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"""Matching polynomials of connected graphs -- numberdb.org/T239
For every connected simple graph G on at most seven vertices, named by its
canonical graph6 string, this stores two forms:
mu(G, x) = sum_k (-1)^k m_k x^(n - 2k),
M(G, x) = sum_k m_k x^k,
where m_k is the number of k-edge matchings of G.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
The generator counts matchings directly from edge subsets. This keeps the
arithmetic in ZZ and gives an independent path to compare with Sage's
matching_polynomial() for the signed form.
"""
import os
import sys
from functools import lru_cache
from itertools import combinations
import numberdb.sage as numberdb
from sage.graphs.graph import Graph
from sage.graphs.graph_generators import graphs
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
T239 = "T239"
MAX_VERTICES = 7
FORMS = ("signed", "generating")
_R = PolynomialRing(ZZ, "x")
_x = _R.gen()
NAMES = {
"@": "complete graph $K_1$",
"A_": "complete graph $K_2$",
"BW": "path $P_3$",
"Bw": "complete graph $K_3$",
"CF": "star $K_{1,3}$",
"CL": "path $P_4$",
"C]": "cycle $C_4$",
"C^": "diamond graph",
"C~": "complete graph $K_4$",
"D?{": "star $K_{1,4}$",
"DBg": "path $P_5$",
"DBk": "bull graph",
"DB{": "dart graph",
"DFw": "complete bipartite graph $K_{2,3}$",
"DK{": "butterfly graph",
"DLo": "cycle $C_5$",
"DN{": "house X graph",
"D]{": "wheel $W_5$",
"Dbk": "house graph",
"D~{": "complete graph $K_5$",
"E?Bw": "star $K_{1,5}$",
"E?~o": "complete bipartite graph $K_{2,4}$",
"E@YO": "path $P_6$",
"EFz_": "complete bipartite graph $K_{3,3}$",
"EIe_": "cycle $C_6$",
"ELrw": "wheel $W_6$",
"E~~w": "complete graph $K_6$",
"F??Fw": "star $K_{1,6}$",
"F?B~o": "complete bipartite graph $K_{2,5}$",
"F?~v_": "complete bipartite graph $K_{3,4}$",
"F@HSO": "path $P_7$",
"FHQSO": "cycle $C_7$",
"FIefw": "wheel $W_7$",
"FjaHw": "Moser spindle",
"F~~~w": "complete graph $K_7$",
}
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
@lru_cache(maxsize=None)
def connected_graphs(max_vertices=MAX_VERTICES):
out = []
for n in range(1, max_vertices + 1):
for graph in graphs.nauty_geng("%d -c" % n):
canonical = graph.canonical_label(algorithm="sage")
out.append((canonical.graph6_string(), canonical))
return tuple(out)
@lru_cache(maxsize=None)
def graph_by_key(max_vertices=MAX_VERTICES):
return {key: graph for key, graph in connected_graphs(max_vertices)}
def graph_from_key(key):
return Graph(str(key))
def _edge_subsets(edges, size):
return combinations(edges, size)
@lru_cache(maxsize=None)
def matching_counts(key):
graph = graph_by_key().get(key)
if graph is None:
graph = graph_from_key(key)
edges = list(graph.edges(labels=False))
limit = graph.num_verts() // 2
counts = [ZZ(0)] * (limit + 1)
counts[0] = ZZ(1)
for size in range(1, limit + 1):
total = ZZ(0)
for chosen in _edge_subsets(edges, size):
used = []
for edge in chosen:
used.extend(edge)
if len(set(used)) == 2 * size:
total += 1
counts[size] = total
return tuple(counts)
@lru_cache(maxsize=None)
def matching_polynomial(key, form):
counts = matching_counts(key)
graph = graph_by_key().get(key)
if graph is None:
graph = graph_from_key(key)
n = graph.num_verts()
if form == "signed":
return sum(
_R((-1) ** k) * _R(count) * _x ** (n - 2 * k)
for k, count in enumerate(counts)
)
if form == "generating":
return sum(_R(count) * _x ** k for k, count in enumerate(counts))
raise ValueError("unknown form %r" % (form,))
def entry_comment(key, form):
del form
if key in NAMES:
return "This is the %s." % NAMES[key]
return ""
class MatchingPolynomialsOfConnectedGraphs(numberdb.Generator):
table = os.environ.get("NUMBERDB_TABLE", T239)
parameters = ("g", "form")
type = "Z[]"
rigour = "exact"
def enumerate(self, max_vertices=MAX_VERTICES):
for key, _graph in connected_graphs(max_vertices):
for form in FORMS:
yield {"g": key, "form": form}
def value(self, params, digits):
key = params["g"]
form = params["form"]
entry = {"number": matching_polynomial(key, form)}
comment = entry_comment(key, form)
if comment:
entry["comment"] = comment
return entry
def fill_draft_once(generator, message):
"""Fill a fresh draft without the empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator),
upsert=False,
run=run,
rigour=generator.rigour,
)
files = _source_files(generator)
stored = []
for name, body in sorted(files.items()):
attach(table, name, body, run=run, message=message,
rigour=generator.rigour)
stored.append(name)
return {
"tid": answer.get("tid", table),
"revision": answer.get("revision"),
"entries": len(entries),
"files": stored,
}
if __name__ == "__main__":
_key_from_stdin()
generator = MatchingPolynomialsOfConnectedGraphs()
if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv:
print(fill_draft_once(
generator,
message="matching polynomials of connected graphs"))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)