U 9v f@s@dZddlZddlZdddgZd ddZd ddZd ddZdS) z5Provides explicit constructions of expander graphs. Nmargulis_gabber_galil_graphchordal_cycle_graph paley_graphcCstjd|tjd}|s"|s0d}t|tjt|ddD]z\}}|d|||f|d|d||f||d||f||d|d|ffD]\}}| ||f||fqqBd|d|j d <|S) aReturns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes. The undirected MultiGraph is regular with degree `8`. Nodes are integer pairs. The second-largest eigenvalue of the adjacency matrix of the graph is at most `5 \sqrt{2}`, regardless of `n`. Parameters ---------- n : int Determines the number of nodes in the graph: `n^2`. create_using : NetworkX graph constructor, optional (default MultiGraph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If the graph is directed or not a multigraph. rdefault0`create_using` must be an undirected multigraph.)repeatzmargulis_gabber_galil_graph()name) nx empty_graph MultiGraph is_directed is_multigraph NetworkXError itertoolsproductrangeadd_edgegraph)n create_usingGmsgxyuvr A/tmp/pip-unpacked-wheel-_lngutwb/networkx/generators/expanders.pyr+s  c Cstjd|tjd}|s"|s0d}t|t|D]T}|d|}|d|}|dkrlt||d|nd}|||fD]}|||qzq8d|d|j d<|S) uReturns the chordal cycle graph on `p` nodes. The returned graph is a cycle graph on `p` nodes with chords joining each vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit) 3-regular expander [1]_. `p` *must* be a prime number. Parameters ---------- p : a prime number The number of vertices in the graph. This also indicates where the chordal edges in the cycle will be created. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If `create_using` indicates directed or not a multigraph. References ---------- .. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and invariant measures", volume 125 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1994. rrrr rzchordal_cycle_graph(r r ) r rrrrrrpowrr) prrrrleftrightZchordrr r r!rUs&    cstjd|tjd}|r(d}t|fddtdD}tD]"}|D]}||||qPqHdd|jd <|S) a%Returns the Paley $\frac{(p-1)}{2}$ -regular graph on $p$ nodes. The returned graph is a graph on $\mathbb{Z}/p\mathbb{Z}$ with edges between $x$ and $y$ if and only if $x-y$ is a nonzero square in $\mathbb{Z}/p\mathbb{Z}$. If $p \equiv 1 \pmod 4$, $-1$ is a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore $x-y$ is a square if and only if $y-x$ is also a square, i.e the edges in the Paley graph are symmetric. If $p \equiv 3 \pmod 4$, $-1$ is not a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore either $x-y$ or $y-x$ is a square in $\mathbb{Z}/p\mathbb{Z}$ but not both. Note that a more general definition of Paley graphs extends this construction to graphs over $q=p^n$ vertices, by using the finite field $F_q$ instead of $\mathbb{Z}/p\mathbb{Z}$. This construction requires to compute squares in general finite fields and is not what is implemented here (i.e `paley_graph(25)` does not return the true Paley graph associated with $5^2$). Parameters ---------- p : int, an odd prime number. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed directed graph. Raises ------ NetworkXError If the graph is a multigraph. References ---------- Chapter 13 in B. Bollobas, Random Graphs. Second edition. Cambridge Studies in Advanced Mathematics, 73. Cambridge University Press, Cambridge (2001). rrz&`create_using` cannot be a multigraph.cs(h|] }|ddkr|dqS)rrr ).0rr#r r! szpaley_graph..r zpaley(r r )r rZDiGraphrrrrr)r#rrrZ square_setrZx2r r'r!rs)  )N)N)N)__doc__rZnetworkxr __all__rrrr r r r!s  # * ?