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Creates a new graph on the set of vertices v_list and with edges e_list.
v_list is a list of vertices ([v1, v2,..., vn]
) or a
list of vertices together with vertex labels ([[v1,l1], [v2,l2],..., [vn,ln]]
).
n is the number of vertices. Vertices will be identified by integers from 0 to n-1.
e_list is a list of edges ([e1, e2,..., em]
) or a list of
edges together with edge-weights ([[e1, w1], ..., [em, wm]]
).
If directed is not false
, a directed graph will be returned.
Example 1: create a cycle on 3 vertices:
(%i1) load ("graphs")$ (%i2) g : create_graph([1,2,3], [[1,2], [2,3], [1,3]])$ (%i3) print_graph(g)$ Graph on 3 vertices with 3 edges. Adjacencies: 3 : 1 2 2 : 3 1 1 : 3 2
Example 2: create a cycle on 3 vertices with edge weights:
(%i1) load ("graphs")$ (%i2) g : create_graph([1,2,3], [[[1,2], 1.0], [[2,3], 2.0], [[1,3], 3.0]])$ (%i3) print_graph(g)$ Graph on 3 vertices with 3 edges. Adjacencies: 3 : 1 2 2 : 3 1 1 : 3 2
Example 3: create a directed graph:
(%i1) load ("graphs")$ (%i2) d : create_graph( [1,2,3,4], [ [1,3], [1,4], [2,3], [2,4] ], 'directed = true)$ (%i3) print_graph(d)$ Digraph on 4 vertices with 4 arcs. Adjacencies: 4 : 3 : 2 : 4 3 1 : 4 3
Returns a copy of the graph g.
Returns the circulant graph with parameters n and d.
Example:
(%i1) load ("graphs")$ (%i2) g : circulant_graph(10, [1,3])$ (%i3) print_graph(g)$ Graph on 10 vertices with 20 edges. Adjacencies: 9 : 2 6 0 8 8 : 1 5 9 7 7 : 0 4 8 6 6 : 9 3 7 5 5 : 8 2 6 4 4 : 7 1 5 3 3 : 6 0 4 2 2 : 9 5 3 1 1 : 8 4 2 0 0 : 7 3 9 1
Returns the Clebsch graph.
Returns the complement of the graph g.
Returns the complete bipartite graph on n+m vertices.
Returns the complete graph on n vertices.
Returns the directed cycle on n vertices.
Returns the cycle on n vertices.
Returns the cuboctahedron graph.
Returns the n-dimensional cube.
Returns the dodecahedron graph.
Returns the empty graph on n vertices.
Returns the flower graph on 4n vertices.
Example:
(%i1) load ("graphs")$ (%i2) f5 : flower_snark(5)$ (%i3) chromatic_index(f5); (%o3) 4
Returns the graph represented by its adjacency matrix A.
Returns the Frucht graph.
Returns the direct product of graphs g1 and g2.
Example:
(%i1) load ("graphs")$ (%i2) grid : graph_product(path_graph(3), path_graph(4))$ (%i3) draw_graph(grid)$
Returns the union (sum) of graphs g1 and g2.
Returns the n x m grid.
Returns the great rhombicosidodecahedron graph.
Returns the great rhombicuboctahedron graph.
Returns the Grotzch graph.
Returns the Heawood graph.
Returns the icosahedron graph.
Returns the icosidodecahedron graph.
Returns the graph induced on the subset V of vertices of the graph g.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) V : [0,1,2,3,4]$ (%i4) g : induced_subgraph(V, p)$ (%i5) print_graph(g)$ Graph on 5 vertices with 5 edges. Adjacencies: 4 : 3 0 3 : 2 4 2 : 1 3 1 : 0 2 0 : 1 4
Returns the line graph of the graph g.
Creates a graph using a predicate function f.
vrt is a list/set of vertices or an integer. If vrt is an integer, then vertices of the graph will be integers from 1 to vrt.
f is a predicate function. Two vertices a and b will
be connected if f(a,b)=true
.
