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Diffstat (limited to '.venv/lib/python3.12/site-packages/networkx/generators/harary_graph.py')
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diff --git a/.venv/lib/python3.12/site-packages/networkx/generators/harary_graph.py b/.venv/lib/python3.12/site-packages/networkx/generators/harary_graph.py new file mode 100644 index 00000000..591587d3 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/networkx/generators/harary_graph.py @@ -0,0 +1,199 @@ +"""Generators 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. + +""" + +import networkx as nx +from networkx.exception import NetworkXError + +__all__ = ["hnm_harary_graph", "hkn_harary_graph"] + + +@nx._dispatchable(graphs=None, returns_graph=True) +def hnm_harary_graph(n, m, create_using=None): + """Returns 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. + """ + + if n < 1: + raise NetworkXError("The number of nodes must be >= 1!") + if m < n - 1: + raise NetworkXError("The number of edges must be >= n - 1 !") + if m > n * (n - 1) // 2: + raise NetworkXError("The number of edges must be <= n(n-1)/2") + + # Construct an empty graph with n nodes first + H = nx.empty_graph(n, create_using) + # Get the floor of average node degree + d = 2 * m // n + + # Test the parity of n and d + if (n % 2 == 0) or (d % 2 == 0): + # Start with a regular graph of d degrees + offset = d // 2 + for i in range(n): + for j in range(1, offset + 1): + H.add_edge(i, (i - j) % n) + H.add_edge(i, (i + j) % n) + if d & 1: + # in case d is odd; n must be even in this case + half = n // 2 + for i in range(half): + # add edges diagonally + H.add_edge(i, i + half) + # Get the remainder of 2*m modulo n + r = 2 * m % n + if r > 0: + # add remaining edges at offset+1 + for i in range(r // 2): + H.add_edge(i, i + offset + 1) + else: + # Start with a regular graph of (d - 1) degrees + offset = (d - 1) // 2 + for i in range(n): + for j in range(1, offset + 1): + H.add_edge(i, (i - j) % n) + H.add_edge(i, (i + j) % n) + half = n // 2 + for i in range(m - n * offset): + # add the remaining m - n*offset edges between i and i+half + H.add_edge(i, (i + half) % n) + + return H + + +@nx._dispatchable(graphs=None, returns_graph=True) +def hkn_harary_graph(k, n, create_using=None): + """Returns 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. + """ + + if k < 1: + raise NetworkXError("The node connectivity must be >= 1!") + if n < k + 1: + raise NetworkXError("The number of nodes must be >= k+1 !") + + # in case of connectivity 1, simply return the path graph + if k == 1: + H = nx.path_graph(n, create_using) + return H + + # Construct an empty graph with n nodes first + H = nx.empty_graph(n, create_using) + + # Test the parity of k and n + if (k % 2 == 0) or (n % 2 == 0): + # Construct a regular graph with k degrees + offset = k // 2 + for i in range(n): + for j in range(1, offset + 1): + H.add_edge(i, (i - j) % n) + H.add_edge(i, (i + j) % n) + if k & 1: + # odd degree; n must be even in this case + half = n // 2 + for i in range(half): + # add edges diagonally + H.add_edge(i, i + half) + else: + # Construct a regular graph with (k - 1) degrees + offset = (k - 1) // 2 + for i in range(n): + for j in range(1, offset + 1): + H.add_edge(i, (i - j) % n) + H.add_edge(i, (i + j) % n) + half = n // 2 + for i in range(half + 1): + # add half+1 edges between i and i+half + H.add_edge(i, (i + half) % n) + + return H |