Miranca Fischermann

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.

Remarks on the bondage number of planar graphs

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. In 1998, J.E. Dunbar, T.W. Haynes, U. Teschner, and L. Volkmann posed the conjecture b(G) ≤ Δ(G) + 1 for every nontrivial connected planar graph G. Two years later, L. Kang and J. Yuan proved b(G) ≤ 8 for every connected planar graph G, and therefore, they confirmed the conjecture for Δ(G) ≥ 7. In this paper we show that this conjecture is valid for all connected planar graphs of girth g(G) ≥ 4 and maximum degree Δ(G) ≥ 5 as well as for all not 3-regular graphs of girth g(G) ≥ 5. Some further related results and open problems are also presented.

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}.

A note on the number of matchings and independent sets in trees

We prove best-possible upper and lower bounds on the number of matchings in a tree in terms of the number of independent sets and the number of 2-independent sets.

Graphs having distance- n domination number half their order

For any positive integer n and any graph G a set D of vertices of G is a distance-n dominating set, if every vertex v ∈ V(G)-D has exactly distance n to at least one vertex in D. The distance-n domination number γ=n(G) is the smallest number of vertices in any distance-n dominating set. If G is a graph of order p and each vertex in G has distance n to at least one vertex in G, then the distance-n domination number has the upper bound p/2 as Ores upper bound on the classical domination number. In this paper, a characterization is given for graphs having distance-n domination number equal to half their order, when the diameter is greater or equal 2n - 1. With this result we confirm a conjecture of Boland, Haynes, and Lawson.

Block graphs with unique minimum dominating sets

For any graph G a set D of vertices of G is a dominating set, if every vertex v ∈ V (G) − D has at least one neighbor in D. The domination number � (G) is the smallest number of vertices in any dominating set. In this paper, a characterization is given for block graphs having a unique minimum dominating set. With this result, we generalize a theorem of Gunther, Hartnell, Markus and Rall for trees. c � 2001 Elsevier Science B.V. All rights reserved.

Maximum graphs with a unique minimum dominating set

We present a conjecture on the maximum number of edges of a graph that has a unique minimum dominating set. We verify our conjecture for some special cases and prove a weakened version of this conjecture in general.

The Numbers of Shared Upper Bounds Determine a Poset

Abstract We prove that a finite poset P=(V,≤) is determined up to some permutation of its elements by the function

Unique minimum domination in trees.

m_{P}\colon\ {V\choose2}\to\mathbf{N}_{0}