upper-bound-conjectures-1.sagews

load("conjecturing-theory.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)

objects = graph_objects

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/structure/sage_object.pyx", line 1001, in sage.structure.sage_object.load (/projects/sage/sage-7.5/src/build/cythonized/sage/structure/sage_object.c:12529) sage.repl.load.load(filename, globals()) File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/repl/load.py", line 227, in load code = compile(f.read(), fpath, 'exec') File "./conjecturing-theory.py", line 433 except: ^ SyntaxError: invalid syntax

#a simpler less comprehensive test
load("conjecturing-theory.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: lovasz_theta(g)

objects = graph_objects[:19]

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file different_degrees <function different_degrees at 0x7fb5fd02f938> diameter <unbound method Graph.diameter> card_KE_part <function card_KE_part at 0x7fb5fd132e60> largest_singular_value <function largest_singular_value at 0x7fb5fd02fe60> vertex_con <function vertex_con at 0x7fb5f59f1578> card_negative_eigenvalues <function card_negative_eigenvalues at 0x7fb5f59bd9b0> randic <function randic at 0x7fb5fd02f1b8> fiedler <function fiedler at 0x7fb5fd132488> independence_number <function independence_number at 0x7fb5fd02fde8> average_distance <function average_distance at 0x7fb5f59f1e60> graph_rank <function graph_rank at 0x7fb5fd1325f0> radius <unbound method Graph.radius> card_independence_irreducible_part <function card_independence_irreducible_part at 0x7fb5f59f1b90> average_vertex_temperature <function average_vertex_temperature at 0x7fb5fd02f668> critical_independence_number <function critical_independence_number at 0x7fb5f7cad050> fractional_alpha <function fractional_alpha at 0x7fb5f549f6e0> degree_sum <function degree_sum at 0x7fb5f59f1ed8> size <unbound method Graph.size> residue <function residue at 0x7fb5f549f7d0> mean_common_neighbors <function mean_common_neighbors at 0x7fb5f549f398> matching_number <function matching_number at 0x7fb5f549f410> card_bridges <function card_bridges at 0x7fb5f59f1668> density <unbound method Graph.density> card_pendants <function card_pendants at 0x7fb5f59f16e0> barrus_bound <function barrus_bound at 0x7fb5fd030320> min_common_neighbors <function min_common_neighbors at 0x7fb5fd030050> degree_variance <function degree_variance at 0x7fb5fd132050> max_degree <function max_degree at 0x7fb5f549ff50> barrus_q <function barrus_q at 0x7fb5f549f1b8> median_degree <function median_degree at 0x7fb5fd02f500> card_cut_vertices <function card_cut_vertices at 0x7fb5f59bd1b8> brinkmann_steffen <function brinkmann_steffen at 0x7fb5f59f1de8> average_degree <function average_degree at 0x7fb5fd132578> order_automorphism_group <function order_automorphism_group at 0x7fb5f59f1050> number_of_triangles <function number_of_triangles at 0x7fb5fd132758> fractional_covering <function fractional_covering at 0x7fb5f549fde8> szekeres_wilf <function szekeres_wilf at 0x7fb5fd02f140> card_periphery <function card_periphery at 0x7fb5f549f938> distinct_degrees <function distinct_degrees at 0x7fb5fd0301b8> maximum_average_degree <unbound method Graph.maximum_average_degree> max_eigenvalue <function max_eigenvalue at 0x7fb5fd02f8c0> sum_temperatures <function sum_temperatures at 0x7fb5fd02f050> annihilation_number <function annihilation_number at 0x7fb5f549f758> clustering_average <unbound method Graph.clustering_average> alon_spencer <function alon_spencer at 0x7fb5f59f1500> szeged_index <unbound method Graph.szeged_index> brooks <function brooks at 0x7fb5fd02f848> card_zero_eigenvalues <function card_zero_eigenvalues at 0x7fb5f59bd230> cvetkovic <function cvetkovic at 0x7fb5f549f500> caro_wei <function caro_wei at 0x7fb5f59f19b0> inverse_degree <function inverse_degree at 0x7fb5fd132500> eulerian_faces <function eulerian_faces at 0x7fb5f549f848> min_degree <function min_degree at 0x7fb5f549f8c0> connected_components_number <unbound method Graph.connected_components_number> cycle_space_dimension <function cycle_space_dimension at 0x7fb5fd4dd1b8> card_positive_eigenvalues <function card_positive_eigenvalues at 0x7fb5f7c84cf8> gutman_energy <function gutman_energy at 0x7fb5fd132848> card_center <function card_center at 0x7fb5f549f2a8> kirchhoff_index <function kirchhoff_index at 0x7fb5fd02fd70> max_even_minus_even_horizontal <function max_even_minus_even_horizontal at 0x7fb5f69c2a28> lovasz_theta <unbound method Graph.lovasz_theta> welsh_powell <function welsh_powell at 0x7fb5fd02f5f0> laplacian_energy <function laplacian_energy at 0x7fb5fd02fed8> spanning_trees_count <unbound method Graph.spanning_trees_count> max_common_neighbors <function max_common_neighbors at 0x7fb5fd030d70> sigma_2 <function sigma_2 at 0x7fb5f59f1758> wiener_index <unbound method Graph.wiener_index> wilf <function wilf at 0x7fb5fd02fcf8> order <unbound method Graph.order> edge_con <function edge_con at 0x7fb5f59f15f0> girth <unbound method Graph.girth> independence_number(x) <= annihilation_number(x) independence_number(x) <= lovasz_theta(x) independence_number(x) <= cvetkovic(x) independence_number(x) <= floor(lovasz_theta(x)) independence_number(x) <= max_even_minus_even_horizontal(x) + min_degree(x) independence_number(x) <= maximum(card_negative_eigenvalues(x), max_even_minus_even_horizontal(x) + 1)