If directed is not false, then the graph will be directed.
Example 1:
(%i1) load("graphs")$ (%i2) g : make_graph(powerset({1,2,3,4,5}, 2), disjointp)$ (%i3) is_isomorphic(g, petersen_graph()); (%o3) true (%i4) get_vertex_label(1, g); (%o4) {1, 2}
Example 2:
(%i1) load("graphs")$ (%i2) f(i, j) := is (mod(j, i)=0)$ (%i3) g : make_graph(20, f, directed=true)$ (%i4) out_neighbors(4, g); (%o4) [8, 12, 16, 20] (%i5) in_neighbors(18, g); (%o5) [1, 2, 3, 6, 9]
Returns the mycielskian graph of the graph g.
Returns the graph with no vertices and no edges.
Returns the directed path on n vertices.
Returns the path on n vertices.
Returns the petersen graph P_{n,d}. The default values for
n and d are n=5
and d=2
.
Returns a random bipartite graph on a+b
vertices. Each edge is
present with probability p.
Returns a random directed graph on n vertices. Each arc is present with probability p.
Returns a random d-regular graph on n vertices. The default
value for d is d=3
.
Returns a random graph on n vertices. Each edge is present with probability p.
Returns a random graph on n vertices and random m edges.
Returns a random network on n vertices. Each arc is present with
probability p and has a weight in the range [0,w]
. The
function returns a list [network, source, sink]
.
Example:
(%i1) load ("graphs")$ (%i2) [net, s, t] : random_network(50, 0.2, 10.0); (%o2) [DIGRAPH, 50, 51] (%i3) max_flow(net, s, t)$ (%i4) first(%); (%o4) 27.65981397932507
Returns a random tournament on n vertices.
Returns a random tree on n vertices.
Returns the small rhombicosidodecahedron graph.
Returns the small rhombicuboctahedron graph.
Returns the snub cube graph.
Returns the snub dodecahedron graph.
Returns the truncated cube graph.
Returns the truncated dodecahedron graph.
Returns the truncated icosahedron graph.
Returns the truncated tetrahedron graph.
Returns the Tutte graph.
Returns the underlying graph of the directed graph g.
Returns the wheel graph on n+1 vertices.
Returns the adjacency matrix of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) c5 : cycle_graph(4)$ (%i3) adjacency_matrix(c5); [ 0 1 0 1 ] [ ] [ 1 0 1 0 ] (%o3) [ ] [ 0 1 0 1 ] [ ] [ 1 0 1 0 ]
Returns the average degree of vertices in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) average_degree(grotzch_graph()); 40 (%o2) -- 11
Returns the (vertex sets of) 2-connected components of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : create_graph( [1,2,3,4,5,6,7], [ [1,2],[2,3],[2,4],[3,4], [4,5],[5,6],[4,6],[6,7] ])$ (%i3) biconnected_components(g); (%o3) [[6, 7], [4, 5, 6], [1, 2], [2, 3, 4]]
Returns a bipartition of the vertices of the graph gr or an empty list if gr is not bipartite.
Example:
(%i1) load ("graphs")$ (%i2) h : heawood_graph()$ (%i3) [A,B]:bipartition(h); (%o3) [[8, 12, 6, 10, 0, 2, 4], [13, 5, 11, 7, 9, 1, 3]] (%i4) draw_graph(h, show_vertices=A, program=circular)$
Returns the chromatic index of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) chromatic_index(p); (%o3) 4
Returns the chromatic number of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) chromatic_number(cycle_graph(5)); (%o2) 3 (%i3) chromatic_number(cycle_graph(6)); (%o3) 2
Removes the weight of the edge e in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : create_graph(3, [[[0,1], 1.5], [[1,2], 1.3]])$ (%i3) get_edge_weight([0,1], g); (%o3) 1.5 (%i4) clear_edge_weight([0,1], g)$ (%i5) get_edge_weight([0,1], g); (%o5) 1
Removes the label of the vertex v in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$ (%i3) get_vertex_label(0, g); (%o3) Zero (%i4) clear_vertex_label(0, g); (%o4) done (%i5) get_vertex_label(0, g); (%o5) false
Returns the (vertex sets of) connected components of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g: graph_union(cycle_graph(5), path_graph(4))$ (%i3) connected_components(g); (%o3) [[1, 2, 3, 4, 0], [8, 7, 6, 5]]
Returns the diameter of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) diameter(dodecahedron_graph()); (%o2) 5
Returns an optimal coloring of the edges of the graph gr.
