Mathematical Graphs, with Embedding Information

**GRAFFITI**
is a dataset directory which
contains 195 mathematical graphs, described
as a collection of nodes, with edges between some pairs of nodes.

The description of each graph includes an "embedding", that is, an assignment of (X,Y) coordinates to each node, so that a plot of the graph can be made.

The files defining each graph are in the GRF file format.

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

GRAPH_REPRESENTATION, a data directory which contains various representations of abstract mathematical graphs.

GRF, a data directory which contains examples of GRF files, an abstract graph file format, 2D graphics;

GRF_DISPLAY, a MATLAB program which reads a GRF file defining a mathematical graph and displays it in the MATLAB graphics window.

GRF_DISPLAY_OPENGL, a C++ program which reads a GRF file defining a mathematical graph and displays it in an OpenGL graphics window.

GRF_IO, a C++ library which reads or writes a GRF file;

GRF_IO, a FORTRAN90 library which reads or writes a GRF file;

GRF_IO, a MATLAB library which reads or writes a GRF file;

GRF_TO_EPS, a FORTRAN90 program which converts a GRF file to EPS forma;

GRF_TO_XYL, a FORTRAN90 program which converts information describing the adjacency and embedding of an abstract graph from GRF to XYL format.

- graff001.grf, K1.
- graff002.grf, the cycle on five vertices and so can be constructed. The circular drawing is the minimum energy configuration for the spring embedding heuristic.
- graff003.grf, two disconnected vertices. Thus it can be constructed.
- graff004.grf, the empty graph on three vertices.
- graff005.grf, four disconnected copies of K_2, embedded as a bipartite graph.
- graff006.grf, the cycle on 29 vertices
- graff007.grf, the cycle on 28 vertices.
- graff008.grf, this circular drawing of graph number 8 is not too enlightening as to its structure.
- graff009.grf, more information will be necessary to provide a reasonable drawing.
- graff010.grf, more information will be necessary to provide a reasonable drawing.
- graff011.grf, an example of a circulant graph.
- graff012.grf, K2, a path on two vertices.
- graff013.grf, K5, the smallest nonplanar graph.
- graff014.grf, K11. Complete graphs and cycles are circulant graphs.
- graff015.grf, the Petersen graph, which is the only three-regular graph of girth 5. The spring embedding does a decent, but not perfect job of showing its structure.
- graff016.grf, of order 64, is too dense for individual edges to appear in displaying its circular embedding. In this case, the complement of the graph is much more informative.
- graff017.grf, Even cycles are bipartite graphs, and this drawing reflects that.
- graff018.grf, a simple cycle on six vertices.
- graff019.grf, a simple cycle on seven vertices.
- graff020.grf, is bipartite, but not regular.
- graff021.grf, is bipartite, but not regular.
- graff022.grf, seems to be K{10,10} less several edges.
- graff023.grf, is the cycle on eight vertices.
- graff024.grf, is a tree on eight vertices. RadialEmbedding has been used to draw the tree so no two edges cross.
- graff025.grf, is a tree on 14 vertices. Again, RadialEmbedding has been used to draw the tree so no two edges cross.
- graff026.grf, has been fed to SpringEmbedding, but it is not at all clear what is going on.
- graff027.grf, is the triangle, or K_3.
- graff028.grf, defines the path on seven vertices, or Path[7].
- graff029.grf, can be constructed by GraphJoin[ EmptyGraph[2], EmptyGraph[3] ].
- graff030.grf, s the complete bipartite graph K_{5,5}.
- graff031.grf, is the circulant graph CirculantGraph[13, {1,5}].
- graff032.grf, is another circulant graph, CirculantGraph[13, {2,3,4,6}].
- graff033.grf, is the star on eleven vertices Star[11].
- graff034.grf, represents K_{10}.
- graff035.grf, is pretty unimpressive under the spring heuristic.
- graff036.grf, consists of K_8 with one vertex incident on path of length two.
- graff037.grf, has two K_4's joined with an isolated vertex, and can be constructed GraphJoin[K[1], GraphUnion[2,K[4]]].
- graff038.grf, is the circulant graph CirculantGraph[6,{1,2}].
- graff039.grf, is a tree with four vertices of degree 1, two of degree 3, and seven of degree 2.
- graff040.grf, is disconnected, GraphUnion[ EmptyGraph[2], K[2] ].
- graff041.grf, is interesting, if not immediately enlightening. An extra vertex is attached to each of the leaves of Star[11].
- graff042.grf, is the circulant graph CirculantGraph[12,{1,2,3}].
- graff043.grf, is a terrible drawing of DeleteEdge[K[4],{1,2}].
- graff044.grf, is a simple path on three vertices.
- graff045.grf, is a simple path on four vertices.
- graff046.grf, is a terrible drawing of DeleteEdge[K[5],{1,2}].
