o ˷e/@s>dZddlZddlZgdZd ddZd ddZd dd ZdS) z5Provides explicit constructions of expander graphs. N)margulis_gabber_galil_graphchordal_cycle_graph paley_graphcCstjd|tjd}|s|sd}t|tjt|ddD]=\}}|d|||f|d|d||f||d||f||d|d|ffD]\}}| ||f||fqOq!d|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 T/var/www/ideatree/venv/lib/python3.10/site-packages/networkx/generators/expanders.pyr+s  rc Cstjd|tjd}|s|sd}t|t|D]*}|d|}|d|}|dkr6t||d|nd}|||fD]}|||q=qd|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) prrrrleftrightchordrr r r!rUs&     rcstjd|tjd}|rd}t|fddtdD}tD]}|D] }||||q(q$dd|jd <|S) asReturns the Paley (p-1)/2-regular graph on p nodes. The returned graph is a graph on Z/pZ with edges between x and y if and only if x-y is a nonzero square in Z/pZ. If p = 1 mod 4, -1 is a square in Z/pZ 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 = 3 mod 4, -1 is not a square in Z/pZ and therefore either x-y or y-x is a square in Z/pZ 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 Z/pZ. 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! s(zpaley_graph..r zpaley(r r )r rDiGraphrrrrr)r#rrr square_setrx2r r(r!rs)  r)N)__doc__rnetworkxr __all__rrrr r r r!s # *?