22,551 research outputs found
On the structure of graphs without short cycles
The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction.
In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles.
In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities.
By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages).
Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction.
By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs.
Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages.
Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs.
As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage.
We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12.
In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q)
Energetics and electronic structures of polymeric all-benzene hollow-cages and planar networks
We studied the energetics and electronic structure of polymerized benzene hollow-cages and sheet using the density functional theory with the generalized gradient approximation. The energetics and electronic structure of the polymeric benzene cages and sheet depend on their size and dimensionality. Because of the symmetric network topology as well as the constituent benzene units, the cages possess highly bunched states around the Fermi level. The energy gap between the highest occupied and the lowest unoccupied states of the cages is approximately proportional to their curvature, owing to the decrease of the strain. The polymerized benzene sheet is a direct gap semiconductor with the gap of 2.4 eV between the less dispersive states of the highest branch of the valence and the lowest branch of the conduction bands
A formulation of a (q+1,8)-cage
Let be a prime power. In this note we present a formulation for
obtaining the known -cages which has allowed us to construct small
--graphs for and . Furthermore, we also obtain smaller
-graphs for even prime power .Comment: 14 pages, 2 figure
A construction of small (q-1)-regular graphs of girth 8
In this note we construct a new infinite family of -regular graphs of
girth and order for all prime powers , which are the
smallest known so far whenever is not a prime power or a prime power plus
one itself.Comment: 8 pages, 2 figure
Electronic structure, vibrational stability, infra-red, and Raman spectra of B24N24 cages
We examine the vibrational stability of three candidate structures for the
B24N24 cage and report their infra-red (IR) and Raman spectra. The candidate
structures considered are a round cage with octahedral O symmetry, a cage with
S_4 symmetry that satisfies the isolated square rule, and a cage of S_8
symmetry, which combines the caps of the (4,4) nanotube, and contains two extra
squares and octagons. The calculations are performed within density functional
theory, at the all electron level, with large basis sets, and within the
generalized gradient approximation. The vertical ionization potential (VIP) and
static dipole polarizability are also reported. The
S_4 and S_8 cages are energetically nearly degenerate and are favored over
the O cage which has six extra octagons and squares. The IR and Raman spectra
of the three clusters show notable differences providing thereby a way to
identify and possibly synthesize the cages.Comment: (Uses Elsevier style file; To appear in Chemical Physics Letters
Dimensional study of the dynamical arrest in a random Lorentz gas
The random Lorentz gas is a minimal model for transport in heterogeneous
media. Upon increasing the obstacle density, it exhibits a growing subdiffusive
transport regime and then a dynamical arrest. Here, we study the dimensional
dependence of the dynamical arrest, which can be mapped onto the void
percolation transition for Poisson-distributed point obstacles. We numerically
determine the arrest in dimensions d=2-6. Comparing the results with standard
mode-coupling theory reveals that the dynamical theory prediction grows
increasingly worse with . In an effort to clarify the origin of this
discrepancy, we relate the dynamical arrest in the RLG to the dynamic glass
transition of the infinite-range Mari-Kurchan model glass former. Through a
mixed static and dynamical analysis, we then extract an improved dimensional
scaling form as well as a geometrical upper bound for the arrest. The results
suggest that understanding the asymptotic behavior of the random Lorentz gas
may be key to surmounting fundamental difficulties with the mode-coupling
theory of glasses.Comment: 9 pages, 6 figure
SiH Aggregates: From Simple Building Blocks to Highly Magnetic Functionalized Materials
Density-functional theory based global geometry optimization is used to
scrutinize the possibility of using endohedrally-doped hydrogenated Si clusters
as building blocks for constructing highly magnetic materials. In contrast to
the known clathrate-type facet-sharing, the clusters exhibit a predisposition
to aggregation through double Si-Si bridge bonds. For the prototypical
CrSiH cluster we show that reducing the degree of hydrogenation
may be used to control the number of reactive sites to which other cages can be
attached, while still preserving the structural integrity of the building block
itself. This leads to a toolbox of CrSiH monomers with
different number of double "docking sites", that allows building network
architectures of any morphology. For (CrSiH) dimer and
[CrSiH](CrSiH) trimer structures we
illustrate that such aggregates conserve the high spin moments of the dopant
atoms and are therefore most attractive candidates for cluster-assembled
materials with unique magnetic properties. The study suggests that the
structural completion of the individual endohedral cages within the
doubly-bridge bonded structures and the high thermodynamic stability of the
obtained aggregates are crucial for potential synthetic polymerization routes
controlled dehydrogenation
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