- graff047.grf, is a cycle of length four, with an edge to an isolated vertex.
- graff048.grf, is two cycles of length four sharing one vertex between them.
- graff049.grf, is a path of length three with both endpoints joined to K_3.
- graff050.grf, is \oi{GraphJoin[K[1], K[3,3]]}.
- graff051.grf, is GraphUnion[K[1],K[4]]. The spring embedding heuristic causes disconnected components to drift apart.
- graff052.grf, is the circulant graph CirculantGraph[10,{1,2,3,4}].
- graff053.grf, is fairly well illustrated by the spring embedding heuristic.
- graff054.grf, is a nicely drawn tree.
- graff055.grf, is a tree with a high degree of symmetry.
- graff056.grf, is a bipartite planar graph.
- graff057.grf, is the union of three even cycles, GraphUnion[Cycle[4], GraphUnion[Cycle[6],Cycle[8]] ].
- graff058.grf, is an odd cycle with two isolated vertices attached on one cycle vertex.
- graff059.grf, looks like K[10,10].
- graff060.grf, has an interesting circular embedding.
- graff061.grf, is another tree.
- graff062.grf, is K_{8,2} minus one edge.
- graff063.grf, is K_{7,2} with one extra vertex and edge.
- graff064.grf, is GraphJoin[K[7],EmptyGraph[2]], and with this drawing looks almost three-dimensional.
- graff065.grf, The structure of graph number 65 is not clear from this drawing.
- graff066.grf, is the complete bipartite graph K_{3,4}.
- graff067.grf, looks like GraphJoin[K[1], K[8]]. but I believe some edges are obscured by the center vertex.
- graff068.grf, has two edges deleted from K_{2,12}.
- graff069.grf, The spring embedding heuristic does a good job showing the symmetry of graph number 69.
- graff070.grf, is a simple tree.
- graff071.grf, is a tree with a longer tail than graph number 70.
- graff072.grf, is a tree with branches consisting of paths on 1, 2, and 3 vertices.
- graff073.grf, has trouble because its embedding had points too close to zero.
- graff074.grf, looks like K_{10} entended by paths on length one, and in one case two.
- graff075.grf, is too dense for its circular embedding to reveal any structure. The complement appears sparser, but it still not understood.
- graff076.grf, is two edges removed from being a regular graph.
- graff077.grf, is the tree Path[29].
- graff078.grf, appears to be two K_{10}'s connected by a path of nine other vertices.
- graff079.grf, is spectacular, whatever it is!
- graff080.grf, is a tree, and so might be better displayed with a radial embedding.
- graff081.grf, appears similar to graph number 78, but is larger.
- graff082.grf, has several clusters of vertices attached to a central path.
- graff083.grf, is another interesting structure but not well understand.
- graff084.grf, The spring embedding of graph number 84 put several vertices too close together.
- graff085.grf, is a tree consisting of stars of 13 and 14 vertices joined at the center.
- graff086.grf, is a path with two cliques dense enough to look ugly.
- graff087.grf, is too dense to be properly displayed in a circular embedding.
- graff088.grf, does not appear to be symmetric, at least from this drawing.
- graff089.grf, looks like it might be a large circulant graph.
- graff090.grf, does not look like a circulant graph, but is quite symmetric.
- graff091.grf, is a tree similar to graph 80.
- graff092.grf, is too dense to properly identify.
- graff093.grf, appears to be a tree.
- graff094.grf, might be a tree similar to graph number 94.
- graff095.grf, is not a tree, which means the previous graphs may not be.
- graff096.grf, continues this trend.
- graff097.grf, contains few cycles.
- graff098.grf, is a cycle on 12 vertices with edges to two additional vertices.
- graff099.grf, is a cycle on fifteen vertices, each of which is connected to a vertex of degree one.
- graff100.grf, seems to be similar to graph number 99, with the cycle replaced by K_{15}.
- graff101.grf, looks like a comb with a little K_4 at the end.
- graff102.grf, is open on one end.
- graff103.grf, resembles a convex polyhedra in this drawing, which means that is it planar.
- graff104.grf, is bipartite, but symmetric enough to suggest that this is not the right embedding.
- graff105.grf, seems to be a product graph.
- graff106.grf, is in the tradition of graphs 94 to 96.
- graff107.grf, looks like another path with loops on the ends.
- graff108.grf, The symmetry of graph number 108 is evident in this spring embedding.
- graff109.grf, consists of two copies of some graph joined in some way.
- graff110.grf, consists of three copies of some graph joined in some way.
- graff111.grf, is K_{2,5}.
- graff112.grf, is K_{3,3} less an edge.
- graff113.grf, is a simple tree.
- graff114.grf, is not K_{4,4} but an incredible simulation.
- graff115.grf, is another path with two loops at the end.
- graff116.grf, is the same flavor as graph number 115.
