wgdZddlZddlmZddgZej ddd dZej ddd d Zy) aEGenerators for Harary graphs This module gives two generators for the Harary graph, which was introduced by the famous mathematician Frank Harary in his 1962 work [H]_. The first generator gives the Harary graph that maximizes the node connectivity with given number of nodes and given number of edges. The second generator gives the Harary graph that minimizes the number of edges in the graph with given node connectivity and number of nodes. References ---------- .. [H] Harary, F. "The Maximum Connectivity of a Graph." Proc. Nat. Acad. Sci. USA 48, 1142-1146, 1962. N) NetworkXErrorhnm_harary_graphhkn_harary_graphT)graphs returns_graphc||dkr td||dz kr td|||dz zdzkDr tdtj||}d|z|z}|dzdk(s|dzdk(r|dz}t|D]F}td|dzD]2}|j |||z |z|j |||z|z4H|dzr*|dz}t|D]}|j |||zd|z|z} | dkDr+t| dzD]}|j |||zdz|S|dz dz}t|D]F}td|dzD]2}|j |||z |z|j |||z|z4H|dz}t|||zz D]}|j |||z|z|S)aReturns the Harary graph with given numbers of nodes and edges. The Harary graph $H_{n,m}$ is the graph that maximizes node connectivity with $n$ nodes and $m$ edges. This maximum node connectivity is known to be floor($2m/n$). [1]_ Parameters ---------- n: integer The number of nodes the generated graph is to contain m: integer The number of edges the generated graph is to contain create_using : NetworkX graph constructor, optional Graph type to create (default=nx.Graph). If graph instance, then cleared before populated. Returns ------- NetworkX graph The Harary graph $H_{n,m}$. See Also -------- hkn_harary_graph Notes ----- This algorithm runs in $O(m)$ time. It is implemented by following the Reference [2]_. References ---------- .. [1] F. T. Boesch, A. Satyanarayana, and C. L. Suffel, "A Survey of Some Network Reliability Analysis and Synthesis Results," Networks, pp. 99-107, 2009. .. [2] Harary, F. "The Maximum Connectivity of a Graph." Proc. Nat. Acad. Sci. USA 48, 1142-1146, 1962. z!The number of nodes must be >= 1!z&The number of edges must be >= n - 1 !z'The number of edges must be <= n(n-1)/2r)rnx empty_graphrangeadd_edge) nm create_usingHdoffsetijhalfrs e/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/networkx/generators/harary_graph.pyrrsZ 1u?@@1q5yDEE1A;! EFF q,'A A A A A aq +A1fqj) + 1q1uk* 1q1uk* + + q56D4[ ( 1a$h' ( EAI q516] . 1a&j1n- . Ha%Aq +A1fqj) + 1q1uk* 1q1uk* + +Avq1v:~& *A JJq1t8q. ) * Hc|dkr td||dzkr td|dk(rtj||}|Stj||}|dzdk(s|dzdk(r|dz}t |D]F}t d|dzD]2}|j |||z |z|j |||z|z4H|dzr*|dz}t |D]}|j |||z|S|dz dz}t |D]F}t d|dzD]2}|j |||z |z|j |||z|z4H|dz}t |dzD]}|j |||z|z|S)aReturns the Harary graph with given node connectivity and node number. The Harary graph $H_{k,n}$ is the graph that minimizes the number of edges needed with given node connectivity $k$ and node number $n$. This smallest number of edges is known to be ceil($kn/2$) [1]_. Parameters ---------- k: integer The node connectivity of the generated graph n: integer The number of nodes the generated graph is to contain create_using : NetworkX graph constructor, optional Graph type to create (default=nx.Graph). If graph instance, then cleared before populated. Returns ------- NetworkX graph The Harary graph $H_{k,n}$. See Also -------- hnm_harary_graph Notes ----- This algorithm runs in $O(kn)$ time. It is implemented by following the Reference [2]_. References ---------- .. [1] Weisstein, Eric W. "Harary Graph." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HararyGraph.html. .. [2] Harary, F. "The Maximum Connectivity of a Graph." Proc. Nat. Acad. Sci. USA 48, 1142-1146, 1962. r z#The node connectivity must be >= 1!z$The number of nodes must be >= k+1 !r r)rr path_graphr r r)krrrrrrrs rrrtsX 1uABB1q5yBCC Av MM!\ * q,'A A A aq +A1fqj) + 1q1uk* 1q1uk* + + q56D4[ ( 1a$h' ( Ha%Aq +A1fqj) + 1q1uk* 1q1uk* + +Avtax *A JJq1t8q. ) * Hr)N) __doc__networkxr networkx.exceptionr__all__ _dispatchablerrrrr$sh", 1 2T2X 3X vT2R 3R r