We give an overview of some of the mathematical results of Dominique de Caen. These include a short proof of König’s theorem, results on Turán numbers, biclique partitions, clique coverings, hypergraphical Steiner systems, p-ranks, stellar permutations, the construction of new distance-regular graphs, large sets of equiangular lines, a bound on the probability of a union, and more.

We are interested in what sizes of cliques are to be found in any arbitrary spanning graph of a Steiner triple system S. In this paper we investigate spanning graphs of projective Steiner triple systems, proving, not surprisingly, that for any positive integer k and any sufficiently large projective Steiner triple system S, every spanning graph of S contains a clique of size k.

The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When a polygon is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. We show that some planar straight-line graphs cannot be triangulated with constant degradation, even with an unbounded number of Steiner points and triangles. T...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

We consider a new Steiner tree problem, called vertex-cover-weighted Steiner tree problem. This problem defines the weight of a Steiner tree as the minimum weight of vertex covers in the tree, and seeks a minimum-weight Steiner tree in a given vertex-weighted undirected graph. Since it is included by the Steiner tree activation problem, the problem admits an O(logn)-approximation algorithm in g...

distance-balanced graphs are introduced as graphs in which every edge uv has the followingproperty: the number of vertices closer to u than to v is equal to the number of vertices closerto v than to u. basic properties of these graphs are obtained. in this paper, we study theconditions under which some graph operations produce a distance-balanced graph.

let $g$ be a connected graph, and let $d[g]$ denote the double graph of $g$. in this paper, we first derive closed-form formulas for different distance based topological indices for $d[g]$ in terms of that of $g$. finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

Given an edge-weighted graph G = (V,E) and a subset R of V , a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In this paper we consider the full Steiner tree problem when G is a unit disk graph. We present a 2...

The Steiner distance dc(S) of a subset S of nodes of a connected graph G is the minimum number of edges in a connected subgraph of G that contains S. We consider the behaviour of the expected value ILk (n) of dT(S) and the probability Pk(n, m) that dT(S) rn lover all subsets S of k nodes of trees T of order n from certain families of trees.