- graff117.grf, is the star Star[32].
- graff118.grf, is the cycle on 30 vertices.
- graff119.grf, is the cycle on 31 vertices
- graff120.grf, is the cycle on 29 vertices.
- graff121.grf, is a path on 30 vertices.
- graff122.grf, is another dense graph with protrusions.
- graff123.grf, is bipartite and symmetrical.
- graff124.grf, is GraphJoin[K[1],Cycle[4]].
- graff125.grf, is two components highly connected.
- graff126.grf, is an edge and vertex added to \oi{Cycle[10]}.
- graff127.grf, is two cycles sharing an edge, and nicely displayed as a spring embedding.
- graff128.grf, is a cycle with two cross-edges.
- graff129.grf, looks like another circulant graph.
- graff130.grf, is not K_4, since it contains 16 vertices.
- graff131.grf, is too dense to identify from this drawing.
- graff132.grf, is two components joined in a star.
- graff133.grf, is the circulant graph CirculantGraph[10,{1,4,5}].
- graff134.grf, is GraphUnion[2,K[3]].
- graff135.grf, is GraphUnion[5,Cycle[5]].
- graff136.grf, is a triangle with a tail.
- graff137.grf, is the union of disjoint cycle of length 6, 10, and 14.
- graff138.grf, is three triangles and a K_2 with a vertex in common.
- graff139.grf, is a cycle on four vertices.
- graff140.grf, is K_{3,2} less an edge.
- graff141.grf, is a nice circulant graph.
- graff142.grf, is too dense to identify from this picture, but it looks like a circulant graph.
- graff143.grf, is two stars whose centers are on a triangle.
- graff144.grf, is a generalization of graph 143.
- graff145.grf, is the path on three vertices.
- graff146.grf, is clearly shown in this spring embedding.
- graff147.grf, is less clearly illustrated by the spring embedding.
- graff148.grf, is even less clearly illustrated by the spring embedding.
- graff149.grf, is too dense to really understand from this drawing.
- graff150.grf, is sparse enough to understand.
- graff151.grf, should not be drawn in such a two-dimensional manner.
- graff152.grf, is sparse enough that a better embedding might be understandable.
- graff153.grf, consists of two large stars joined at the centers.
- graff154.grf, might be clearer with a better drawing.
- graff155.grf, is GraphUnion[2,Star[6]].
- graff156.grf, is GraphUnion[Cycle[4],Cycle[5]].
- graff157.grf, is the circulant graph CirculantGraph[7,{2,3}].
- graff158.grf, is GraphUnion[K[2],K[3]].
- graff159.grf, is GraphUnion[2,K[3]].
- graff160.grf, is GraphUnion[K[1],K[2]].
- graff161.grf, is Star[53].
- graff162.grf, is not a cycle, but has a dense set of interconnections to its neighborhood.
- graff163.grf, is similar to the above, but with a thicker neighborhood.
- graff164.grf, Since graph number 164 the largest graph in the collection, an almost complete graph on 144 vertices, we display its complement instead.
- graff165.grf, is a ring but not a cycle.
- graff166.grf, is the path on 90 vertices.
- graff167.grf, appears in the tradition of graph number 122.
- graff168.grf, is bipartite, but otherwise indistinguishable.
- graff169.grf, is GraphUnion[ GraphUnion[7,K[2]], GraphUnion[2, Star[6]] ].
- graff170.grf, is GraphUnion[K[2], GraphUnion[2,Cycle[4]]].
- graff171.grf, is dense, but has a clearly identifiable structure.
- graff172.grf, is a regular-looking structure with some attachments.
- graff173.grf, seems to be two vertices of degree n-1 and n-2 vertices of degree two.
- graff174.grf, is another dense graph with attachments.
- graff175.grf, is another fairly dense graph.
- graff176.grf, is a cycle on four vertices.
- graff177.grf, is an eight vertex component unioned with EmptyGraph[2].
- graff178.grf, is a cycle of length four with an attached path of length two.
- graff179.grf, is like graph number 178 with an attached path of length three.
- graff180.grf, is like graph number 178 with an attached path of length four.
- graff181.grf, is GraphUnion[ Cycle[5], K[2] ].
- graff182.grf, is the union of GraphUnion[6,K[2]]} and \oi{Cycle[5].
- graff183.grf, is two 4-cycles sharing one vertex. Didn't this one occur earlier?
- graff184.grf, is properly drawn using the spring embedding heuristic.
- graff185.grf, is GraphUnion[K[1], K[2]]. Didn't this one occur earlier?
- graff186.grf, has fifteen vertices overall.
- graff187.grf, is another large dense graph.
- graff188.grf, is another disconnected graph.
- graff189.grf, looks like graph number 187.
- graff190.grf, is at least small enough that the spring embedding heurstic can be applied.
- graff191.grf, looks like it will prosper with a different embedding.
- graff192.grf, also looks like it has possibilities.
- graff193.grf, looks like a candidate for the graph editor.
- graff194.grf, is another black hole.
- graff195.grf, is another ugly to end this exercise.

You can go up one level to the DATASETS directory.