Barbell Graph
Author
Eric W. Weisstein
Title
Barbell Graph
Description
There are several different definitions of the barbell graph. Most commonly and in this work, the n-barbell graph is the simple graph obtained by connecting two copies of a complete graph K_n by a bridge (Ghosh et al. 2006, Herbster and Pontil 2006). Precomputed properties of barbell graphs are available in the Wolfram Language as GraphData[{"Barbell", n}]. By definition, the n-barbell graph has cycle polynomial is given by C_n(x)=2C_(K_n)(x), (1) where C_(K_n)(x) is the cycle...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z3r8fm/
DOI
https://notebookarchive.org/2019-07-0z3r8fm
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
1.07 megabytes
Supplements
Rights
Redistribution rights reserved
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Barbell Graph
Barbell Graph
Author
Author
Eric W. Weisstein
September 11, 2017
September 11, 2017
©2017 Wolfram Research, Inc. except for portions noted otherwise
Figures
Figures
Graphs.m
Graphs.m
GraphicsGrid[Partition[Table[Show[BarbellGraph[n,VertexSize{"Scaled",.05},System`VertexStyleRed,System`EdgeStyleBlack],PlotLabelStyle[HoldForm[n]n,FontFamily"Times"]],{n,3,8}],UpTo[3]],DividersAll,ImageSize500,Spacings{-50,20}]
Special Cases
Special Cases
Sort[{"Barbell"/.GraphData[#,"NotationRules"],GraphData[#,"Name"]}&/@DeleteCases[GraphData["Barbell"],{"Barbell",_}]]
{}
Properties
Properties
Acyclic
Acyclic
AdjacencyMatrixCount
AdjacencyMatrixCount
Arboricity
Arboricity
ArcTransitive
ArcTransitive
ArcTransitcivity
ArcTransitcivity
AutomorphismCount
AutomorphismCount
BalabanIndex
BalabanIndex
Bandwidth
Bandwidth
BridgeCount
BridgeCount
BurningNumber
BurningNumber
CharacteristicPolynomial
CharacteristicPolynomial
Chordal
Chordal
ChordCount
ChordCount
Chordless
Chordless
ChordlessCycleCount
ChordlessCycleCount
Chords
Chords
ChromaticallyUnique
ChromaticallyUnique
ChromaticInvariant
ChromaticInvariant
ChromaticNumber
ChromaticNumber
ChromaticPolynomial
ChromaticPolynomial
Circumference
Circumference
Classes
Classes
CliqueCount
CliqueCount
CliqueCoveringNumber
CliqueCoveringNumber
CliqueNumber
CliqueNumber
CliquePolynomial
CliquePolynomial
CochromaticGraphNames
CochromaticGraphNames
ComplementOddChordlessCycleCount
ComplementOddChordlessCycleCount
ConnectedComponentCount
ConnectedComponentCount
ConnectedDominatingSetCount
ConnectedDominatingSetCount
ConnectedDominationNumber
ConnectedDominationNumber
ConnectedDominationPolynomial
ConnectedDominationPolynomial
ConnectedInducedSubgraphCount
ConnectedInducedSubgraphCount
ConnectedInducedSubgraphPolynomial
ConnectedInducedSubgraphPolynomial
CrossingNumber
CrossingNumber
CycleCount
CycleCount
CyclePolynomial
CyclePolynomial
Cyclic
Cyclic
CyclomaticNumber
CyclomaticNumber
DetourIndex
DetourIndex
DetourMatrix
DetourMatrix
DetourPolynomial
DetourPolynomial
Diameter
Diameter
DistanceMatrix
DistanceMatrix
DistancePolynomial
DistancePolynomial
DistinguishingNumber
DistinguishingNumber
DomaticNumber
DomaticNumber
DominatingSetCount
DominatingSetCount
DominationNumber
DominationNumber
DominationPolynomial
DominationPolynomial
EdgeChromaticNumber
EdgeChromaticNumber
EdgeConnectivity
EdgeConnectivity
EdgeCount
EdgeCount
EdgeCoverCount
EdgeCoverCount
EdgeCoverNumber
EdgeCoverNumber
EdgeCoverPolynomial
EdgeCoverPolynomial
EulerianCycleCount
EulerianCycleCount
FaceCount
FaceCount
FlowPolynomial
FlowPolynomial
FractionalChromaticNumber
FractionalChromaticNumber
FractionalEdgeChromaticNumber
FractionalEdgeChromaticNumber
Girth
Girth
Gonality
Gonality
HamiltonConnected
HamiltonConnected
