Arne Hoffmann

Wiener index versus maximum degree in trees

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper, we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two different degrees.

Extremal Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

A linear-programming approach to the generalized Randić index

The generalized Randic´ index Rα(G) of a graph G is the sum of (dG(u)dG(v))α over all edges uv of G. Using a linear-programming approach, we establish results on graphs with a given number of vertices and edges and a bounded maximum degree that are of minimum generalized Randic´ index for α ∈ {- ½, - 1}.

On unique k-factors and unique [1,k]-factors in graphs

In the first part we examine bipartite graphs with a unique regular factor and present upper bounds for their number of edges. The second part deals with the maximum number of edges in graphs having a unique [1,k]-factor.

Extremal bipartite graphs with a unique k-factor

Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size has exactly 2k vertices of degree k and 2k vertices of degree |V (G)| 2 . As our main result we show that for k ≥ 1, p ≡ t (mod k), 0 ≤ t < k, ∗The results were proved while the author was working at the Lehrstuhl C für Mathematik, RWTH-Aachen. 182 A. Hoffmann, E. Sidorowicz and L. Volkmann a bipartite graph G of order 2p with a unique k-factor meets 2|E(G)| ≤ p(p + k)− t(k− t). Furthermore, we present the structure of extremal graphs.

Regular factors in regular multipartite graphs

Abstract We present sufficient conditions for a regular multipartite graph to have a regular factor. These conditions are best possible

Structural Remarks on Bipartite Graphs with Unique f-Factors

In this note we will derive some structural results for a bipartite graph G with a unique f-factor. Two necessary conditions will be that G is saturated, meaning that the addition of any edge leads to a second f-factor, and that fA, fB≥1. Here fA and fB are defined as the minimum of f over the vertices in the two partite sets A and B of G, respectively. Our main result states that G has at least fA + fB vertices for which dG (v) = f(v) holds.

Maximal Sets of 2-Factors in Complete Equipartite Graphs

Abstract We present a sufficient condition for a regular p -partite graph to have a 2-factor. Using this result, we then determine the 2-spectrum of the class Or p r of complete p -partite graphs with r vertices in each part. At which the 2-spectrum is defined by Sp 2 ( p , r ) = { m : there exists a maximal set of m egde-disjoint 2-factors of Or p r }.