+ jB< Rt^RIt^RIt.ROt]P !RRR7RRl4t]P !RRR7RRl4t]P !RRR7RRl4t]PPPR4]P !RRR7RRR ^dRR/R l44t RRR ^dRR/R lt ]PPR 4]PPR 4]P !RR^//R7R^/Rl444t]PPPR4]P !RRR7R^RRR ^dRR/Rl44tR#)z3Provides explicit constructions of expander graphs.NT)graphs returns_graphcl\P!^V\PR7pVP4'gVP 4'gRp\P !V4h\ P!\V4^R7FwrEV^V,,V,V3V^V,^,,V,V3WE^V,,V,3WE^V,^,,V,33FwrgVPWE3Wg34K K RV R2VPR&V#)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. default0`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&& ^/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/generators/expanders.pymargulis_gabber_galil_graphr2s4 q, >A}}aoo//@s##!!%(15!a%i1_a 1q519o "A & QUa a!eaiA% &  DA JJvv &  65QCq9AGGFO Hc\P!^V\PR7pVP4'gVP 4'gRp\P !V4h\ V4F[pV^, V,pV^,V,pV^8d\W@^, V4M^pWVV3FpVPWH4K K] RV R2VPR&V#)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]_ when viewed as a simple graph. The default return type is a MultiGraph, which has 6-regular nodes due to the symmetric nature of the edge additions. Use ``nx.Graph(nx.chordal_cycle_graph(p))`` to obtain the 3-regular simple graph. `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. rrzchordal_cycle_graph(r r ) r r r rrrrpowrr) prrrrleftrightchordrs && rchordal_cycle_graphr']sT q, >A}}aoo//@s## 1XA{Q! %&EA1ua qu%A JJq & -QCq1AGGFO Hr c\P!^V\PR7pVP4'dRp\P!V4h\ ^V4Uu0uF)qD^,V,^8wgKV^,V,kK+ pp\ V4F+pVF"pVP WDV,V,4K$ K- RV R2VPR&V#uupi)aReturns 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). rz&`create_using` cannot be a multigraph.zpaley(r r )r r DiGraphrrrrr)r#rrrr square_setx2s&& r paley_graphr,sX q, ;A6s## ',AqkEkdaZ1_*1a41**kJE 1XB JJqr6Q, 'qcmAGGFO H Fs $C$?C$seedr max_triesc^RIpV^8d\P!R4hV^8g\P!R4hV^,^8Xg\P!R4hV^, V8g%\P!RV^, RV R24h\P!W4pV^8dV#.p\ 4p\ V^,4EFp Tp \ V4V ^,V,8wgK&V ^,p VPV^, 4P4p V PV^, 4\PPV RR 7U U u0uFwrW3V9gKW3V9gKW3kK pp p \ V4V8Xd#VPV 4VPV4V ^8XgKR p\P!V4h VPV4V#uup p i) aUtility for creating a random regular expander. Returns a random $d$-regular graph on $n$ nodes which is an expander graph with very good probability. Parameters ---------- n : int The number of nodes. d : int The degree of each node. create_using : Graph Instance or Constructor Indicator of type of graph to return. If a Graph-type instance, then clear and use it. If a constructor, call it to create an empty graph. Use the Graph constructor by default. max_tries : int. (default: 100) The number of allowed loops when generating each independent cycle seed : (default: None) Seed used to set random number generation state. See :ref`Randomness`. Notes ----- The nodes are numbered from $0$ to $n - 1$. The graph is generated by taking $d / 2$ random independent cycles. Joel Friedman proved that in this model the resulting graph is an expander with probability $1 - O(n^{-\tau})$ where $\tau = \lceil (\sqrt{d - 1}) / 2 \rceil - 1$. [1]_ Examples -------- >>> G = nx.maybe_regular_expander_graph(n=200, d=6, seed=8020) Returns ------- G : graph The constructed undirected graph. Raises ------ NetworkXError If $d % 2 != 0$ as the degree must be even. If $n - 1$ is less than $ 2d $ as the graph is complete at most. If max_tries is reached See Also -------- is_regular_expander random_regular_expander_graph References ---------- .. [1] Joel Friedman, A Proof