# Ticket #8513: trac_8513_graph_theory_documentation-abm.patch

File trac_8513_graph_theory_documentation-abm.patch, 8.4 KB (added by abmasse, 11 years ago)

Adds digraph.py and generic_graph.py in the doctree

• ## doc/en/reference/graphs.rst

# HG changeset patch
# User Alexandre Blondin Masse < alexandre.blondin.masse at gmail.com>
# Date 1269287092 -3600
# Node ID fb26ae3b55b2a1491727483c8d1ed5ed845f760c
# Parent  2b6f82c8b0075babce925e6f0cc90a792c7b8515
#8513 Including the files digraph.py and generic_graph.py in the doctree.
Also removes all the warnings displayed when the documentation is generated by Sphinx.

diff --git a/doc/en/reference/graphs.rst b/doc/en/reference/graphs.rst
 a Graph Theory :maxdepth: 2 sage/graphs/cliquer sage/graphs/digraph sage/graphs/generic_graph sage/graphs/graph sage/graphs/graph_coloring sage/graphs/graph_generators
• ## sage/graphs/generic_graph.py

diff --git a/sage/graphs/generic_graph.py b/sage/graphs/generic_graph.py
 a class GenericGraph(GenericGraph_pyx): INPUT: - root_vertex -- integer (default: the first vertex) This is the vertex that will be used as root for all spanning out-trees if the graph is a directed graph. This argument is ignored if the graph is not a digraph. - root_vertex -- integer (default: the first vertex) This is the vertex that will be used as root for all spanning out-trees if the graph is a directed graph. This argument is ignored if the graph is not a digraph. REFERENCES: -  http://mathworld.wolfram.com/MatrixTreeTheorem.html -  Lih-Hsing Hsu, Cheng-Kuan Lin, "Graph Theory and Interconnection Networks"  http://mathworld.wolfram.com/MatrixTreeTheorem.html  Lih-Hsing Hsu, Cheng-Kuan Lin, "Graph Theory and Interconnection Networks" AUTHORS: - Anders Jonsson (2009-10-10) EXAMPLES:: sage: G = graphs.PetersenGraph() class GenericGraph(GenericGraph_pyx): possible maximum outdegree of the current graph. Given a Graph G, is is polynomial to compute an orientation D of the edges of G such that the maximum out-degree in D is minimized. This problem, though, is NP-complete in the D of the edges of G such that the maximum out-degree in D is minimized. This problem, though, is NP-complete in the weighted case [AMOZ06]_. INPUT: - use_edge_labels (boolean) - When set to True, uses edge labels as weights to compute the orientation and assumes a weight of 1 when there is no value available for a given edge. - When set to False (default), gives a weight of 1 to all the edges. - When set to True, uses edge labels as weights to compute the orientation and assumes a weight of 1 when there is no value available for a given edge. - When set to False (default), gives a weight of 1 to all the edges. EXAMPLE: Given a complete bipartite graph K_{n,m}, the maximum out-degree of an optimal orientation is \left\lceil \frac {nm} {n+m}\right\rceil:: of an optimal orientation is \left\lceil \frac {nm} {n+m}\right\rceil:: sage: g = graphs.CompleteBipartiteGraph(3,4) sage: o = g.minimum_outdegree_orientation() # optional - requires GLPK or CBC sage: max(o.out_degree()) == ceil((4*3)/(3+4)) # optional - requires GLPK or CBC True REFERENCES: .. [AMOZ06] Asahiro, Y. and Miyano, E. and Ono, H. and Zenmyo, K. class GenericGraph(GenericGraph_pyx): Proceedings of the 12th Computing: The Australasian Theroy Symposium Volume 51, page 20 Australian Computer Society, Inc. 2006 """ if self.is_directed(): class GenericGraph(GenericGraph_pyx): Returns the Wiener index of the graph. The Wiener index of a graph G can be defined in two equivalent ways [KRG96]_ : ways  : - W(G) = \frac 1 2 \sum_{u,v\in G} d(u,v) where d(u,v) denotes the distance between vertices u and v. class GenericGraph(GenericGraph_pyx): EXAMPLE: From [GYLL93]_, cited in [KRG96]_:: From , cited in :: sage: g=graphs.PathGraph(10) sage: w=lambda x: (x*(x*x -1)/6) class GenericGraph(GenericGraph_pyx): REFERENCE: .. [KRG96] Klavzar S., Rajapakse A., Gutman I. (1996). The Szeged and the Wiener index of graphs . Applied Mathematics Letters, 9 (5), pp. 45-49.  Klavzar S., Rajapakse A., Gutman I. (1996). The Szeged and the Wiener index of graphs. Applied Mathematics Letters, 9 (5), pp. 45-49.  I Gutman, YN Yeh, SL Lee, YL Luo (1993), Some recent results in the theory of the Wiener number. INDIAN JOURNAL OF CHEMISTRY SECTION A PUBLICATIONS & INFORMATION DIRECTORATE, CSIR """ return sum([sum(v.itervalues()) for v in self.distance_all_pairs().itervalues()])/2 def average_distance(self): r""" Returns the average distance between vertices of the graph. Formally, for a graph G this value is equal to \frac 1 {n(n-1)} \sum_{u,v\in G} d(u,v) where d(u,v) denotes the distance between vertices u and v and n is the number of vertices in G. EXAMPLE: From [GYLL93]_:: sage: g=graphs.PathGraph(10) sage: w=lambda x: (x*(x*x -1)/6)/(x*(x-1)/2) sage: g.average_distance()==w(10) True REFERENCE: .. [GYLL93] I Gutman, YN Yeh, SL Lee, YL Luo (1993), Some recent results in the theory of the Wiener number. INDIAN JOURNAL OF CHEMISTRY SECTION A PUBLICATIONS & INFORMATION DIRECTORATE, CSIR """ return sum([sum(v.itervalues()) for v in self.distance_all_pairs().itervalues()])/2 def average_distance(self): r""" Returns the average distance between vertices of the graph. Formally, for a graph G this value is equal to \frac 1 {n(n-1)} \sum_{u,v\in G} d(u,v) where d(u,v) denotes the distance between vertices u and v and n is the number of vertices in G. EXAMPLE: From [GYLL93]_:: sage: g=graphs.PathGraph(10) sage: w=lambda x: (x*(x*x -1)/6)/(x*(x-1)/2) sage: g.average_distance()==w(10) True REFERENCE: .. [GYLL93] I Gutman, YN Yeh, SL Lee, YL Luo (1993), Some recent results in the theory of the Wiener number. INDIAN JOURNAL OF CHEMISTRY SECTION A PUBLICATIONS & INFORMATION DIRECTORATE, CSIR """ class GenericGraph(GenericGraph_pyx): For any uv\in E(G), let `N_u(uv) = \{w\in G:d(u,w)