The nine finite, planar, 3-connected, edge-transitive graphs have been known and studied for many centuries. The infinite, locally finite, planar, 3-connected, edge-transitive graphs can be ...classified according to the number of their ends (the supremum of the number of infinite components when a finite subgraph is deleted). Prior to this study the 1-ended graphs in this class were identified by Grunbaum and Shephard as 1-skeletons of tessellations of the hyperbolic plane; Watkins characterized the 2-ended members. Any remaining graphs in this class must have uncountably many ends. In this work, infinite-ended members of this class are shown to exist. A more detailed classification scheme in terms of the tupes of Petrie walks in the graphs in this class and the local structure of their automorphism groups is presented. Explicit constructions ar devised for all of the graphs in most of the classes under this new classification. Also included are partial results toward the complete description of the graphs in the few remaining classes.
Coxeter's classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each ...geodesic dome is characterized by its signature: a plane graph on twelve vertices with labeled angles and edges. In the case of the Coxeter geodesic domes, the plane graph is the icosahedron, all angles are labeled one, and all edges are labeled by the same pair of integers (p,q). Edges with these "Coxeter coordinates" correspond to straight line segments joining twovertices of $\Lambda$, the regular triangular tessellation of the plane, and the faces of the icosahedron are filled in with equilateral triangles from $\Lambda$ whose sides have coordinates (p,q). We describe the construction of the signature for any geodesic dome. In turn, we describe how each geodesic dome may be reconstructed from its signature: the angle and edge labels around each face of the signature identify that face with a polygonal region of $\Lambda$ and, when the faces are filled by the corresponding regions, the geodesic dome is reconstituted. The signature of a fullerene is the signature of its dual. For each fullerene, the separation of its pentagons, the numbers of its vertices, faces, and edges, and its symmetry structure are easily computed directly from its signature. Also, it is easy to identify nanotubes by their signatures.
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A mathematical note in reference to Donald E. Campbell & Jerry S. Kelly, "Social Welfare Functions that Satisfy Pareto, Anonymity, and Neutrality, but Not Independence of Irrelevant Alternatives," ...appearing in the same issue of Social Choice & Welfare. Adapted from the source document.
Kekuléan benzenoids Graver, Jack E.; Hartung, Elizabeth J.
Journal of mathematical chemistry,
03/2014, Letnik:
52, Številka:
3
Journal Article
Recenzirano
A
Kekulé structure
for a benzenoid or a fullerene
Γ
is a set of edges
K
such that each vertex of
Γ
is incident with exactly one edge in
K
, i.e. a perfect matching. All fullerenes admit a Kekulé ...structure; however, this is not true for benzenoids. In this paper, we develop methods for deciding whether or not a given benzenoid admits a Kekulé structure by constructing Kekulé structures that have a high density of benzene rings. The
benzene rings
of the Kekulé structure
K
are the faces in
Γ
that have exactly three edges in
K
. The
Fries number
of
Γ
is the maximum number of benzene rings over all possible Kekulé structures for
Γ
and the set of benzene rings giving the Fries number is called a
Fries set
. The
Clar number
is the maximum number of independent benzene rings over all possible Kekulé structures for
Γ
and the set of benzene rings giving the Clar number is called a
Clar set
. Our method of constructing Kekulé structures for benzenoids generally gives good estimates for the Clar and Fries numbers, often the exact values.
Self-Dual Spherical Grids Graver, Jack E.; Hartung, Elizabeth J.
The Electronic journal of combinatorics,
02/2014, Letnik:
21, Številka:
1
Journal Article
Recenzirano
Self-dual plane graphs have been studied extensively. C. A. B Smith and W. T. Tutte published A class of self-dual maps in 1950; in 1992, Archdeacon and Richter described a method for constructing ...all self-dual plane graphs and a second construction was produced by Servatius and Christopher in 1992. Both constructions are inductive. In this paper, we produce four templates from which all self-dual plane graphs with maximum degree 4 (self-dual spherical grids) can be constructed. The self-dual spherical grids are further subdivided into 27 basic automorphism classes. Self-dual spherical grids in the same automorphism class have similar architecture. A smallest example of each class is constructed.
Clar and Fries numbers for benzenoids Graver, Jack E.; Hartung, Elizabeth J.; Souid, Ahmed Y.
Journal of mathematical chemistry,
09/2013, Letnik:
51, Številka:
8
Journal Article
Recenzirano
A
Kekulé structure
of a benzenoid or a fullerene
Γ
is a set of edges
K
such that each vertex of
Γ
is incident with exactly one edge in
K
. The set of faces in
Γ
that have exactly three edges in
K
are ...called the
benzene faces
of
K
. The Fries number of
Γ
is the maximum number of benzene faces over all possible Kekulé structures for
Γ
. The Clar number is the maximum number of independent benzene faces over all possible Kekulé structures for
Γ
. It is often assumed, but never proved, that some set of independent benzene faces giving the Clar number is a subset of a set of benzene faces giving the Fries number. In Hartung (The Clar structure of fullerenes, Ph.D. Dissertation. Syracuse University,
2012
) it is shown that this assumption is false for a large class of fullerenes. In this paper, we prove that this assumption is valid for a large a class of benzenoids.
We explore the relationship between Kekulé structures and maximum face independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. For the class of leap-frog ...fullerenes, we show that a maximum face independence set corresponds to a Kekulé structure with a maximum number of benzene rings and may be constructed by partitioning the pentagonal faces into pairs and 3-coloring the faces with the exception of a very few faces along paths joining paired pentagons. We also obtain some partial results for non-leap-frog fullerenes.
We explore the structure of the maximum vertex independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. At the same time, we will consider benzenoids: plane graphs ...with hexagonal faces and one large outer face. In the case of fullerenes, a maximum vertex independence set may constructed as follows:
(i)
Pair up the pentagonal faces.
(ii)
Delete the edges of a shortest path in the dual joining the paired faces to get a bipartite subgraph of the fullerene.
(iii)
Each of the deleted edges will join two vertices in the same cell of the bipartition; eliminating one endpoint of each of the deleted edges results in two independent subsets.
The main part of this paper is devoted to showing that for a properly chosen pairing, the larger of these two independent subsets will be a maximum independent set. We also prove that the construction of a maximum vertex independence set in a benzenoid is similar with the dual paths between pentagonal faces replaced by dual circuits through the outside face. At the end of the paper, we illustrate this method by computing the independence number for each of the icosahedral fullerenes.
With as few as eight individuals and five alternatives, there are 561, 304, 372, 286, 875, 579, 077, 983 strategy-proof social choice rules.
► Counting the number of strategy-proof rules. ► ...Connection with Dedekind numbers. ► Counting strategy-proof rules satisfying additional criteria: neutrality, anonymity, full range.
Fullerene patches II Graver, Jack E.; Graves, Christina; Graves, Stephen J.
Ars mathematica contemporanea,
01/2014, Letnik:
7, Številka:
2
Journal Article
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