Wolfram Data Repository

Immediate Computable Access to Curated Contributed Data

All Unlabeled Cubic Graphs of Order 32 with Girth at Least 7

Source Notebook

A compactly encoded dataset containing all 30,368 unlabeled connected 3-regular graphs on 32 vertices with girth at least 7

Details

In graph theory, a cubic graph is a graph in which every vertex has degree three—equivalently, a 3-regular graph. Such graphs are also known as trivalent graphs.
This dataset employs differential encoding to compress graphs by storing only their differences. The compressed format uses 77.9% less storage space than storing full adjacency list representations for all 30,368 graphs.
Shortcode representation: Each graph is encoded as a sequence of vertex numbers representing edges (only higher-numbered neighbors per vertex). For instance, decoding (2 3 4 3 4 4) for a 4-vertex, 3-regular graph:
Vertex 1reads 2, 3, 4Edge: 1-2, Edge: 1-3, Edge: 1-4
Vertex 2reads 3, 4Edge: 2-3, Edge: 2-4
Vertex 3read 4Edge: 3-4
Vertex 4read nothingNo more entries
Prefix compression: each graph stores one byte indicating how many leading bytes match the previous graph, and only the remaining differing bytes.Example: The the two adjacency lists are ( 2 3 4 5 3 4 5 6 7 6 7 6 7 7 ) and (2 3 4 5 3 4 6 5 7 6 7 6 7 7). This compression is very effective when consecutive graphs share long common prefixes, which happens frequently in the systematic generation process.
The conversion algorithm leverages NumericArray for efficient data storage and faster processing.

Examples

Basic Examples (1) 

Retrieve the data as a NumericArray:

In[1]:=
ResourceData[\!\(\* TagBox["\"\<All Unlabeled Cubic Graphs of Order 32 with Girth at Least 7\>\"", #& , BoxID -> "ResourceTag-All Unlabeled Cubic Graphs of Order 32 with Girth at Least 7-Input", AutoDelete->True]\)]
Out[1]=

Scope & Additional Elements (3) 

The following steps demonstrate how to convert the list into graphs. Start with the base cubic graph:

In[2]:=
baseRG = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 19, 17, 21, 16, 20, 18, 23, 22, 24, 25, 26, 27, 28, 29, 30, 27, 30, 25, 31, 32, 28, 29, 31, 32, 31, 32}; nEdges = 32*3/2;
In[3]:=
data = ResourceData[\!\(\* TagBox["\"\<All Unlabeled Cubic Graphs of Order 32 with Girth at Least 7\>\"", #& , BoxID -> "ResourceTag-All Unlabeled Cubic Graphs of Order 32 with Girth at Least 7-Input", AutoDelete->True]\)]; samebitFlagPos = NumericArray[ Most[NestWhileList[With[{pos = #[[1]] + nEdges - #[[2]] + 1}, {pos, If[pos > ndp, -1, data[[pos]]]}] &, {1, 0}, #[[2]] != -1 &] ][[All, 1]], "UnsignedInteger32"];
In[4]:=
adjacencyLists = NumericArray[Rest@Module[{k = 0, m = 0, dupLen, dupPos}, NestList[(k++; dupPos = samebitFlagPos[[k]]; dupLen = data[[dupPos]]; #[[;; dupLen]]~Join~ data[[dupPos + 1 ;; dupPos + nEdges - dupLen]]) &, baseRG, Length[samebitFlagPos]]], "UnsignedInteger8"];

Convert the data into adjacency lists:

In[5]:=
AdjacencyListToGraph[l_] := Module[{dsStk, ctArray, graphEdges, larger}, dsStk = CreateDataStructure["Stack", Reverse[l]]; ctArray = CreateDataStructure["FixedArray", ConstantArray[3, 32]]; graphEdges = CreateDataStructure["DynamicArray"]; Do[While[(! dsStk["EmptyQ"]) && ctArray[[k]] > 0 && k < dsStk["Peek"], ctArray["SetPart", k, ctArray[[k]] - 1]; larger = dsStk["Pop"]; ctArray["SetPart", larger, ctArray[[larger]] - 1]; graphEdges["Append", {k, larger}];], {k, 47}]; Graph[UndirectedEdge @@@ graphEdges["Elements"]]]

Choose several random cubic graphs from our database:

In[6]:=
Partition[With[{db = adjacencyLists}, g1 = AdjacencyListToGraph /@ (db[[ RandomInteger[{1, Length[db]}, 9]]] // Normal) ], 3] // Grid
Out[6]=

Analysis (2) 

The graphs in the database are 3-regular. For instances:

