivity of a graph

In this paper, we consider the concept of the average connectivity of a graph, defined to be the average, over all pairs of vertices, of the maximum number of internally disjoint paths connecting these vertices. We establish sharp bounds for this parameter in terms of the average degree and improve one of these bounds for bipartite graphs with perfect matchings. Sharp upper bounds for planar and outerplanar graphs and cartesian products of graphs are established. Nordhaus-Gaddum-type results for this parameter and relationships between the clique number and chromatic number of a graph are also established.

Minimum average distance of strong orientations of graphs

The average distance of a graph (strong digraph) G, denoted by µ(G) is the average, among the distances between all pairs (ordered pairs) of vertices of G. If G is a 2-edge-connected graph, then µ→min(G) is the minimum average distance taken over all strong orientations of G. A lower bound for µ→min(G) in terms of the order, size, girth and average distance of G is established and shown to be sharp for several complete multipartite graphs. It is shown that there is no upper bound for µ→min(G) in terms of µ(G). However, if every edge of G lies on 3-cycle, then it is shown that µ→min(G) ≤ 7/4 µ(G). This bound is improved for maximal planar graphs to 5/3 µ(G) and even further to 3/2 µ(G) for eulerian maximal planar graphs and for outerplanar graphs with the property that every edge lies on 3-cycle. In the last case the bound is shown to be sharp.

On strong distances in oriented graphs

Let D be a strongly connected digraph. The strong distance between two vertices u and v in D, denoted by sdD(u,v) is the minimum size of a strongly connected subdigraph of D containing u and v. The strong eccentricity, se(u), of a vertex u of D, is the strong distance between u and a vertex farthest from u. The minimum strong eccentricity among the vertices of D is the strong radius, srad(D), and the maximum strong eccentricity is the strong diameter, sdiam(D). For asymmetric digraphs (that is, oriented graphs) we present bounds on the strong radius in terms of order and on the strong diameter in terms of order, girth and connectivity.