The function returns the chromatic index and a list representing the coloring of the edges of gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) [ch_index, col] : edge_coloring(p); (%o3) [4, [[[0, 5], 3], [[5, 7], 1], [[0, 1], 1], [[1, 6], 2], [[6, 8], 1], [[1, 2], 3], [[2, 7], 4], [[7, 9], 2], [[2, 3], 2], [[3, 8], 3], [[5, 8], 2], [[3, 4], 1], [[4, 9], 4], [[6, 9], 3], [[0, 4], 2]]] (%i4) assoc([0,1], col); (%o4) 1 (%i5) assoc([0,5], col); (%o5) 3
Returns the list of vertex degrees of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) degree_sequence(random_graph(10, 0.4)); (%o2) [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]
Returns the edge-connectivity of the graph gr.
See also min_edge_cut
.
Returns the list of edges (arcs) in a (directed) graph gr.
Example:
(%i1) load ("graphs")$ (%i2) edges(complete_graph(4)); (%o2) [[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]
Returns the weight of the edge e in the graph gr.
If there is no weight assigned to the edge, the function returns 1. If the edge is not present in the graph, the function signals an error or returns the optional argument ifnot.
Example:
(%i1) load ("graphs")$ (%i2) c5 : cycle_graph(5)$ (%i3) get_edge_weight([1,2], c5); (%o3) 1 (%i4) set_edge_weight([1,2], 2.0, c5); (%o4) done (%i5) get_edge_weight([1,2], c5); (%o5) 2.0
Returns the label of the vertex v in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$ (%i3) get_vertex_label(0, g); (%o3) Zero
Returns the characteristic polynomial (in variable x) of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) graph_charpoly(p, x), factor; 5 4 (%o3) (x - 3) (x - 1) (x + 2)
Returns the center of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : grid_graph(5,5)$ (%i3) graph_center(g); (%o3) [12]
Returns the eigenvalues of the graph gr. The function returns
eigenvalues in the same format as maxima eigenvalues
function.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) graph_eigenvalues(p); (%o3) [[3, - 2, 1], [1, 4, 5]]
Returns the periphery of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : grid_graph(5,5)$ (%i3) graph_periphery(g); (%o3) [24, 20, 4, 0]
Returns the number of edges in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) graph_size(p); (%o3) 15
Returns the number of vertices in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) graph_order(p); (%o3) 10
Returns the length of the shortest cycle in gr.
Example:
(%i1) load ("graphs")$ (%i2) g : heawood_graph()$ (%i3) girth(g); (%o3) 6
Returns the Hamilton cycle of the graph gr or an empty list if gr is not hamiltonian.
Example:
(%i1) load ("graphs")$ (%i2) c : cube_graph(3)$ (%i3) hc : hamilton_cycle(c); (%o3) [7, 3, 2, 6, 4, 0, 1, 5, 7] (%i4) draw_graph(c, show_edges=vertices_to_cycle(hc))$
Returns the Hamilton path of the graph gr or an empty list if gr does not have a Hamilton path.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) hp : hamilton_path(p); (%o3) [0, 5, 7, 2, 1, 6, 8, 3, 4, 9] (%i4) draw_graph(p, show_edges=vertices_to_path(hp))$
Returns a an isomorphism between graphs/digraphs gr1 and gr2. If gr1 and gr2 are not isomorphic, it returns an empty list.
