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Net structure and cages

C.W. Evans
1979 Discrete Mathematics  
An embeddable g net of valency k corresponds to a graph of girth g which can be drawn on a surface in such a way that every face is bounded by a circuit of length g.  ...  A g cage of valency k is a regu' -bU. graph of girth g and valency k with the minimum number of vertices consistent with this conciition [12].  ...  The former for discussions on residual finiteness resulting in the proof of Theorem 11 and the latter for supervising him during the period when the major part of this work was produced.  ... 
doi:10.1016/0012-365x(79)90110-9 fatcat:34i5bbwsxvelfohb7gfk5t62pq

Isomorphism of graphs of bounded valence can be tested in polynomial time

Eugene M. Luks
1982 Journal of computer and system sciences (Print)  
It is shown that testing isomorphism of graphs of bounded valance is polynomial-time reducible to the color automorphism problem for groups with composition factors of bounded order.  ...  In the trivalent case p = 2, P = G and the analysis requires only elementary notions. For higher valence, the justification requires some new observations about primitive permutation groups.  ...  THE BOUNDED VALENCE CASE The Groups That Arise We now consider graphs of valence <t where t is, henceforth, fixed.  ... 
doi:10.1016/0022-0000(82)90009-5 fatcat:dwxhyw4o7ng33owfqwnu7rayoa

On the graph labellings arising from phylogenetics

Weronika Buczyńska, Jarosław Buczyński, Kaie Kubjas, Mateusz Michałek
2013 Open Mathematics  
Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.  ...  AbstractWe study semigroups of labellings associated to a graph.  ...  The authors would like to thank Christian Haase and Andreas Paffenholz for helpful discussions, Alexander Kasprzyk and Zach Teitler for their comments that have helped to improve the presentation.  ... 
doi:10.2478/s11533-013-0263-3 fatcat:arsdk6s5obc4lnjksrsu7zmnza

Graph curves

Dave Bayer, David Eisenbud
1991 Advances in Mathematics  
For any set A of vertices of a graph G of valency 1, define a retraction operator rA by rA(G) = G-A. If A is the set of vertices of valency 1, then we will write r(G) for rA(G).  ...  Bayer and D. Eisenbud introduce the combinatorial Clifford index for trivalent graphs and prove that this is the same as the resolution Clifford index for plane trivalent graphs.  ...  For the same reason, {cl, c,_~} is the set of vertices of G3, each of which is incident to one of the pairs of vertices {u,, uz} and {u,-,, u,}. We may assume that ci is incident to vr.  ... 
doi:10.1016/0001-8708(91)90034-5 fatcat:f3divaxzdnhrhmy3fkt4xhq2xy

Page 2754 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
The author defines three special families of metacir- culant graphs of valency 4 and then shows that any non-Cayley metacirculant graph of valency 4 is isomorphic to a union of disjoint copies of a graph  ...  As corollaries, asymptotics of the form cj(n) ~ ajn ge (called the normal rate) as n — oo are derived for the corresponding numbers of maps c;(”), where x; satisfies an algebraic equation of degree 6,  ... 

Small vertex-transitive graphs of given degree and girth

Robert Jajcay, Jozef Širáň
2011 Ars Mathematica Contemporanea  
We prove the existence of k-valent Cayley graphs of girth g for every pair of parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs for certain  ...  Small vertex-transitive graphs of given degree and girth. Ars Mathematica Contemporanea, 4(2) pp. 375-384. For guidance on citations see FAQs.  ...  For every pair of parameters k ≥ 2, g ≥ 3, there exists a finite Cayley graph C(Γ, X) of valence k and girth g. Proof.  ... 
doi:10.26493/1855-3974.124.06d fatcat:u57sdizqh5fsvexqkqys4zog6e

Foreword

G. B. Shabat
2009 Journal of Mathematical Sciences  
However, as soon as we change the abstract graphs for the ribbon graphs or, equivalently, fix an embedding of a graph into an oriented surface, we find that such an object represents a complicated arithmetic  ...  The first publication where some of the constructions of the Esquisse were represented in the traditional mathematical form was the paper of 1990 by G. Shabat and V.  ...  Artamkin's paper gives a beautiful, nontrivial, and self-contained description of the 2-faced dessins with trivalent graphs.  ... 
doi:10.1007/s10958-009-9367-5 fatcat:mztvsoitujgqbfjrrrbv3pkeqy

Invariants of theta-curves and other graphs in 3-space

Louis Kauffman, Jonathan Simon, Keith Wolcott, Peiyi Zhao
1993 Topology and its Applications  
It turns out that &curves and &-graphs are the unique graphs for which this approach works.  ...  We consider, in particular, surfaces that contain the graph as a spine and that are canonical in the sense of having trivial Seifert linking form.  ...  Acknowledgement We thank the referee for an excellent job proofreading the manuscript, for correcting an error in a link polynomial computation, and for supplying us with a computation that provides an  ... 
doi:10.1016/0166-8641(93)90110-y fatcat:srswnzeoxbdpze3lps2ci5cklq

Page 5294 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
Summary: “Leland and Solomon introduced a family of trivalent graphs, called Mobius graphs, and because of their small diameter suggested that they be used as an effective interconnection scheme for multiprocessor  ...  He establishes a new sufficient condition for infinite, locally finite, planar, 3-connected graphs: a finite face and odd valence.  ... 

