Peter E. Pd Dr. rer. nat. habil. John

Spectra of toroidal graphs

An n-fold periodic locally finite graph in the Euclidean n-space may be considered the parent of an infinite class of n-dimensional toroidal finite graphs. An elementary method is developed that allows the characteristic polynomials of these graphs to be factored, in a uniform manner, into smaller polynomials, all of the same size. Applied to the hexagonal tessellation of the plane (the graphite sheet), this method enables the spectra and corresponding orthonormal eigenvector systems for all toroidal fullerenes and (3, 6)-cages to be explicitly calculated. In particular, a conjecture of P.W. Fowler on the spectra of (3, 6)-cages is proved.

On Statistics of Graph Energy

Abstract The energy EG of a graph G is the sum of the absolute values of the eigenvalues of G. In the case whene G is a molecular graph, EG is closely related to the total π-electron energy of the corresponding conjugated molecule. We determine the average value of the difference between the energy of two graphs, randomly chosen from the set of all graphs with n vertices and m edges. This result provides a criterion for deciding when two (molecular) graphs are almost coeneigetic.

An Algorithm for Counting Spanning Trees in Labeled Molecular Graphs Homeomorphic to Cata-Condensed Systems

The algorithmic method of Gutman and Mallion (1993), for calculating the number of spanning trees in the (labeled) molecular graphs of cata-condensed systems containing rings of only one size, was subsequently generalized by John and Mallion (1996) to make it applicable to such systems comprising rings of more than one size; this latter algorithm is thus generally valid for enumerating the spanning trees in the molecular graphs of any cata-condensed system. This algorithmic philosophy is extended here in order to devise a procedure that is suitable for an even more general class of molecular graphsnamely, those homeomorphic to the molecular graphs of cata-condensed systems. An example of its use is illustrated by explicitly computing the numerical value for the complexity of a (hypothetical) pentacyclic network consisting of two four-membered rings, two five-membered rings, and a nine-membered ring, giving rise to a spanning-tree count entirely in accord with that predicted via the theorem of Gutman, Mall...

Contributions to the theory of benzenoid isomers. Part 2: Building-up principles

Abstract The fundamental building-up principle states that three types of addition of hexagons are sufficient for generating all benzenoids. The principle is applied to benzenoid CnHs isomers with the result of another principle, which implies the three addition units C4H2, C3H and C2.