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Fibonacci Index and Stability Number of Graphs: a Polyhedral Study [article]

Véronique Bruyère, Hadrien Mélot
2008 arXiv   pre-print
We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order n.  ...  The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory.  ...  The Theory of Graphs. Dover Publications, New York, 2001.  ... 
arXiv:0811.1449v1 fatcat:6npw4h6tfzhqxoeluesdm73f6e

Fibonacci index and stability number of graphs: a polyhedral study

Véronique Bruyère, Hadrien Mélot
2009 Journal of combinatorial optimization  
We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order n.  ...  The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory.  ...  The Theory of Graphs. Dover Publications, New York, 2001.  ... 
doi:10.1007/s10878-009-9228-7 fatcat:mzfdrb3i3zbdfei2exegfxbnpm

On the expected number of perfect matchings in cubic planar graphs [article]

Marc Noy, Clément Requilé, Juanjo Rué
2021 arXiv   pre-print
We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.  ...  Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγ^n, where c>0 and γ∼ 1.14196 is an explicit algebraic number.  ...  The third author acknowledges preliminary discussions on this topic with Mihyun Kang, Michael Mosshammer and Philipp Sprüssel during a visit to the Technical University of Graz in 2015, and wishes to thank  ... 
arXiv:2005.13821v2 fatcat:vrqwkhpdnver7hhzzizl63pho4

Mathematical aspects of fullerenes

Vesna Andova, František Kardoš, Riste Škrekovski
2016 Ars Mathematica Contemporanea  
The study from graph theoretical point of view has been motivated by a search for invariants that will correlate with their stability as a chemical compound.  ...  A number of graph-theoretical invariants were examined as potential stability predictors with various degrees of success [42, 29] .  ...  [73] studied the Balaban index for cubic graphs, and determined that the Balaban index tends to zero as the number of vertices increases. More precisely, they showed the following.  ... 
doi:10.26493/1855-3974.834.b02 fatcat:h76vff3ybna3tkwujtz53nxe4u

SYMMETRY IN CHEMICAL STRUCTURES AND REACTIONS [chapter]

ALEXANDRU T. BALABAN
1986 Symmetry  
A few excursions into mathematics involve solid angles, heuristics of (3, -/)-cages, and visualization of symmetry operations for graphs in three-and four-dimensional Euclidean spaces.  ...  theorem; Jahn-Teller effects) and at the supramolecular level (reaction graphs for rearrangements and automerizations; repeating sequences in polymer chains; conservation of orbital symmetry in chemical  ...  Interestingly, for a given number n of monomer molecules in the repeating unit, the number N of solutions is seen to form the Fibonacci sequence (Fi = Fi-1 "~-F;_ 2) for n ~< 8 (but not for higher n values  ... 
doi:10.1016/b978-0-08-033986-3.50068-9 fatcat:6zxfi3agbne2bcxmiq5li6ebnu

Symmetry in chemical structures and reactions

Alexandru T. Balaban
1986 Computers and Mathematics with Applications  
A few excursions into mathematics involve solid angles, heuristics of (3, -/)-cages, and visualization of symmetry operations for graphs in three-and four-dimensional Euclidean spaces.  ...  theorem; Jahn-Teller effects) and at the supramolecular level (reaction graphs for rearrangements and automerizations; repeating sequences in polymer chains; conservation of orbital symmetry in chemical  ...  Interestingly, for a given number n of monomer molecules in the repeating unit, the number N of solutions is seen to form the Fibonacci sequence (Fi = Fi-1 "~-F;_ 2) for n ~< 8 (but not for higher n values  ... 
doi:10.1016/0898-1221(86)90437-2 fatcat:jzgft7ecfnbxtffpor6y2y57ky

The Farey Maps Modulo N [article]

David Singerman, James Strudwick
2018 arXiv   pre-print
We study the quotients of the Farey map by the principal congruence subgroups of the modular group. We also study the structure of the underlying graphs of these quotients.  ...  For n > 2, Γ 1 (n) is a normal subgroup of Γ of index n 2 2 Π p|n (1 − 1 p 2 ) (2) and the index of Γ(2) in Γ is equal to 6, Hence the number of vertices of M 3 (n) is given by (2) if n > 2 and the number  ...  We study the quotients of the Farey map by the principal congruence subgroups of the modular group. We also study the structure of the underlying graphs of these quotients.  ... 
arXiv:1803.08851v1 fatcat:eo6p4s2o2fblff4oi27j2fytu4

