Tomaz Pisanski

The 10-cages and derived configurations

Symmetry properties of the three 10-cages on 70 vertices are investigated. Being bipartite, these graphs are Levi graphs of triangle- and quadrangle-free (353) configurations. For each of these graphs a Hamilton cycle is given via the associated LCF notation. Furthermore, the automorphism groups of respective orders 80, 120, and 24 are computed. A special emphasis is given to the Balaban 10-cage, the first known example of a 10-cage (Rev. Roumaine Math. Pure Appl. 18 (1973) 1033-1043), and the corresponding Balaban configuration. It is shown that the latter is linear, that is, it can be realized as a geometric configuration of points and lines in the Euclidean plane. Finally, based on the Balaban configuration, an infinite series of linear triangle-free and quadrangle-free ((7n)3) configurations is produced for each odd integer n?5.

Cyclic Haar graphs

For a given group Γ with a generating set A, a dipole with |A| parallel arcs (directed edges) labeled by elements of A gives rise to a voltage graph whose covering graph, denoted by H(Γ,A) is a bipartite, regular graph, called a bi-Cayley graph. In the case when Γ is abelian we refer to H(Γ,A) as the Haar graph of Γ with respect to the symbol A. In particular for Γ cyclic the above graph is referred to as a cyclic Haar graph. A basic theory of cyclic Haar graphs is presented.

Small Triangle-Free Configurations of Points and Lines

AbstractIn the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (183) triangle-free configuration and its Levi graph is the generalized Petersen graph G(18,5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (203) and the unique point transitive triangle-free configuration (213).

The Gray graph revisited

Let G = (V,E) be a graph. A set S ⊆ V is a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination number of G, denoted by Υt(G), is the minimum cardinality of a total dominating set of G. We establish a property of minimum total dominating sets in graphs. If G is a connected graph of order n ≥ 3, then (see [3]) Υt(G) ≤ 2n-3. We show that if G is a connected graph of order n with minimum degree at least 2, then either Υt(G) ≤ 4n-7 or G e {C3, C5, C6, C10}. A characterization of those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G) e 2 and Υt(G) ≥ 4n-7 is obtained. We establish that if G is a connected graph of size q with minimum degree at least 2, then Υt(G) ≤(q + 2)-2. Connected graphs G of size q with minimum degree at least 2 satisfying Υt(G) > q-2 are characterized.

Semisymmetric graphs from polytopes

Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

Determining membrane permeability of giant phospholipid vesicles from a series of videomicroscopy images

A technique for determining the permeability of a phospholipid membrane on a single giant unilamellar vesicle (GUV) is described, which complements the existing methods utilizing either a planar black lipid membrane or sub-micrometre-sized liposomes. A single GUV is transferred using a micropipette from a solution of a nonpermeable solute into an iso-osmolar solution of a solute with a higher membrane permeability. Osmotical swelling of the vesicle is monitored with a CCD camera mounted on a phase contrast microscope, and a sequence of images is obtained. On each image, the points on the vesicle contour are determined using Sobel filtering with adaptive binarization threshold, and from these, the vesicle radius is calculated with great accuracy. From the time dependence of the vesicle radius, the membrane permeability is obtained. Using a test set of data, the method provided a consistent estimate of the POPC membrane permeability for glycerol, P = 1.7 × 10−8 m s−1, with individual samples ranging from P = 1.61 × 10−8 m s−1 to P = 1.98 × 10−8 m s−1. This value is ≈40% lower than the one obtained on similar systems. Possible causes for this discrepancy are discussed.

Generation of various classes of trivalent graphs

It turns out that there exist numerous useful classes of cubic graphs. Some are needed in connection with maps, hypermaps, configurations, polytopes, or covering graphs. In this paper, we briefly explore these connections and give motivation why some classes of cubic graphs should be generated. Then we describe the algorithms we used to generate these classes. The results are presented in various tables.

On numerical characterization of cyclicity

We propose characterizing the cyclicity of molecular graphs by considering their D/DD matrix. Each nondiagonal element of D/DD is a quotient of the corresponding elements of the distance matrix D and the detour matrix DD of a graph. In particular, we are using the leading eigenvalue of the D/DD matrix as a descriptor of cyclicity and are investigating for monocyclic graphs Cn how this eigenvalue depends on the number of vertexes n, as n approaches infinity.

An algorithm for K-convex closure and an application

An algorithm for computing the K-convex closure of a subgraph relative to a given equivalence relation R among edges of a graph is given.For general graph and arbitrary relation R the time complexity is O(q n 2 + mn), where n is the number of vertices, m is the number of edges and q is the number of equivalence classes of R.A special case is an O(mn) algorithm for the usual k-convexity.We also show that Cartesian graph bundles over triangle free bases can be recognized in O(mn) time and that all representations of such graphs as Cartesian graph bundles can be found in O(mn 2) time.

ISOPERIMETRIC QUOTIENT FOR FULLERENES AND OTHER POLYHEDRAL CAGES

The notion of Isoperimetric Quotient (IQ) of a polyhedron has been already introduced by Polya. It is a measure that tells us how spherical is a given polyhedron. If we are given a polyhedral graph it can be drawn in a variety of ways in 3D space. As the coordinates of vertices belonging to the same face may not be coplanar the usual definition of IQ fails. Therefore, a method based on a proper triangulation (obtained from omni-capping) is developed that enables one to extend the definition of IQ and compute it for any 3D drawing. The IQs of fullerenes and other polyhedral cages are computed and compared for their NiceGraph and standard Laplacian 3D drawings. It is shown that the drawings with the maximal IQ values reproduce well the molecular mechanics geometries in the case of fullerenes and exact geometries for Platonic and Archimedean polyhedra.