HamiltonDecompositionCount
HamiltonDecompositionCount
Hamiltonian
Hamiltonian
HamiltonianCycleCount
HamiltonianCycleCount
HamiltonianNumber
HamiltonianNumber
HamiltonianPathCount
HamiltonianPathCount
HamiltonianWalkCount
HamiltonianWalkCount
HararyIndex
HararyIndex
HexagonCount
HexagonCount
IdiosyncraticPolynomial
IdiosyncraticPolynomial
IndependenceNumber
IndependenceNumber
IndependencePolynomial
IndependencePolynomial
IndependenceRatio
IndependenceRatio
IndependentEdgeSetCount
IndependentEdgeSetCount
IndependentVertexSetCount
IndependentVertexSetCount
IntersectionNumber
IntersectionNumber
IrredundancePolynomial
IrredundancePolynomial
IrredundantSetCount
IrredundantSetCount
KirchhoffIndex
KirchhoffIndex
KirchhoffSumIndex
KirchhoffSumIndex
LaplacianMatrix
LaplacianMatrix
LaplacianPolynomial
LaplacianPolynomial
LeafCount
LeafCount
LongestCycleCount
LongestCycleCount
LovaszNumber
LovaszNumber
MatchingGeneratingPolynomial
MatchingGeneratingPolynomial
MatchingNumber
MatchingNumber
MatchingPolynomial
MatchingPolynomial
MaximalCliqueCount
MaximalCliqueCount
MaximalIndependentEdgeSetCount
MaximalIndependentEdgeSetCount
MaximalIndependentEdgeSetSignature
MaximalIndependentEdgeSetSignature
MaximalIndependentVertexSetCount
MaximalIndependentVertexSetCount
MaximalIrredundantSetCount
MaximalIrredundantSetCount
MaximumCliqueCount
MaximumCliqueCount
MaximumIndependentEdgeSetCount
MaximumIndependentEdgeSetCount
MaximumIndependentVertexSetCount
MaximumIndependentVertexSetCount
MaximumIrredundantSetCount
MaximumIrredundantSetCount
seq=Table[GraphData[{"Barbell",n},"MaximumIrredundantSetCount"],{n,3,20}]
{9,16,25,36,49,64,81,100,121,144,169,Missing[NotAvailable],Missing[NotAvailable],Missing[NotAvailable],Missing[NotAvailable],Missing[NotAvailable],Missing[NotAvailable],Missing[NotAvailable]}
FindSequenceFunction[DeleteMissing@seq,n-2]//Simplify
2
n
MaximumLeafNumber
MaximumLeafNumber
MaximumVertexDegree
MaximumVertexDegree
MeanDistance
MeanDistance
MinimalDominatingSetCount
MinimalDominatingSetCount
MinimalDominatingSetSignature
MinimalDominatingSetSignature
MinimalEdgeCoverCount
MinimalEdgeCoverCount
MinimumCliqueCoveringCount
MinimumCliqueCoveringCount
MinimumConnectedDominatingSetCount
MinimumConnectedDominatingSetCount
MinimumDominatingSetCount
MinimumDominatingSetCount
MinimumEdgeCoverCount
MinimumEdgeCoverCount
MinimumVertexDegree
MinimumVertexDegree
MolecularTopologicalIndex
MolecularTopologicalIndex
Nonhamiltonian
Nonhamiltonian
NormalizedLaplacianMatrix
NormalizedLaplacianMatrix
OddChordlessCycleCount
OddChordlessCycleCount
PathCount
PathCount
PathPolynomial
PathPolynomial
PentagonCount
PentagonCount
Perfect
Perfect
ProjectivePlaneCrossingNumber
ProjectivePlaneCrossingNumber
Radius
Radius
RankPolynomial
RankPolynomial
RectilinearCrossingNumber
RectilinearCrossingNumber
ReliabilityPolynomial
ReliabilityPolynomial
SigmaPolynomial
SigmaPolynomial
SpanningTreeCount
SpanningTreeCount
SquareCount
SquareCount
StabilityIndex
StabilityIndex
TopologicalIndex
TopologicalIndex
ToroidalCrossingNumber
ToroidalCrossingNumber
Traceable
Traceable
TriangleCount
TriangleCount
TuttePolynomial
TuttePolynomial
Untraceable
Untraceable
VertexConnectivity
VertexConnectivity
VertexCount
VertexCount
VertexCoverNumber
VertexCoverNumber
VertexCoverPolynomial
VertexCoverPolynomial
VertexDegrees
VertexDegrees
VertexTransitive
VertexTransitive
WienerIndex
WienerIndex
WienerSumIndex
WienerSumIndex
GraphData
GraphData
Cite this as: Eric W. Weisstein, "Barbell Graph" from the Notebook Archive (2017), https://notebookarchive.org/2019-07-0z3r8fm
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