Example:
(%i1) load ("graphs")$ (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$ (%i3) isomorphism(clk5, petersen_graph()); (%o3) [9 -> 0, 2 -> 1, 6 -> 2, 5 -> 3, 0 -> 4, 1 -> 5, 3 -> 6, 4 -> 7, 7 -> 8, 8 -> 9]
Returns the list of in-neighbors of the vertex v in the directed graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : path_digraph(3)$ (%i3) in_neighbors(2, p); (%o3) [1] (%i4) out_neighbors(2, p); (%o4) []
Returns true
if gr is 2-connected and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_biconnected(cycle_graph(5)); (%o2) true (%i3) is_biconnected(path_graph(5)); (%o3) false
Returns true
if gr is bipartite (2-colorable) and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_bipartite(petersen_graph()); (%o2) false (%i3) is_bipartite(heawood_graph()); (%o3) true
Returns true
if the graph gr is connected and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_connected(graph_union(cycle_graph(4), path_graph(3))); (%o2) false
Returns true
if gr is a directed graph and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_digraph(path_graph(5)); (%o2) false (%i3) is_digraph(path_digraph(5)); (%o3) true
Returns true
if e is an edge (arc) in the (directed) graph g
and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) c4 : cycle_graph(4)$ (%i3) is_edge_in_graph([2,3], c4); (%o3) true (%i4) is_edge_in_graph([3,2], c4); (%o4) true (%i5) is_edge_in_graph([2,4], c4); (%o5) false (%i6) is_edge_in_graph([3,2], cycle_digraph(4)); (%o6) false
Returns true
if gr is a graph and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_graph(path_graph(5)); (%o2) true (%i3) is_graph(path_digraph(5)); (%o3) false
Returns true
if gr is a graph or a directed graph and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_graph_or_digraph(path_graph(5)); (%o2) true (%i3) is_graph_or_digraph(path_digraph(5)); (%o3) true
Returns true
if graphs/digraphs gr1 and gr2 are isomorphic
and false
otherwise.
See also isomorphism
.
Example:
(%i1) load ("graphs")$ (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$ (%i3) is_isomorphic(clk5, petersen_graph()); (%o3) true
Returns true
if gr is a planar graph and false
otherwise.
The algorithm used is the Demoucron’s algorithm, which is a quadratic time algorithm.
Example:
(%i1) load ("graphs")$ (%i2) is_planar(dodecahedron_graph()); (%o2) true (%i3) is_planar(petersen_graph()); (%o3) false (%i4) is_planar(petersen_graph(10,2)); (%o4) true
Returns true
if the directed graph gr is strongly connected and
false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_sconnected(cycle_digraph(5)); (%o2) true (%i3) is_sconnected(path_digraph(5)); (%o3) false
Returns true
if v is a vertex in the graph g and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) c4 : cycle_graph(4)$ (%i3) is_vertex_in_graph(0, c4); (%o3) true (%i4) is_vertex_in_graph(6, c4); (%o4) false
Returns true
if gr is a tree and false
otherwise.
Example:
(%i1) load ("graphs")$ (%i2) is_tree(random_tree(4)); (%o2) true (%i3) is_tree(graph_union(random_tree(4), random_tree(5))); (%o3) false
Returns the laplacian matrix of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) laplacian_matrix(cycle_graph(5)); [ 2 - 1 0 0 - 1 ] [ ] [ - 1 2 - 1 0 0 ] [ ] (%o2) [ 0 - 1 2 - 1 0 ] [ ] [ 0 0 - 1 2 - 1 ] [ ] [ - 1 0 0 - 1 2 ]
Returns a maximum clique of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : random_graph(100, 0.5)$ (%i3) max_clique(g); (%o3) [6, 12, 31, 36, 52, 59, 62, 63, 80]
Returns the maximal degree of vertices of the graph gr and a vertex of maximal degree.
Example:
(%i1) load ("graphs")$ (%i2) g : random_graph(100, 0.02)$ (%i3) max_degree(g); (%o3) [6, 79] (%i4) vertex_degree(95, g); (%o4) 2
Returns a maximum flow through the network net with the source s and the sink t.
