Tomaž Pisanski

On the wiener index of a graph

A modification of the Weiner index which properly takes into account the symmetry of a graph is proposed. The explicit formulae for the modified Wiener index of path, cycle, complete bipartite, cube and lattice graphs are derived and compared with their standard Wiener index.

Use of the Szeged index and the revised Szeged index for measuring network bipartivity

We have revisited the Szeged index (Sz) and the revised Szeged index (Sz^*), both of which represent a generalization of the Wiener number to cyclic structures. Unexpectedly we found that the quotient of the two indices offers a novel measure for characterization of the degree of bipartivity of networks, that is, offers a measure of the departure of a network, or a graph, from bipartite networks or bipartite graphs, respectively. This is because the two indices assume the same values for bipartite graphs and different values for non-bipartite graphs. We have proposed therefore the quotient Sz/Sz^* as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient Sz/Sz^* illustrated on a number of smaller graphs as models of networks.

Graphical representation of proteins as four-color maps and their numerical characterization.

We put forward a novel compact 2-D graphical representation of proteins based on the concept of virtual genetic code and a four-color map. The novel graphical representation uniquely represents proteins and allows one to easily and quickly visually observe and inspect similarity/dissimilarity between them. It also leads to a novel protein descriptor, a 10-dimensional vector derived from a novel structure matrix S associated with the map. The introduced numerical characterization of proteins is not only useful for their comparative study, but also for cataloguing information on a single protein. The approach is illustrated with the A chain of human insulin and the A chain of human insulin analogue glargine.

Leapfrog transformations and polyhedra of Clar type

The so-called leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentagonal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spherical polyhedra can be classified according to their leapfrog order. A polyhedron is said to be of Clar type if there exists a set of faces that cover each vertex exactly once. It is shown that a fullerence is of Clar type if and only if it is a leapfrog transform of another fullerene. Several basic transformations on maps are defined by means of which the leapfrog and other transformations can be accomplished.

Counting symmetric configurations v 3

Abstract In this article we give tables of configurations v 3 for v ⩽18 and triangle-free configurations for v ⩽21 together with some statistics about some properties of the structures like transitivity, self-duality or self-polarity.

Characterizing graph drawing with eigenvectors

Two definitions of the problem of graph drawing are considered, and an analytical solution is provided for each of them. The solutions obtained make use of the eigenvectors of the Laplacian matrix of a related structure. The procedures give good results for symmetrical graphs, and they have already been used for drawing fullerene molecules in the literature. The analysis characterizes precisely what problems the two procedures are solving. It also illuminates why they can perform unsatisfactorily on asymmetrical graphs.

Computing the diameter in multiple-loop networks

Abstract A multiple-loop network is an undirected Cayley graph of a cyclic group. Algorithms for computing the diameter of multiple-loop networks are discussed. By a geometrical approach the problem of computing the diameter of a multiple-loop network (with k hops) is transformed to an equivalent problem in the integer lattice Z k . In the special case of double-loop networks this leads to an algorithm with running time O (log n ). This is an undirected analogue to the result of Cheng and Hwang, who recently found an O (log n ) algorithm for computing diameters of directed double-loop networks.

The matching polynomial of a polygraph

Abstract Polygraphs are introduced in order to describe and generalize the chemical notion of polymers. A general method for determining the matching polynomial of a polygraph is presented.

Weakly Flag-transitive Configurations and Half-arc-transitive Graphs

A configuration is weakly flag-transitive if its group of automorphisms acts intransitively on flags but the group of all automorphisms and anti-automorphisms acts transitively on flags. It is shown that weakly flag-transitive configurations are in one-to-one correspondence with bipartite12-arc-transitive graphs of girth not less than 6. Several infinite families of weakly flag-transitive configurations are given via their Levi graphs. Among others an infinite family of non-self-polar weakly flag-transitive configurations is constructed. The smallest known weakly flag-transitive configuration has 27 points and the smallest known non-self-polar weakly flag-transitive configuration has 34 points.

Edge-contributions of some topological indices and arboreality of molecular graphs

Some graph invariants can be computed by summing certain values, called edge-contributions over all edges of graphs. In this note we use edge-contributions to study relationships among three graph invariants, also known as topological indices in mathematical chemistry: Wiener index, Szeged index and recently introduced revised Szeged index. We also use the quotient between the Wiener index and the revised Szeged index to study tree-likeness of graphs.