Mikhail I Shtogrin

Pentaheptite Modifications of the Graphite Sheet

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-packed to hexagonal-close-packed packings in 3D. Symmetries realizable for these pentaheptite layers are three strip groups (periodic in one dimension), p1a1, p112, and p111, and five Fedorov groups (periodic in two dimensions), cmm, pgg, pg, p2, and p1. All can be constructed by simultaneous rotation of the central bonds of pyrene tilings of the graphite sheet. The unique lattice of cmm symmetry corresponds to the previously proposed pentaheptite carbon metal. Analogous pentagon-heptagon tilings on other surfaces including the torus, Klein bottle, and cylinder, face-regular tilings of pentagons and b-gons, and a full characterization of tilings involving isolated pairs and/or triples of pentagons are presented. The Kelvin paradigm of a continuum of structures arising from propagation of two original motifs has many potential applications in 2D and 3D.

4-valent plane graphs with 2-, 3- and 4-gonal faces

Call i-hedrite any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and p2+p3 = i. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any i-hedrite is a projection of an alternating link, whose components correspond to its central circuits. Call an i-hedrite irreducible, if it has no rail-road, i.e. a circuit of 4-gonal faces, in which every 4-gon is adjacent to two of its neighbors on opposite edges. We present the list of all i-hedrites with at most 15 vertices. Examples of other results: (i) All i-hedrites, which are not 3-connected, are identified. (ii) Any irreducible i-hedrite has at most i − 2 central circuits. (iii) All i-hedrites without self-intersecting central circuits are listed. (iv) All symmetry group of i-hedrites are listed.

Version of Zones and Zigzag Structure in Icosahedral Fullerenes and Icosadeltahedra

A circuit of faces in a polyhedron is called a zone if each face is attached to its two neighbors by opposite edges. (For odd-sized faces, each edge has a left and a right opposite partner.) Zones are called alternating if, when odd faces (if any) are encountered, left and right opposite edges are chosen alternately. Zigzag (Petrie) circuits in cubic (= trivalent) polyhedra correspond to alternating zones in their deltahedral duals. With these definitions, a full analysis of the zone and zigzag structure is made for icosahedral centrosymmetric fullerenes and their duals. The zone structure provides hypercube embeddings of these classes of polyhedra which preserve all graph distances (subject to a scale factor of 2) up to a limit that depends on the vertex count. These embeddings may have applications in nomenclature, atom/vertex numbering schemes, and in calculation of distance invariants for this subclass of highly symmetric fullerenes and their deltahedral duals.

Special ℓ1-graphs

In the first half of this chapter we follow [DePa01], where proofs of results below can be found. A graph G is an equicut graph if it admit an l 1-embedding, such that the equality holds in the left-hand side of the inequality (1.2) of chapter 1, concerning the size s(d G) of this embedding. Below s(d G) means the size of such equicut embedding. This means that, for such a graph, every S in the equality (1.1) of chapter 1 corresponds to an equicut δ(S), i.e. satisfy a S = 0 if and only if S partitions V into parts of size n 2 and n 2 , where n = |V |. Remind that a connected graph is called 2-connected (or 2-vertex-connected) if it remains connected after deletion of any vertex. Lemma 14.1 An equicut graph with at least four vertices is 2-connected.