The function returns the value of the maximal flow and a list representing the weights of the arcs in the optimal flow.
Example:
(%i1) load ("graphs")$ (%i2) net : create_graph( [1,2,3,4,5,6], [[[1,2], 1.0], [[1,3], 0.3], [[2,4], 0.2], [[2,5], 0.3], [[3,4], 0.1], [[3,5], 0.1], [[4,6], 1.0], [[5,6], 1.0]], directed=true)$ (%i3) [flow_value, flow] : max_flow(net, 1, 6); (%o3) [0.7, [[[1, 2], 0.5], [[1, 3], 0.2], [[2, 4], 0.2], [[2, 5], 0.3], [[3, 4], 0.1], [[3, 5], 0.1], [[4, 6], 0.3], [[5, 6], 0.4]]] (%i4) fl : 0$ (%i5) for u in out_neighbors(1, net) do fl : fl + assoc([1, u], flow)$ (%i6) fl; (%o6) 0.7
Returns a maximum independent set of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) d : dodecahedron_graph()$ (%i3) mi : max_independent_set(d); (%o3) [0, 3, 5, 9, 10, 11, 18, 19] (%i4) draw_graph(d, show_vertices=mi)$
Returns a maximum matching of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) d : dodecahedron_graph()$ (%i3) m : max_matching(d); (%o3) [[5, 7], [8, 9], [6, 10], [14, 19], [13, 18], [12, 17], [11, 16], [0, 15], [3, 4], [1, 2]] (%i4) draw_graph(d, show_edges=m)$
Returns the minimum degree of vertices of the graph gr and a vertex of minimum degree.
Example:
(%i1) load ("graphs")$ (%i2) g : random_graph(100, 0.1)$ (%i3) min_degree(g); (%o3) [3, 49] (%i4) vertex_degree(21, g); (%o4) 9
Returns the minimum edge cut in the graph gr.
See also edge_connectivity
.
Returns the minimum vertex cover of the graph gr.
Returns the minimum vertex cut in the graph gr.
See also vertex_connectivity
.
Returns the minimum spanning tree of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : graph_product(path_graph(10), path_graph(10))$ (%i3) t : minimum_spanning_tree(g)$ (%i4) draw_graph(g, show_edges=edges(t))$
Returns the list of neighbors of the vertex v in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) neighbors(3, p); (%o3) [4, 8, 2]
Returns the length of the shortest odd cycle in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : graph_product(cycle_graph(4), cycle_graph(7))$ (%i3) girth(g); (%o3) 4 (%i4) odd_girth(g); (%o4) 7
Returns the list of out-neighbors of the vertex v in the directed graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : path_digraph(3)$ (%i3) in_neighbors(2, p); (%o3) [1] (%i4) out_neighbors(2, p); (%o4) []
Returns the list of facial walks in a planar embedding of gr and
false
if gr is not a planar graph.
The graph gr must be biconnected.
The algorithm used is the Demoucron’s algorithm, which is a quadratic time algorithm.
