Gd:dZddlZddlZgdZddZddZddZdS)z5Provides explicit constructions of expander graphs. N)margulis_gabber_galil_graphchordal_cycle_graph paley_graphctjd|tj}|s|sd}tj|t jt|dD]]\}}|d|zz|z|f|d|zdzz|z|f||d|zz|zf||d|zdzz|zffD]\}}| ||f||f ^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_usingGmsgxyuvs =lib/python3.11/site-packages/networkx/generators/expanders.pyrr+s.2 q, >>>A}}$aoo//$@s###!%((1555''1!a%i1_a 1q519o "A & QUa a!eaiA% &   ' 'DAq JJ1v1v & & & &  ':Q999AGFO Hctjd|tj}|s|sd}tj|t |D]L}|dz |z}|dz|z}|dkrt||dz |nd}|||fD]}|||Md|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. rrr r r zchordal_cycle_graph(r r) rrrrrrrpowrr) prrrrleftrightchordrs r"rrUsL q, >>>A}}$aoo//$@s### 1XXA{Q! %&EEAq1ua   qu%  A JJq!     1Q111AGFO Hr#cXtjd|tj}|rd}tj|fdt dD}t D]#}|D]}||||zz$dd|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.c8h|]}|dzzdk|dzzS)r r).0rr&s r" zpaley_graph..s.EEEadaZ1__1a41*___r#r zpaley(r r)rrDiGraphrrrrr)r&rrr square_setrx2s` r"rrsR q, ;;;A$6s### FEEEeAqkkEEEJ 1XX(( ( (B JJq1r6Q, ' ' ' ' (#qmmmAGFO Hr#)N)__doc__rnetworkxr__all__rrrr,r#r"r5s} O O OF' ' ' ' T< < < < ~7 7 7 7 7 7 r#