Wolfram Data Repository
Immediate Computable Access to Curated Contributed Data
All Unlabeled Cubic Graphs of Order 18 with Girth 3
A compactly encoded dataset containing all 41,301 unlabeled connected 3-regular graphs on 18 vertices with girth 3
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 77.5% less storage space than storing full adjacency list representations for all 41,301 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 1: reads 2 3 4 -> Edge: 1-2, Edge: 1-3, Edge: 1-4
Vertex 2: reads 3 4 -> Edge: 2-3, Edge: 2-4
Vertex 3: reads 4 -> Edge: 3-4
Vertex 4: reads nothing (no 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 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
![ResourceData[\!\(\*
TagBox["\"\<All Unlabeled Cubic Graphs of Order 18 with Girth 3\>\"",
#& ,
BoxID -> "ResourceTag-All Unlabeled Cubic Graphs of Order 18 with Girth 3-Input",
AutoDelete->True]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/72a9c8f610094608.png){kind=small}
Scope & Additional Elements (4)
The following steps demonstrate how to convert the list into graphs. Start with the base cubic graph:
| In[2]:= |  |
Convert the data into adjacency lists:
| In[3]:= | ![data = ResourceData[\!\(\*
TagBox["\"\<All Unlabeled Cubic Graphs of Order 18 with Girth 3\>\"",
#& ,
BoxID -> "ResourceTag-All Unlabeled Cubic Graphs of Order 18 with Girth 3-Input",
AutoDelete->True]\)]; ndp = Length[data]; nEdges = 18*3/2; With[{samebitFlagPos = NumericArray[Most[NestWhileList[
With[{pos = #[[1]] + nEdges - #[[2]] + 1}, {pos, If[pos > 250329, -1, data[[pos]]]}] &, {1, 0},
#[[2]] != -1 &]][[All, 1]], "UnsignedInteger32"]},
adjacencyLists = NumericArray@Module[{k = 0, m = 0, dupLen, dupPos},
NestList[(k++;
dupPos = samebitFlagPos[[k]];
dupLen = data[[dupPos]];
#[[;; dupLen ]]~Join~
data[[dupPos + 1 ;; dupPos + 27 - dupLen]]) &,
baseRG, Length[samebitFlagPos]]]
]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/21ffa64fc8026e2b.png) |
Define a function to convert the given adjacency list to the corresponding graph:
| In[4]:= | ![AdjacencyListToGraph[l_] := Module[{dsStk, ctArray, graphEdges, larger},
dsStk = CreateDataStructure["Stack", Reverse[l]];
ctArray = CreateDataStructure["FixedArray", ConstantArray[3, 18]];
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, 17}];
Graph[UndirectedEdge @@@ graphEdges["Elements"]]
]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/497b2431b395c889.png) |
Choose several random cubic graphs from our database:
| In[5]:= | ![SeedRandom["CubicGraph"];
Partition[With[{db = adjacencyLists},
AdjacencyListToGraph /@
(db[[RandomInteger[{1, Length[db]}, 9]]] // Normal)
], 3] // Grid](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/078a0536c2093633.png) |
| Out[6]= |  |
Analysis (2)
The graphs in the database are 3-regular. For instances:
| In[7]:= | ![g = {\!\(\*
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All vertices have degree three:
| In[8]:= |  |
| Out[8]= |  |
| In[9]:= | ![ResourceFunction["RegularGraphQ"] /@ g](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/250caf60900bbe78.png){kind=small} |
| Out[9]= |  |
The girths or the lengths of shortest cycle in the graphs are three:
| In[10]:= | ![ResourceFunction["Girth"] /@ g](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/2fcf5124d858fa38.png){kind=small} |
| Out[10]= |  |
The graphs are pair-wisely non-isomorphic or structurally inequivalent:
| In[11]:= | ![IsomorphicGraphQ @@ (g[[{1, 2}]])](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/25c8ea19ce600a63.png){kind=small} |
| Out[11]= |  |
The graphs may have different centrality:
| In[12]:= | ![GraphicsRow[With[{bc = BetweennessCentrality[#]},
HighlightGraph[#, VertexList[#], VertexSize -> Thread[VertexList[#] -> Rescale[bc]]]] & /@ g]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/7c8d902687e39a28.png){kind=small} |
| Out[12]= |  |
| In[13]:= | ![GraphicsRow[With[{bc = ClosenessCentrality[#]},
HighlightGraph[#, VertexList[#], VertexSize -> Thread[VertexList[#] -> Rescale[bc]]]] & /@ g]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/02bf126a737f2b2e.png){kind=small} |
| Out[13]= |  |
The cubic graphs may share large isomorphic subgraphs:
| In[14]:= | ![g2 = {\!\(\*
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ImageSize->{196.4921875, Automatic}]\)};](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/4375a376beca1963.png) |
A subgraph of the second entry in the list above:
| In[15]:= | ![g2s = Graph[{7 \[UndirectedEdge] 11, 5 \[UndirectedEdge] 8, 2 \[UndirectedEdge] 5, 9 \[UndirectedEdge] 11, 11 \[UndirectedEdge] 15, 16 \[UndirectedEdge] 17, 3 \[UndirectedEdge] 6, 1 \[UndirectedEdge] 4, 10 \[UndirectedEdge] 13, 10 \[UndirectedEdge] 14, 16 \[UndirectedEdge] 18, 15 \[UndirectedEdge] 18, 12 \[UndirectedEdge] 14, 4 \[UndirectedEdge] 7, 7 \[UndirectedEdge] 8, 1 \[UndirectedEdge] 3, 6 \[UndirectedEdge] 10, 15 \[UndirectedEdge] 17, 9 \[UndirectedEdge] 13, 4 \[UndirectedEdge] 5, 14 \[UndirectedEdge] 18}]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/0157a05e80ce4b71.png) |
| Out[15]= |  |
The subgraph contains 100% vertices and 77.8% edges of the graphs above:
| In[16]:= | ![PercentForm[Length /@ {VertexList[g2s], EdgeList[g2s]}/{18., 27.}]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/766d5a9d9a79daef.png){kind=small} |
| Out[16]= |  |
Show that the graph is indeed the subgraph of the both entries:
| In[17]:= | ![Length[FindIsomorphicSubgraph[#, g2s]] > 0 & /@ g2](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/071461cf1c177c12.png){kind=small} |
| Out[17]= |  |
Highlight the isomorphic sub-structure of the first entry with dashed lines:
| In[18]:= | ![With[{g = g2[[1]]}, HighlightGraph[g, FindIsomorphicSubgraph[g, g2s], GraphHighlightStyle -> {Blue, "Thick", "Dashed"}]]](https://www.wolframcloud.com/obj/resourcesystem/images/f4e/f4e61847-04d2-404a-8b1d-05c121af0022/37111fcdfc3e25da.png){kind=small} |
| Out[18]= |  |
External Links
Bibliographic Citation
Shenghui Yang, "All Unlabeled Cubic Graphs of Order 18 with Girth 3" from the Wolfram Data Repository (2026)