Example:
(%i1) load ("graphs")$ (%i2) planar_embedding(grid_graph(3,3)); (%o2) [[3, 6, 7, 8, 5, 2, 1, 0], [4, 3, 0, 1], [3, 4, 7, 6], [8, 7, 4, 5], [1, 2, 5, 4]]
Prints some information about the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) c5 : cycle_graph(5)$ (%i3) print_graph(c5)$ Graph on 5 vertices with 5 edges. Adjacencies: 4 : 0 3 3 : 4 2 2 : 3 1 1 : 2 0 0 : 4 1 (%i4) dc5 : cycle_digraph(5)$ (%i5) print_graph(dc5)$ Digraph on 5 vertices with 5 arcs. Adjacencies: 4 : 0 3 : 4 2 : 3 1 : 2 0 : 1 (%i6) out_neighbors(0, dc5); (%o6) [1]
Returns the radius of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) radius(dodecahedron_graph()); (%o2) 5
Assigns the weight w to the edge e in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : create_graph([1, 2], [[[1,2], 1.2]])$ (%i3) get_edge_weight([1,2], g); (%o3) 1.2 (%i4) set_edge_weight([1,2], 2.1, g); (%o4) done (%i5) get_edge_weight([1,2], g); (%o5) 2.1
Assigns the label l to the vertex v in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : create_graph([[1, "One"], [2, "Two"]], [[1,2]])$ (%i3) get_vertex_label(1, g); (%o3) One (%i4) set_vertex_label(1, "oNE", g); (%o4) done (%i5) get_vertex_label(1, g); (%o5) oNE
Returns the shortest path from u to v in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) d : dodecahedron_graph()$ (%i3) path : shortest_path(0, 7, d); (%o3) [0, 1, 19, 13, 7] (%i4) draw_graph(d, show_edges=vertices_to_path(path))$
Returns the length of the shortest weighted path and the shortest weighted path from u to v in the graph gr.
The length of a weighted path is the sum of edge weights of edges in the path. If an edge has no weight, then it has a default weight 1.
Example:
(%i1) load ("graphs")$ (%i2) g: petersen_graph(20, 2)$ (%i3) for e in edges(g) do set_edge_weight(e, random(1.0), g)$ (%i4) shortest_weighted_path(0, 10, g); (%o4) [2.575143920268482, [0, 20, 38, 36, 34, 32, 30, 10]]
Returns the strong components of a directed graph gr.
Example:
(%i1) load ("graphs")$ (%i2) t : random_tournament(4)$ (%i3) strong_components(t); (%o3) [[1], [0], [2], [3]] (%i4) vertex_out_degree(3, t); (%o4) 3
Returns a topological sorting of the vertices of a directed graph dag or an empty list if dag is not a directed acyclic graph.
Example:
(%i1) load ("graphs")$ (%i2) g:create_graph( [1,2,3,4,5], [ [1,2], [2,5], [5,3], [5,4], [3,4], [1,3] ], directed=true)$ (%i3) topological_sort(g); (%o3) [1, 2, 5, 3, 4]
Returns the vertex connectivity of the graph g.
See also min_vertex_cut
.
Returns the degree of the vertex v in the graph gr.
Returns the length of the shortest path between u and v in the (directed) graph gr.
Example:
(%i1) load ("graphs")$ (%i2) d : dodecahedron_graph()$ (%i3) vertex_distance(0, 7, d); (%o3) 4 (%i4) shortest_path(0, 7, d); (%o4) [0, 1, 19, 13, 7]
Returns the eccentricity of the vertex v in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g:cycle_graph(7)$ (%i3) vertex_eccentricity(0, g); (%o3) 3
Returns the in-degree of the vertex v in the directed graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p5 : path_digraph(5)$ (%i3) print_graph(p5)$ Digraph on 5 vertices with 4 arcs. Adjacencies: 4 : 3 : 4 2 : 3 1 : 2 0 : 1 (%i4) vertex_in_degree(4, p5); (%o4) 1 (%i5) in_neighbors(4, p5); (%o5) [3]
Returns the out-degree of the vertex v in the directed graph gr.
Example:
(%i1) load ("graphs")$ (%i2) t : random_tournament(10)$ (%i3) vertex_out_degree(0, t); (%o3) 2 (%i4) out_neighbors(0, t); (%o4) [7, 1]
Returns the list of vertices in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) vertices(complete_graph(4)); (%o2) [3, 2, 1, 0]
Returns an optimal coloring of the vertices of the graph gr.
The function returns the chromatic number and a list representing the coloring of the vertices of gr.
Example:
(%i1) load ("graphs")$ (%i2) p:petersen_graph()$ (%i3) vertex_coloring(p); (%o3) [3, [[0, 2], [1, 3], [2, 2], [3, 3], [4, 1], [5, 3], [6, 1], [7, 1], [8, 2], [9, 2]]]
Returns the Wiener index of the graph gr.
