How to Exhibit Toroidal Maps in Space

Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.     

An algebraic characterization of planar graphs

A cycle in a graph is a set of edges that covers each vertex an even number of times. A cocycle is a collection of edges that intersects each cycle in an even number of edges. A bicycle is a collection of edges that is both a cycle and a cocycle. The cycles, cocycles, and bicycles each form a vector space over the integers modulo two when addition is defined as symmetric difference of sets. In this paper we examine the relationship between the left-right paths in a planar graph and the cycle space, cocylce space, and bicycle space. We show that planar graphs are characterized by the existence of a diagonal—a double cover by tours that interacts with the cycle space, cocycle space, and bicycle space in a special manner. This generalizes a result of Rosenstiehl and Read that characterized those planar graphs with no nonempty bicycles. © 1995 John Wiley & Sons, Inc.     

Planar embeddings with infinite faces

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K₂ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003     

Influence of interparticle interactions on the kinetics of self-assembly and mechanical strength of nanoparticulate aggregates

Computer simulations based on Discrete Element Method have been performed in order to investigate the influence of interparticle interactions on the kinetics of self-assembly and the mechanical strength of nanoparticle aggregates.Three different systems have been considered.In the first system the interaction between particles has been simulated using the JKR (Johnson,Kendall and Roberts) contact theory,while in the second and third systems the interaction between particles has been simulated using van der Waals and electrostatic forces respectively.In order to compare the mechanical behaviour of the three systems,the magnitude of the maximum attractive force between particles has been kept the same in all cases.However,the relationship between force and separation distance differs from case to case and thus,the range of the interparticle force.The results clearly indicate that as the range of the interparticle force increases,the self-assembly process is faster and the work required to produce the mechanical failure of the assemblies increases by more than one order of magnitude.     

Obstruction sets for outer‐cylindrical graphs

A graph is outer‐cylindrical if it embeds in the sphere so that there are two distinct faces whose boundaries together contain all the vertices. The class of outer‐cylindrical graphs is closed under minors. We give the complete set of 38 minor‐minimal non‐outer‐cylindrical graphs, or equivalently, an excluded minor characterization of outer‐cylindrical graphs. We also give the obstruction sets under the related topological ordering and Y Δ‐ordering. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 42–64, 2001     

Characterization of synthetic and commercial trisiloxane surfactant materials

The organosilicone surfactant Silwet L‐77^® (L‐77), used as an agrochemical adjuvant, is a mixture comprised predominantly of [(CH₃)₃SiO]₂? (CH₃)Si? (CH₂)₃? (OCH₂CH₂)_n? OCH₃ oligomers (n = 3–16, average n ≈ 7.5). The commercially available L‐77 mixture was purified by reversed‐phase high‐performance liquid chromatography (HPLC) to obtain individual trisiloxane surfactant components. Pure oligomers (n = 3, 6 and 9) were also synthesized. Synthesis was achieved by hydrosilylation of monomeric ethoxylate monomethyl ether starting reagents. Pure hexa‐ and nona‐ethylene glycols were produced by condensation of smaller oligomers. Atmospheric‐pressure ionization mass spectrometry (MS) methods were used to characterize fully the commercial L‐77 product and synthesized or isolated components. The application of Fourier‐transform ion cyclotron resonance MS and online HPLC–electrospray ionization MS techniques to the analysis of this surfactant are described here. The application of these analytical techniques also enabled elucidation of the synthetic by‐products present in the commercial formulation. In addition, physico‐chemical properties specific to agrochemical uses, such as droplet spread areas on plant foliage and surface tension for the different oligomer solutions, are also reported. Copyright © 2004 John Wiley & Sons, Ltd.     

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2‐colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α‐size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α‐size of trees with maximum degree three. Let α₃(n) be the smallest α‐size among all trees with n vertices, each of degree at most three. We prove that α₃(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α₃(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999     

The Relative Maximum Genus of a Graph

A relative embedding of a graph in a surface with respect to a set of closed walks is one where each of the prescribed closed walks bounds a face of the embedding In the special case where the set of closed walks is empty, this amounts to the usual concept of a graph embedding. We present a formula for the maximum (orientable) genus of the surface on which a graph has a relative embedding with respect to a set of closed walks.     

Utilisation of electrospray time-of-flight mass spectrometry for solving complex fragmentation patterns: application to benzoxazinone derivatives

In this paper we describe the application of electrospray time-of-flight mass spectrometry (ESI-TOFMS) to structural elucidation of the fragment ions formed from a range of natural and synthetic allelochemical derivatives. The extensive mass spectrometric characterisation of ten non-glucosylated benzoxazinone derivatives using this method is described here for the first time. The analytes include six naturally occurring 1,4-benzoxazin-3(4H)-one derivatives, including the hydroxamic acids DIMBOA [2,4-dihydroxy-7-methoxy-2H-1,4-benzoxazin-3(4H)-one] and DIBOA [2,4-dihydroxy-2H-1,4-benzoxazin-3(4H)-one], lactams HBOA [2-hydroxy-2H-1,4-benzoxazin-3(4H)-one] and HMBOA [2-hydroxy-7-methoxy-2H-1,4-benzoxazin-3(4H)-one], benzoxazolinones BOA [benzoxazolin-2(3H)-one] and MBOA [6-methoxy-benzoxazolin-2(3H)-one] and four synthetic variations, 2'H-DIBOA [4-hydroxy-2H-1,4-benzoxazin-3(4H)-one], 2'OMe-DIBOA [2-methoxy-4-hydroxy-2H-1,4-benzoxazin-3(4H)-one], 2'H-HBOA [2H-1,4-benzoxazin-3(4H)-one] and 2'OMe-HBOA [2-methoxy-2H-1,4-benzoxazin-3(4H)-one]. Assignments of the mass spectral fragments were aided by elemental composition calculation results, comparison of structural analogues and background literature, and acquired knowledge regarding feasible structures for the compounds. The influence of substituents on the chemical reactivity of the compounds with respect to the observed MS behaviour over varying nozzle potentials is addressed and, through comparison of the structural analogues, generic fragmentation patterns have also been identified.     

Geometric Realization of a Triangulation on the Projective Plane with One Face Removed

Let M be a map on a surface F ². A geometric realization of M is an embedding of F ² into a Euclidean 3-space ?³ such that each face of M is a flat polygon. We shall prove that every triangulation G on the projective plane has a face f such that the triangulation of the Möbius band obtained from G by removing the interior of f has a geometric realization.