of Alon's Second Eigenvalue Conjecture and Related Problems, 2004 https://arxiv.org/abs/cs/0405020 Nzn must be a positive integerz$d must be greater than or equal to 2zd must be evenzNeed n-1>= d to have room for z independent cycles with z nodesT)cyclicz3Too many iterations in maybe_regular_expander_graph)numpyr rr setrlen permutationtolistappendutilspairwiseupdateadd_edges_from)rdrr.r-nprcyclesedgesi iterationscyclerr new_edgesrs&&$$$ rmaybe_regular_expander_graphrCs~1u=>> FEFF EQJ/00 EQJ,Q!VH4MaSPV W   q'A1u F EE16] %jQUaK' !OJ$$QU+224E LLQ HH--eD-AADA6&,-6+>A 9~" e$ Y'QK&&s++/2U H#s# G04G0>G0cV^RIpVPR\^R7\WW#VR7#)z .. deprecated:: 3.6 `maybe_regular_expander` is a deprecated alias for `maybe_regular_expander_graph`. Use `maybe_regular_expander_graph` instead. NzQmaybe_regular_expander is deprecated, use `maybe_regular_expander_graph` instead.)category stacklevelrr.r-)warningswarnDeprecationWarningrC)rr;rr.r-rHs&&$$$ rmaybe_regular_expanderrKSs9 MM 6#  ( <4 r directed multigraphrweight)preserve_edge_attrsepsilonc^RIp^RIpV^8d\P!R4h\P!V4'gR#\P P VP4wrE\P!V\R7pVPPPVR^RR7p\V4p\\V4^VP!V^, 4,V,84#)aDetermines whether the graph G is a regular expander. [1]_ An expander graph is a sparse graph with strong connectivity properties. More precisely, this helper checks whether the graph is a regular $(n, d, \lambda)$-expander with $\lambda$ close to the Alon-Boppana bound and given by $\lambda = 2 \sqrt{d - 1} + \epsilon$. [2]_ In the case where $\epsilon = 0$ then if the graph successfully passes the test it is a Ramanujan graph. [3]_ A Ramanujan graph has spectral gap almost as large as possible, which makes them excellent expanders. Parameters ---------- G : NetworkX graph epsilon : int, float, default=0 Returns ------- bool Whether the given graph is a regular $(n, d, \lambda)$-expander where $\lambda = 2 \sqrt{d - 1} + \epsilon$. Examples -------- >>> G = nx.random_regular_expander_graph(20, 4) >>> nx.is_regular_expander(G) True See Also -------- maybe_regular_expander_graph random_regular_expander_graph References ---------- .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph Nzepsilon must be non negativeF)dtypeLM)whichkreturn_eigenvectors)r1scipyr r is_regularr7arbitrary_elementdegreeadjacency_matrixfloatsparselinalgeigshminboolabssqrt) rrPr<sp_r;Alamslambda2s &$ ris_regular_expanderrigsb{=>> ==   88 % %ahh /DA AU+A 99   ! !!41% ! PD$iG G q2771q5>1G;; <`. Raises ------ NetworkXError If max_tries is reached Examples -------- >>> G = nx.random_regular_expander_graph(20, 4) >>> nx.is_regular_expander(G) True Notes ----- This loops over `maybe_regular_expander_graph` and can be slow when $n$ is too big or $\epsilon$ too small. See Also -------- maybe_regular_expander_graph is_regular_expander References ---------- .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph rG)rP)rr;rr.r-z4Too many iterations in random_regular_expander_graph)rCrir r)rr;rPrr.r-rr@s&&$$$$ rrandom_regular_expander_graphrkshr % <4 AJ!!55a (<4  ?""F  Hr )rr'r,rKrCrirk)N)__doc__rnetworkxr __all__ _dispatchablerr'r,r7 decoratorsnp_random_staterCrKnot_implemented_forrirkr rrts9 VT2' 3' TT2? 3? DT29 39 x$$V,T2q tq sq QUq 3-q h4(j)l+sXqM&:;@=a@=<,*@=F$$V,T2G G %)G 58G ?CG 3-G r