Euler's enumerations

Brian Hopkins, Saint Peter's University
2021 Enumerative Combinatorics and Applications  
The topics are derangements, Catalan numbers, Latin squares, two disjoint contributions to what eventually became graph theory, and integer partitions along with the associated sum of divisors function  ...  We conclude with a sampling of rich internet resources that enable further historical investigations.  ...  Graph theory Two famous results of Euler, the Königsberg bridges problem and the polyhedral formula, are widely considered to be the beginnings of graph theory.  ... 
doi:10.54550/eca2021v1s1h1 fatcat:wozsfafjffdtnkhjxmxzoeruce

Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

Henrique F. da Cruz, Rosário Fernandes, Susana Furtado
2017 Linear Algebra and its Applications  
graph theory and combinatorics.  ...  We have received 150 abstracts for 20-minute contributed talks which will run in four parallel sessions and cover a large variety of areas within graph theory and combinatorics.  ...  The boundary-type sets are rather trivial for trees and are all subsets of the leaves. We provide a characterization of ∂(G), Ct(G), and Ext(G) for the family of MOPs.  ... 
doi:10.1016/j.laa.2017.05.014 fatcat:zlcp5rb2sjeqvlyaajvf2nyoby

On the Theory of a Non-Periodic Quasilattice Associated with the Icosahedral Group

P. Kramer
1985 Zeitschrift fur Naturforschung A-A Journal of Physical Sciences  
The construction is based on a general theory of periodic and non-periodic space filling by projection from hypergrids.  ...  A quasilattice in the euclidean space E3 is generated by dualization of a hexagrid.  ...  Kramer for the computation and graphic representation given in Figs. 3. 4 and 6.  ... 
doi:10.1515/zna-1985-0801 fatcat:ca3t56xvnzhpplq7li5k462lxa

Notes on the quantum tetrahedron [article]

Robert Coquereaux CERN, TH Division, Geneva, Switzerland
2001 arXiv   pre-print
We recall the concept of essential matrices (intertwiners) for a graph and describe their module properties with respect to right and left actions of fusion algebras.  ...  square of its fusion algebra (the tensor product should be taken over a particular subalgebra); this is the Cayley graph for the two generators of the twelve dimensional algebra E6 \otimes_A3 E6 and E6  ...  given by Catalan numbers (like for SU (2)), but by the sum of the squares of two consecutive Fibonacci number (so again a Fibonacci number).  ... 
arXiv:math-ph/0011006v5 fatcat:vca4vmwegrhe5nt2cmuybh5vwu

Notes on the Quantum Tetrahedron

R. Coquereaux
2002 Moscow Mathematical Journal  
We recall the concept of essential matrices (intertwiners) for a graph and describe their module properties with respect to right and left actions of fusion algebras.  ...  square of its fusion algebra (the tensor product should be taken over a particular subalgebra); this is the Cayley graph for the two generators of the twelve-dimensional algebra E6 ⊗A 3 E6 (here A3 and  ...  This set of notes was elaborated during my stay at CERN and at the Mathematics Department of the University of Geneva during the academic year 1999/2000.  ... 
doi:10.17323/1609-4514-2002-2-1-41-80 fatcat:2ct4cqgvizdcnnhqm7l4hn4kau

Robust Visibility Surface Determination in Object Space via Plücker Coordinates

Alessandro Rossi, Marco Barbiero, Paolo Scremin, Ruggero Carli
2021 Journal of Imaging  
Industrial 3D models are usually characterized by a large number of hidden faces and it is very important to simplify them.  ...  More specifically, we propose a strategy that, in a finite number of steps, determines if each face of the mesh is globally visible or not.  ...  Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/jimaging7060096 fatcat:dqpoleaggbhcloyxg6np7h7abq

Towards Fast-Convergence, Low-Delay and Low-Complexity Network Optimization [article]

Sinong Wang, Ness Shroff
2017 arXiv   pre-print
The new technique builds on a kind of inexact Uzawa method in the Alternating Directional Method of Multiplier, and provides a new theoretical path to prove global and linear convergence rate of such a  ...  Distributed network optimization has been studied for well over a decade.  ...  Recall that the set-valued mapping F is piecewise polyhedral if the graph of F is the union of finitely many polyhedral sets.  ... 
arXiv:1707.05398v1 fatcat:ylzvlejb2zfc5kolnc6hdcs2r4

Stochastic and deterministic characteristics of orbits in chaotically looking dynamical systems

V. I. Arnold
2009 Transactions of the Moscow Mathematical Society  
We study finite length sequences of numbers which, at the first glance, look like realizations of a random variable (for example, sequences of fractional parts of arithmetic and geometric progressions,  ...  last digits of sequences of prime numbers, and incomplete periodic continuous fractions).  ...  Thinking about the stability of the solar system, H.  ... 
doi:10.1090/s0077-1554-09-00180-0 fatcat:6e626dy3kvcbziyibuwanild44
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