300 research outputs found
Fullerenes with the maximum Clar number
The Clar number of a fullerene is the maximum number of independent resonant
hexagons in the fullerene. It is known that the Clar number of a fullerene with
n vertices is bounded above by [n/6]-2. We find that there are no fullerenes
whose order n is congruent to 2 modulo 6 attaining this bound. In other words,
the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is
bounded above by [n/6]-3. Moreover, we show that two experimentally produced
fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a
graph-theoretical characterization for fullerenes, whose order n is congruent
to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3
(respectively, [n/6]-2)
Resonance graphs of plane bipartite graphs as daisy cubes
We characterize all plane bipartite graphs whose resonance graphs are daisy
cubes and therefore generalize related results on resonance graphs of benzenoid
graphs, catacondensed even ring systems, as well as 2-connected outerplane
bipartite graphs. Firstly, we prove that if is a plane elementary bipartite
graph other than , then the resonance graph is a daisy cube if and
only if the Fries number of equals the number of finite faces of , which
in turn is equivalent to being homeomorphically peripheral color
alternating. Next, we extend the above characterization from plane elementary
bipartite graphs to all plane bipartite graphs and show that the resonance
graph of a plane bipartite graph is a daisy cube if and only if is
weakly elementary bipartite and every elementary component of other than
is homeomorphically peripheral color alternating. Along the way, we prove
that a Cartesian product graph is a daisy cube if and only if all of its
nontrivial factors are daisy cubes
Maximizing the minimum and maximum forcing numbers of perfect matchings of graphs
Let be a simple graph with vertices and a perfect matching. The
forcing number of a perfect matching of is the smallest
cardinality of a subset of that is contained in no other perfect matching
of . Among all perfect matchings of , the minimum and maximum values
of are called the minimum and maximum forcing numbers of , denoted
by and , respectively. Then . Che and Chen
(2011) proposed an open problem: how to characterize the graphs with
. Later they showed that for bipartite graphs , if and
only if is complete bipartite graph . In this paper, we solve the
problem for general graphs and obtain that if and only if is a
complete multipartite graph or ( with arbitrary additional
edges in the same partite set). For a larger class of graphs with
we show that is -connected and a brick (3-connected and
bicritical graph) except for . In particular, we prove that the
forcing spectrum of each such graph is continued by matching 2-switches and
the minimum forcing numbers of all such graphs form an integer interval
from to
Relations between global forcing number and maximum anti-forcing number of a graph
The global forcing number of a graph G is the minimal cardinality of an edge
subset discriminating all perfect matchings of G, denoted by gf(G). For any
perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M
such that G-S has a unique perfect matching is called the anti-forcing number
of M,denoted by af(G, M). The maximum anti-forcing number of G among all
perfect matchings is denoted by Af(G). It is known that the maximum
anti-forcing number of a hexagonal system equals the famous Fries number.
We are interested in some comparisons between the global forcing number and
the maximum anti-forcing number of a graph. For a bipartite graph G, we show
that gf(G)is larger than or equal to Af(G). Next we mainly extend such result
to non-bipartite graphs, which is the set of all graphs with a perfect matching
which contain no two disjoint odd cycles such that their deletion results in a
subgraph with a perfect matching. For any such graph G, we also have gf(G) is
larger than or equal to Af(G) by revealing further property of non-bipartite
graphs with a unique perfect matching. As a consequence, this relation also
holds for the graphs whose perfect matching polytopes consist of non-negative
1-regular vectors. In particular, for a brick G, de Carvalho, Lucchesi and
Murty [4] showed that G satisfying the above condition if and only if G is
solid, and if and only if its perfect matching polytope consists of
non-negative 1-regular vectors.
Finally, we obtain tight upper and lower bounds on gf(G)-Af(G). For a
connected bipartite graph G with 2n vertices, we have that 0 \leq gf(G)-Af(G)
\leq 1/2 (n-1)(n-2); For non-bipartite case, -1/2 (n^2-n-2) \leq gf(G)-Af(G)
\leq (n-1)(n-2).Comment: 19 pages, 11 figure
Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics
Materials science and the study of the electronic properties of solids are a
major field of interest in both physics and engineering. The starting point for
all such calculations is single-electron, or non-interacting, band structure
calculations, and in the limit of strong on-site confinement this can be
reduced to graph-like tight-binding models. In this context, both
mathematicians and physicists have developed largely independent methods for
solving these models. In this paper we will combine and present results from
both fields. In particular, we will discuss a class of lattices which can be
realized as line graphs of other lattices, both in Euclidean and hyperbolic
space. These lattices display highly unusual features including flat bands and
localized eigenstates of compact support. We will use the methods of both
fields to show how these properties arise and systems for classifying the
phenomenology of these lattices, as well as criteria for maximizing the gaps.
Furthermore, we will present a particular hardware implementation using
superconducting coplanar waveguide resonators that can realize a wide variety
of these lattices in both non-interacting and interacting form
Quantum protocols for few-qubit devices
Quantum computers promise to dramatically speed up certain algorithms, but remain challenging to build in practice. This thesis focuses on near-term experiments, which feature a small number (say, 10-200) of qubits that lose the stored information after a short amount of time. We propose various theoretical protocols that can get the best out of such highly limited computers. For example, we construct logical operations, the building blocks of algorithms, by exploiting the native physical behavior of the machine. Moreover, we describe how quantum information can be sent between qubits that are only indirectly connected
Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods
Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym-
metry consisting of only pentagons and hexagons faces. It has been the object
of interest for chemists and mathematicians due to its widespread application
in various fields, namely including electronic and optic engineering, medical sci-
ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity
of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751
isomers. The Fries and Clar numbers are stability predictors of a Fullerene
molecule. These number can be computed by solving a (possibly N P -hard)
combinatorial optimization problem. We propose several ILP formulation of
such a problem each yielding a solution algorithm that provides the exact value
of the Fries and Clar numbers. We compare the performances of the algorithm
derived from the proposed ILP formulations. One of this algorithm is used to
find the Clar isomers, i.e., those for which the Clar number is maximum among
all isomers having a given size. We repeated this computational experiment for
all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774
isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path
following (PDIPF) interior point (IP) algorithm for separable convex quadratic
minimum cost flow network problem. In each iteration of PDIPF algorithm, the
main computational effort is solving the underlying Newton search direction
system. We concentrated on finding the solution of the corresponding linear
system iteratively and inexactly. We assumed that all the involved inequalities
can be solved inexactly and to this purpose, we focused on different approaches
for distributing the error generated by iterative linear solvers such that the
convergences of the PDIPF algorithm are guaranteed. As a result, we achieved
theoretical bases that open the path to further interesting practical investiga-
tion
Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems
AbstractIt is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number
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