Example:
(%i2) wiener_index(dodecahedron_graph()); (%o2) 500
Adds the edge e to the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : path_graph(4)$ (%i3) neighbors(0, p); (%o3) [1] (%i4) add_edge([0,3], p); (%o4) done (%i5) neighbors(0, p); (%o5) [3, 1]
Adds all edges in the list e_list to the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : empty_graph(3)$ (%i3) add_edges([[0,1],[1,2]], g)$ (%i4) print_graph(g)$ Graph on 3 vertices with 2 edges. Adjacencies: 2 : 1 1 : 2 0 0 : 1
Adds the vertex v to the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : path_graph(2)$ (%i3) add_vertex(2, g)$ (%i4) print_graph(g)$ Graph on 3 vertices with 1 edges. Adjacencies: 2 : 1 : 0 0 : 1
Adds all vertices in the list v_list to the graph gr.
Connects all vertices from the list v_list with the vertices in the list u_list in the graph gr.
v_list and u_list can be single vertices or lists of vertices.
Example:
(%i1) load ("graphs")$ (%i2) g : empty_graph(4)$ (%i3) connect_vertices(0, [1,2,3], g)$ (%i4) print_graph(g)$ Graph on 4 vertices with 3 edges. Adjacencies: 3 : 0 2 : 0 1 : 0 0 : 3 2 1
Contracts the edge e in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g: create_graph( 8, [[0,3],[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])$ (%i3) print_graph(g)$ Graph on 8 vertices with 7 edges. Adjacencies: 7 : 4 6 : 4 5 : 4 4 : 7 6 5 3 3 : 4 2 1 0 2 : 3 1 : 3 0 : 3 (%i4) contract_edge([3,4], g)$ (%i5) print_graph(g)$ Graph on 7 vertices with 6 edges. Adjacencies: 7 : 3 6 : 3 5 : 3 3 : 5 6 7 2 1 0 2 : 3 1 : 3 0 : 3
Removes the edge e from the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) c3 : cycle_graph(3)$ (%i3) remove_edge([0,1], c3)$ (%i4) print_graph(c3)$ Graph on 3 vertices with 2 edges. Adjacencies: 2 : 0 1 1 : 2 0 : 2
Removes the vertex v from the graph gr.
Exports the graph into the file fl in the DIMACS format. Optional comments will be added to the top of the file.
Returns the graph from file fl in the DIMACS format.
Returns the graph encoded in the graph6 format in the string str.
Returns a string which encodes the graph gr in the graph6 format.
Exports graphs in the list gr_list to the file fl in the graph6 format.
Returns a list of graphs from the file fl in the graph6 format.
Returns the graph encoded in the sparse6 format in the string str.
Returns a string which encodes the graph gr in the sparse6 format.
Exports graphs in the list gr_list to the file fl in the sparse6 format.
Returns a list of graphs from the file fl in the sparse6 format.
Draws the graph using the draw package.
The algorithm used to position vertices is specified by the optional
argument program. The default value is
program=spring_embedding
. draw_graph can also use the
graphviz programs for positioning vertices, but graphviz must be
installed separately.