In[7]:=
g = {\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, {Null, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 7}, {3, 8}, {4, 9}, {4, 10}, {5, 11}, {5, 12}, {6, 13}, {6, 14}, {7, 15}, {7, 16}, {8, 17}, {8, 18}, {9, 19}, {9, 20}, {10, 21}, {10, 22}, {11, 15}, {11, 19}, {12, 17}, {12, 21}, {13, 16}, {13, 23}, {14, 24}, {14, 25}, {15, 26}, { 16, 20}, {17, 27}, {18, 22}, {18, 23}, {19, 28}, {20, 29}, {21, 30}, {22, 24}, {23, 31}, {24, 29}, {25, 28}, {25, 30}, {26, 30}, {26, 31}, {27, 28}, {27, 32}, {29, 32}, { 31, 32}}}]]}, TagBox[GraphicsGroupBox[GraphicsComplexBox[CompressedData[" 1:eJwBEQLu/SFib1JlAgAAACAAAAACAAAA/RMWjhbWAEAGqPe34LYCQEUeXbl/ 0ANAqvsFiQV9BUCbiuGtd1r3P3KgIlpquQBAHjnd58OgAEAhbv94h7r4P8rv gGQPVAxAVTo6vnnwAUCIX9KbW3rzPyMvSgKV1gNA+PUHLGcD9j9S6WW5YOYJ 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All vertices have degree three:

In[8]:=
AllSameBy[#, v |-> v == 3] &@*VertexDegree /@ g
Out[8]=
In[9]:=
ResourceFunction["RegularGraphQ"] /@ g
Out[9]=

The girths or the lengths of shortest cycle in the graphs are at least seven:

In[10]:=
ResourceFunction["Girth"] /@ g
Out[10]=

The graphs are pair-wisely non-isomorphic or structurally inequivalent:

In[11]:=
IsomorphicGraphQ @@ (g[[{1, 2}]])
Out[11]=

The graphs may have different centrality:

In[12]:=
GraphicsRow[ With[{bc = BetweennessCentrality[#]}, HighlightGraph[#, VertexList[#], VertexSize -> Thread[VertexList[#] -> Rescale[bc]]]] & /@ g]
Out[12]=
In[13]:=
GraphicsRow[ With[{bc = ClosenessCentrality[#]}, HighlightGraph[#, VertexList[#], VertexSize -> Thread[VertexList[#] -> Rescale[bc]]]] & /@ g]
Out[13]=

The cubic graphs may share large isomorphic subgraphs:

In[14]:=
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A subgraph of the second entry in the list above:

In[15]:=
g2s = Graph[{ UndirectedEdge[1, 2], UndirectedEdge[1, 3], UndirectedEdge[1, 4], UndirectedEdge[2, 5], UndirectedEdge[2, 6], UndirectedEdge[3, 7], UndirectedEdge[3, 8], UndirectedEdge[4, 9], UndirectedEdge[4, 10], UndirectedEdge[5, 11], UndirectedEdge[5, 12], UndirectedEdge[6, 13], UndirectedEdge[6, 14], UndirectedEdge[7, 15], UndirectedEdge[7, 16], UndirectedEdge[8, 17], UndirectedEdge[8, 18], UndirectedEdge[9, 19], UndirectedEdge[9, 20], UndirectedEdge[10, 21], UndirectedEdge[10, 22], UndirectedEdge[11, 15], UndirectedEdge[11, 19], UndirectedEdge[12, 17], UndirectedEdge[12, 23], UndirectedEdge[13, 18], UndirectedEdge[13, 20], UndirectedEdge[14, 24], UndirectedEdge[14, 25], UndirectedEdge[15, 21], UndirectedEdge[16, 24], UndirectedEdge[16, 26], UndirectedEdge[17, 27], UndirectedEdge[18, 28], UndirectedEdge[19, 29], UndirectedEdge[20, 30], UndirectedEdge[21, 25], UndirectedEdge[22, 27], UndirectedEdge[22, 31], UndirectedEdge[23, 25], UndirectedEdge[23, 32], UndirectedEdge[24, 31], UndirectedEdge[26, 30], UndirectedEdge[26, 32], UndirectedEdge[27, 30], UndirectedEdge[28, 29]}]
Out[15]=

The subgraph contains 100% vertices and 95.8% edges of the graphs above:

In[16]:=
PercentForm[Length /@ {VertexList[g2s], EdgeList[g2s]}/{32., 48.}]
Out[16]=

Show that the graph is indeed the subgraph of both entries above:

In[17]:=
Length[FindIsomorphicSubgraph[#, g2s]] > 0 & /@ g2
Out[17]=

Highlight the maximal isomorphic sub-structure of both entries with dashed lines:

In[18]:=
GraphicsRow[ With[{g = #}, HighlightGraph[ Graph[g, EdgeStyle -> Directive[Thick, StandardBlue]], FindIsomorphicSubgraph[g, g2s], GraphHighlightStyle -> {"Thick", "Dashed"}] ] & /@ g2]
Out[18]=

Shenghui Yang, "All Unlabeled Cubic Graphs of Order 32 with Girth at Least 7" from the Wolfram Data Repository (2026)  

Data Resource History

  • Date Created:

Source Metadata

  • Citation:
    • Weisstein, Eric W. "Cubic Graph." From MathWorld--A Wolfram Resource

See Also

Data Downloads

Publisher Information

  • Prepared for the Wolfram Data Repository By: Shenghui Yang
  • Publisher of Record: Shenghui Yang