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## Filters

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Net structure and cages

1979
*
Discrete Mathematics
*

An embeddable g net

doi:10.1016/0012-365x(79)90110-9
fatcat:34i5bbwsxvelfohb7gfk5t62pq
*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. ...##
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Isomorphism of graphs of bounded valence can be tested in polynomial time

1982
*
Journal of computer and system sciences (Print)
*

It is shown that testing isomorphism

doi:10.1016/0022-0000(82)90009-5
fatcat:dwxhyw4o7ng33owfqwnu7rayoa
*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. ...##
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On the graph labellings arising from phylogenetics

2013
*
Open Mathematics
*

Our main theorem

doi:10.2478/s11533-013-0263-3
fatcat:arsdk6s5obc4lnjksrsu7zmnza
*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. ...##
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Graph curves

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. ...

##
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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

2011
*
Ars Mathematica Contemporanea
*

We prove the existence

doi:10.26493/1855-3974.124.06d
fatcat:u57sdizqh5fsvexqkqys4zog6e
*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. ...##
###
Foreword

2009
*
Journal of Mathematical Sciences
*

However, as soon as we change the abstract

doi:10.1007/s10958-009-9367-5
fatcat:mztvsoitujgqbfjrrrbv3pkeqy
*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*. ...##
###
Invariants of theta-curves and other graphs in 3-space

1993
*
Topology and its Applications
*

It turns out that &curves

doi:10.1016/0166-8641(93)90110-y
fatcat:srswnzeoxbdpze3lps2ci5cklq
*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 ...##
###
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]

2022
*
arXiv
*
pre-print

Hence, the Legendrian links

arXiv:2101.10306v2
fatcat:fh5ngjh7hjch5hujkdkggdtgwi
*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. ...##
###
The Iso-Competition Point for Counterion Competition Binding to DNA: Calculated Multivalent Versus Monovalent Cation Binding Equivalence

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

doi:10.1016/s0006-3495(99)76876-1
pmid:10388744
pmcid:PMC1300316
fatcat:ww23coa26vecfm6ixeyren5sbu
*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. ...##
###
Worldsheet formulations of gauge theories and gravity
[article]

1994
*
arXiv
*
pre-print

The evolution operator

arXiv:gr-qc/9412035v1
fatcat:4a3rrdzqufhztkojc4vw62nkpq
*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. ...##
###
Morita classes in the homology of automorphism groups of free groups

2004
*
Geometry and Topology
*

Topol. 3 (2003) 1167--1224], we reinterpret

doi:10.2140/gt.2004.8.1471
fatcat:6xstzgajtzgijidkfilmwmlh6i
*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 . ...##
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The classical evaluation of relativistic spin networks
[article]

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*. ...

##
###
Low complexity algorithms in knot theory
[article]

2018
*
arXiv
*
pre-print

We also show, that the equivalence problem

arXiv:1803.04908v2
fatcat:tli55rg6i5aydoubuw5up4uupi
*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. ...
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