Firecracker Graph

Author

Eric W. Weisstein
August 31, 2020

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/FirecrackerGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FirecrackerGraph.html.

©2020 Wolfram Research, Inc. except for portions noted otherwise

Graphics

GraphData default

GraphicsGrid[Table[Show[GraphData[{"Firecracker",{k,n}}]//StyleGraphs,LabelStyle->Directive[Black,FontFamily->"Times"],PlotLabel->Subscript[Style["F",Italic],Sequence@@{k,n}]],{n,4,7},{k,2,4}],ImageSize->600,Alignment->Center]

GraphData "UnitDistance"

GraphicsGrid[Table[Show[GraphData[{"Firecracker",{k,n}},"Graph","UnitDistance"][[1]]//StyleGraphs,LabelStyle->Directive[Black,FontFamily->"Times"],PlotLabel->Subscript[Style["F",Italic],Sequence@@{k,n}]],{n,4,7},{k,2,4}],ImageSize->600,Alignment->Center]

"StarEmbedding"

GraphicsGrid[Table[Show[FirecrackerGraph[k,n,GraphLayout->"StarEmbedding"]//StyleGraphs[#,VertexSize->.15k]&,LabelStyle->Directive[Black,FontFamily->"Times"],PlotLabel->Subscript[Style["F",Italic],Sequence@@{k,n}]],{n,2,7},{k,2,5}],ImageSize->600,Alignment->Center,Spacings->{-70,30}]

"UnitDistanceEmbedding"

GraphicsGrid[Table[Show[FirecrackerGraph[k,n,GraphLayout->"UnitDistanceEmbedding"]//StyleGraphs,LabelStyle->Directive[Black,FontFamily->"Times"],PlotLabel->Subscript[Style["F",Italic],Sequence@@{k,n}]],{n,4,7},{k,2,5}],ImageSize->600,Alignment->Center]

Counts

Edges

Subtract@@@({{n,k}=("Firecracker"/.GraphData[#,"NotationRules"]);(k-1)n+n-1,GraphData[#,"M"]}&/@GraphData["Firecracker"])//Union

{n,k}=.;

(k-1)n+n-1//Simplify

Vertices

Subtract@@@({Times@@("Firecracker"/.GraphData[#,"NotationRules"]),GraphData[#,"V"]}&/@GraphData["Firecracker"])//Union

Special cases

DeleteCases[Flatten[Table[{{n,k},RecognizeGraph[FirecrackerGraph[n,k]]},{n,2,8},{k,2,8}],1],{_,{}}|{_,{"Firecracker",_}}]//Column

GraphData

GraphClassString

GraphClassString[GraphData["Firecracker"]]

GraphDataString

Do[Print[GraphDataString[FirecrackerGraph[n,k],
"Name"->"("<>ToString[n]<>","<>ToString[k]<>")-firecracker graph",
"StandardName"->{"Firecracker",{n,k}},
"Classes"->{"Firecracker"},
"NotationRules"->{"Firecracker"->{n,k}},
"Skip"->{"Perfect"},
TimeConstraint->120
]],{n,2,8},{k,2,8}]