Weave Realizability for D-type [article]

James Hughes
2022 arXiv   pre-print
Hence, the Legendrian links of D_n-type have at least as many Hamiltonian isotopy classes of Lagrangian fillings as cluster seeds in the D_n-type cluster algebra, and their geometric exchange graph for  ...  Lagrangian disk surgeries contains the cluster exchange graph of D_n-type.  ...  Thanks also to Youngjin Bae and Eric Zaslow for helpful conversations, and to the anonymous referee for insightful comments.  ... 
arXiv:2101.10306v2 fatcat:fh5ngjh7hjch5hujkdkggdtgwi

The Iso-Competition Point for Counterion Competition Binding to DNA: Calculated Multivalent Versus Monovalent Cation Binding Equivalence

Anzhi Z. Li, Kenneth A. Marx
1999 Biophysical Journal  
The ionic strength effect on ICP can be expressed by a semiquantitative equation as ICP Za /ICP Zb ϭ (I a /I b ) Z , where I a , I b refers to the instance of ionic strengths and Z indicates the valence  ...  At fixed ionic strength, the total DNA charge neutralization fractions ICP are equal, no matter whether the higher valence cation is divalent, trivalent, or tetravalent.  ...  Manning for his insightful reading of our manuscript and Haiyan Huang for her biophysical experimental support. The authors acknowledge support from the Center for Intelligent Biomaterials.  ... 
doi:10.1016/s0006-3495(99)76876-1 pmid:10388744 pmcid:PMC1300316 fatcat:ww23coa26vecfm6ixeyren5sbu

Worldsheet formulations of gauge theories and gravity [article]

Michael Reisenberger (Institute for theoretical physics, University of Utrecht)
1994 arXiv   pre-print
The evolution operator for states of gauge theories in the graph representation (closely related to the loop representation of Gambini and Trias, and Rovelli and Smolin) is formulated as a weighted sum  ...  As examples, lattice SU(2) BF and Yang-Mills theories are expressed as worldsheet theories, and (formal) worldsheet forms of several continuum U(1) theories are given.  ...  The graph states are normalized by multiplying by a factor √ 2j + 1 for each edge, including the fictitious ones in the expansion of vertices into trivalent trees.  ... 
arXiv:gr-qc/9412035v1 fatcat:4a3rrdzqufhztkojc4vw62nkpq

Morita classes in the homology of automorphism groups of free groups

James Conant, Karen Vogtmann
2004 Geometry and Topology  
Topol. 3 (2003) 1167--1224], we reinterpret and generalize Morita's cycles, obtaining an unstable cycle for every connected odd-valent graph.  ...  In this setting, a further generalization becomes apparent, giving cycles for objects more general than odd-valent graphs. Some of these cycles lie in the stable range.  ...  For each possible pairing π , glue the paired leaves together to form a trivalent graph G π , and use the interiors of the X i to form a forest F π in G π .  ... 
doi:10.2140/gt.2004.8.1471 fatcat:6xstzgajtzgijidkfilmwmlh6i

The classical evaluation of relativistic spin networks [article]

John W. Barrett
1999 arXiv   pre-print
For the graph determined by a 4-simplex this gives the evaluation as an integral over a space of geometries for a 4-simplex.  ...  The evaluation of a relativistic spin network for the classical case of the Lie group SU(2) is given by an integral formula over copies of SU(2).  ...  Since the contraction formula is the same for both invariants, they coincide for graphs with vertices of any valence.  ... 
arXiv:math/9803063v3 fatcat:jwzyjqnx7rbwzerc2lwykzxpce

Low complexity algorithms in knot theory [article]

Olga Kharlampovich, Alina Vdovina
2018 arXiv   pre-print
We also show, that the equivalence problem for such knots with n crossings has time complexity n (n) and is in Logspace(n) and TC^0 complexity classes.  ...  We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace(n).  ...  For the proof, and later, we will need the following additional structure on a trivalent graph with Bieulerian path. Definition 5.4.  ... 
arXiv:1803.04908v2 fatcat:tli55rg6i5aydoubuw5up4uupi
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