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

2008
*
arXiv
*
pre-print

We also make

arXiv:0811.1449v1
fatcat:6npw4h6tfzhqxoeluesdm73f6e
*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. ...##
###
Fibonacci index and stability number of graphs: a polyhedral study

2009
*
Journal of combinatorial optimization
*

We also make

doi:10.1007/s10878-009-9228-7
fatcat:mzfdrb3i3zbdfei2exegfxbnpm
*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. ...##
###
On the expected number of perfect matchings in cubic planar graphs
[article]

2021
*
arXiv
*
pre-print

We also compute the expected

arXiv:2005.13821v2
fatcat:vrqwkhpdnver7hhzzizl63pho4
*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 ...##
###
Mathematical aspects of fullerenes

2016
*
Ars Mathematica Contemporanea
*

The

doi:10.26493/1855-3974.834.b02
fatcat:h76vff3ybna3tkwujtz53nxe4u
*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. ...##
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SYMMETRY IN CHEMICAL STRUCTURES AND REACTIONS
[chapter]

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 ...

##
###
Symmetry in chemical structures and reactions

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 ...

##
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The Farey Maps Modulo N
[article]

2018
*
arXiv
*
pre-print

We

arXiv:1803.08851v1
fatcat:eo6p4s2o2fblff4oi27j2fytu4
*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. ...##
###
Euler's enumerations

2021
*
Enumerative Combinatorics and Applications
*

The topics are derangements, Catalan

doi:10.54550/eca2021v1s1h1
fatcat:wozsfafjffdtnkhjxmxzoeruce
*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. ...##
###
Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

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. ...

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

1985
*
Zeitschrift fur Naturforschung A-A Journal of Physical Sciences
*

The construction is based on

doi:10.1515/zna-1985-0801
fatcat:ca3t56xvnzhpplq7li5k462lxa
*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. ...##
###
Notes on the quantum tetrahedron
[article]

2001
*
arXiv
*
pre-print

We recall the concept

arXiv:math-ph/0011006v5
fatcat:vca4vmwegrhe5nt2cmuybh5vwu
*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*). ...##
###
Notes on the Quantum Tetrahedron

2002
*
Moscow Mathematical Journal
*

We recall the concept

doi:10.17323/1609-4514-2002-2-1-41-80
fatcat:2ct4cqgvizdcnnhqm7l4hn4kau
*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. ...##
###
Robust Visibility Surface Determination in Object Space via Plücker Coordinates

2021
*
Journal of Imaging
*

Industrial 3D models are usually characterized by

doi:10.3390/jimaging7060096
fatcat:dqpoleaggbhcloyxg6np7h7abq
*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. ...##
###
Towards Fast-Convergence, Low-Delay and Low-Complexity Network Optimization
[article]

2017
*
arXiv
*
pre-print

The new technique builds on

arXiv:1707.05398v1
fatcat:ylzvlejb2zfc5kolnc6hdcs2r4
*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. ...##
###
Stochastic and deterministic characteristics of orbits in chaotically looking dynamical systems

2009
*
Transactions of the Moscow Mathematical Society
*

We

doi:10.1090/s0077-1554-09-00180-0
fatcat:6e626dy3kvcbziyibuwanild44
*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. ...
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