Stephan G. Wagner

The matching energy of a graph

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G. We define the matching energy (ME) of the graph G as the sum of the absolute values of the zeros of the matching polynomial of G, and determine its basic properties. It is pointed out that the chemical applications of ME go back to the 1970s.

Chemical Trees Minimizing Energy and Hosoya Index

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.

Correlation of Graph-Theoretical Indices

The correlation of graph characteristics, such as the number of independent vertex or edge subsets, the number of connected subsets, or the sum of distances, which also play a role in combinatorial chemistry, is studied by a generating function approach and asymptotic analysis. It is shown how an asymptotic formula for the correlation coefficient can be obtained when simply generated families of trees are investigated. For rooted ordered trees, the calculations are done explicitly. Further feasible correlation measures are discussed.

On the number of spanning trees on various lattices

We consider the number of spanning trees in lattices; for a lattice , one defines the bulk limit , where NST(G) is the number of spanning trees in a finite section G of . Explicit values for are known in various special cases. In this note we describe a simple yet effective method to deduce relations between the values of for different lattices by means of electrical network theory.

Enumeration problems for classes of self-similar graphs

We describe a general construction principle for a class of self-similar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical self-similar graphs such as the Sierpinski graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets.

Graphs, partitions and Fibonacci numbers

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number >2^n^-^1+5 have diameter = ~. This is proved by using a natural correspondence between partitions of integers and star-like trees.

Asymptotic enumeration of extensional acyclic digraphs

The enumeration of extensional acyclic digraphs, which have the property that the outneighbourhoods are pairwise distinct, was considered in a recent article of Policriti and Tomescu. Several asymptotic questions were left as open problems. In this article, we determine the asymptotic number of such digraphs and show that a number of distributional results can be carried over from ordinary acyclic digraphs. In particular, we consider the distribution of the number of sources, the number of arcs, the maximum rank and the number of vertices of maximum rank, thereby also proving some conjectures made by Policriti and Tomescu. Finally, we study a very similar class of acyclic digraphs and provide analogous distributional results.

Hitting Times, Cover Cost, and the Wiener Index of a Tree

We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees, we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are “easier to reach” by a random walk, but “more difficult to get out of.” We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behavior or random walk on a graph to its eigenvalues in a new perspective.

Enumeration of matchings in families of self-similar graphs

The number of matchings of a graph G is an important graph parameter in various contexts, notably in statistical physics (dimer-monomer model). Following recent research on graph parameters of this type in connection with self-similar, fractal-like graphs, we study the asymptotic behavior of the number of matchings in families of self-similar graphs that are constructed by a very general replacement procedure. Under certain conditions on the geometry of the graphs, we are able to prove that the number of matchings generally follows a doubly exponential growth. The proof depends on an independence theorem for the number of matchings that has been used earlier to treat the special case of Sierpinski graphs. We also further extend the technique to the matching-generating polynomial (equivalent to the partition function for the dimer-monomer model) and provide a variety of examples.

Enumeration of the adjunctive hierarchy of hereditarily finite sets

Hereditarily finite sets (sets which are finite and have only hereditarily finite sets as members) are basic mathematical and computational objects, and also stand at the basis of some programming languages. This raises the need for efficient representation of such sets, for example by numbers. In 2008, Kirby proposed an adjunctive hierarchy of hereditarily finite sets, based on the fact that they can also be seen as built up from the empty set by repeated adjunction, that is, by the addition of a new single element drawn from the already existing sets to an already existing set. Determining the cardinality