Example 1:
(%i1) load ("graphs")$ (%i2) g:grid_graph(10,10)$ (%i3) m:max_matching(g)$ (%i4) draw_graph(g, spring_embedding_depth=100, show_edges=m, edge_type=dots, vertex_size=0)$
Example 2:
(%i1) load ("graphs")$ (%i2) g:create_graph(16, [ [0,1],[1,3],[2,3],[0,2],[3,4],[2,4], [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11], [8,10],[11,10],[8,9],[11,12],[9,15],[12,13], [10,14],[15,14],[13,14] ])$ (%i3) t:minimum_spanning_tree(g)$ (%i4) draw_graph( g, show_edges=edges(t), show_edge_width=4, show_edge_color=green, vertex_type=filled_square, vertex_size=2 )$
Example 3:
(%i1) load ("graphs")$ (%i2) g:create_graph(16, [ [0,1],[1,3],[2,3],[0,2],[3,4],[2,4], [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11], [8,10],[11,10],[8,9],[11,12],[9,15],[12,13], [10,14],[15,14],[13,14] ])$ (%i3) mi : max_independent_set(g)$ (%i4) draw_graph( g, show_vertices=mi, show_vertex_type=filled_up_triangle, show_vertex_size=2, edge_color=cyan, edge_width=3, show_id=true, text_color=brown )$
Example 4:
(%i1) load ("graphs")$ (%i2) net : create_graph( [0,1,2,3,4,5], [ [[0,1], 3], [[0,2], 2], [[1,3], 1], [[1,4], 3], [[2,3], 2], [[2,4], 2], [[4,5], 2], [[3,5], 2] ], directed=true )$ (%i3) draw_graph( net, show_weight=true, vertex_size=0, show_vertices=[0,5], show_vertex_type=filled_square, head_length=0.2, head_angle=10, edge_color="dark-green", text_color=blue )$
Example 5:
(%i1) load("graphs")$ (%i2) g: petersen_graph(20, 2); (%o2) GRAPH (%i3) draw_graph(g, redraw=true, program=planar_embedding); (%o3) done
Example 6:
(%i1) load("graphs")$ (%i2) t: tutte_graph(); (%o2) GRAPH (%i3) draw_graph(t, redraw=true, fixed_vertices=[1,2,3,4,5,6,7,8,9]); (%o3) done
Default value: spring_embedding
The default value for the program used to position vertices in
draw_graph
program.
Default value: false
If true then ids of the vertices are displayed.
Default value: false
If true then labels of the vertices are displayed.
Default value: center
Determines how to align the labels/ids of the vertices. Can
be left
, center
or right
.
Default value: false
If true then weights of the edges are displayed.
Default value: circle
Defines how vertices are displayed. See the point_type option for
the draw
package for possible values.
The size of vertices.
The color used for displaying vertices.
Default value: []
Display selected vertices in the using a different color.
Defines how vertices specified in show_vertices are displayed.
See the point_type option for the draw
package for possible
values.
The size of vertices in show_vertices.
The color used for displaying vertices in the show_vertices list.
Default value: []
A partition [[v1,v2,...],...,[vk,...,vn]]
of the vertices of the
graph. The vertices of each list in the partition will be drawn in a
different color.
Specifies coloring of the vertices. The coloring col must be specified in the format as returned by vertex_coloring.
The color used for displaying edges.
The width of edges.
Defines how edges are displayed. See the line_type option for the
draw
package.
Display edges specified in the list e_list using a different color.
The color used for displaying edges in the show_edges list.
The width of edges in show_edges.
Defines how edges in show_edges are displayed. See the
line_type option for the draw
package.
A partition [[e1,e2,...],...,[ek,...,em]]
of edges of the
graph. The edges of each list in the partition will be drawn using a
different color.
The coloring of edges. The coloring must be specified in the format as returned by the function edge_coloring.
Default value: false
If true
, vertex positions are recomputed even if the positions
have been saved from a previous drawing of the graph.
Default value: 15
The angle for the arrows displayed on arcs (in directed graphs).
Default value: 0.1
The length for the arrows displayed on arcs (in directed graphs).
Default value: 50
The number of iterations in the spring embedding graph drawing algorithm.
The terminal used for drawing (see the terminal option in the
draw
package).
The filename of the drawing if terminal is not screen.
Defines the program used for positioning vertices of the graph. Can be
one of the graphviz programs (dot, neato, twopi, circ, fdp),
circular, spring_embedding or
planar_embedding. planar_embedding is only available for
2-connected planar graphs. When program=spring_embedding
, a set
of vertices with fixed position can be specified with the
fixed_vertices option.
Specifies a list of vertices which will have positions fixed along a regular polygon.
Can be used when program=spring_embedding
.
Converts a list v_list of vertices to a list of edges of the path defined by v_list.
Converts a list v_list of vertices to a list of edges of the cycle defined by v_list.
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