#a simpler less comprehensive test
load("conjecturing-theory.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)

objects = graph_objects[:9]
invariants = efficiently_computable_invariants + [independence_number]
theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm]

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file

#a simpler less comprehensive test
load("conjecturing.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: lovasz_theta(g)

objects = graph_objects[:19]

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file independence_number(x) <= annihilation_number(x) independence_number(x) <= lovasz_theta(x) independence_number(x) <= cvetkovic(x) independence_number(x) <= floor(lovasz_theta(x)) independence_number(x) <= max_even_minus_even_horizontal(x) + min_degree(x) independence_number(x) <= maximum(card_negative_eigenvalues(x), max_even_minus_even_horizontal(x) + 1)

#a simpler less comprehensive test
load("conjecturing.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: lovasz_theta(g)

objects = graph_objects

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file independence_number(x) <= annihilation_number(x) independence_number(x) <= size(x) independence_number(x) <= fractional_alpha(x) independence_number(x) <= cvetkovic(x) independence_number(x) <= barrus_bound(x) independence_number(x) <= floor(fractional_alpha(x)) independence_number(x) <= ceil(lovasz_theta(x)) independence_number(x) <= 10^order_automorphism_group(x) independence_number(x) <= maximum(residue(x), floor(lovasz_theta(x))) independence_number(x) <= card_positive_eigenvalues(x) + max_even_minus_even_horizontal(x) independence_number(x) <= maximum(max_degree(x), floor(lovasz_theta(x))) independence_number(x) <= card_negative_eigenvalues(x) + max_even_minus_even_horizontal(x) independence_number(x) <= e^(cosh(max_degree(x) - 1)) independence_number(x) <= (2*diameter(x))^order_automorphism_group(x) independence_number(x) <= card_center(x) + max_even_minus_even_horizontal(x) + 1 independence_number(x) <= brinkmann_steffen(x) + max_even_minus_even_horizontal(x) + 1

load("conjecturing-theory.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: lovasz_theta(g)

objects = graph_objects

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file different_degrees <function different_degrees at 0x7fb5f59c36e0> diameter <unbound method Graph.diameter> card_KE_part <function card_KE_part at 0x7fb5f54a7938> largest_singular_value <function largest_singular_value at 0x7fb5f54a72a8> vertex_con <function vertex_con at 0x7fb5f59f17d0> card_negative_eigenvalues <function card_negative_eigenvalues at 0x7fb5fd044d70> randic <function randic at 0x7fb5f59c3848> fiedler <function fiedler at 0x7fb5f59c30c8> independence_number <function independence_number at 0x7fb5f54a7ed8> average_distance <function average_distance at 0x7fb5f59f1f50> graph_rank <function graph_rank at 0x7fb5fd044ed8> radius <unbound method Graph.radius> card_independence_irreducible_part <function card_independence_irreducible_part at 0x7fb5f59f1b18> average_vertex_temperature <function average_vertex_temperature at 0x7fb5f59c3c08> critical_independence_number <function critical_independence_number at 0x7fb5f7c84d70> fractional_alpha <function fractional_alpha at 0x7fb5f5614a28> degree_sum <function degree_sum at 0x7fb5f59f10c8> size <unbound method Graph.size> residue <function residue at 0x7fb5f5614aa0> mean_common_neighbors <function mean_common_neighbors at 0x7fb5fd1326e0> matching_number <function matching_number at 0x7fb5f5614668> card_bridges <function card_bridges at 0x7fb5f59f1410> density <unbound method Graph.density> card_pendants <function card_pendants at 0x7fb5f59f1848> barrus_bound <function barrus_bound at 0x7fb5fd132a28> min_common_neighbors <function min_common_neighbors at 0x7fb5fd132578> degree_variance <function degree_variance at 0x7fb5fd044938> max_degree <function max_degree at 0x7fb5f59b6a28> barrus_q <function barrus_q at 0x7fb5f5614320> median_degree <function median_degree at 0x7fb5f59bd1b8> card_cut_vertices <function card_cut_vertices at 0x7fb5fd044aa0> brinkmann_steffen <function brinkmann_steffen at 0x7fb5f59f1cf8> average_degree <function average_degree at 0x7fb5f59c3140> order_automorphism_group <function order_automorphism_group at 0x7fb5f59f1a28> number_of_triangles <function number_of_triangles at 0x7fb5fd044cf8> fractional_covering <function fractional_covering at 0x7fb5f5614578> szekeres_wilf <function szekeres_wilf at 0x7fb5f59c3320> card_periphery <function card_periphery at 0x7fb5f5614cf8> distinct_degrees <function distinct_degrees at 0x7fb5fd132c08> maximum_average_degree <unbound method Graph.maximum_average_degree> max_eigenvalue <function max_eigenvalue at 0x7fb5f54a7aa0> sum_temperatures <function sum_temperatures at 0x7fb5f59c3938> annihilation_number <function annihilation_number at 0x7fb5f5614758> clustering_average <unbound method Graph.clustering_average> alon_spencer <function alon_spencer at 0x7fb5f59f12a8> szeged_index <unbound method Graph.szeged_index> brooks <function brooks at 0x7fb5f54a7488> card_zero_eigenvalues <function card_zero_eigenvalues at 0x7fb5fd044f50> cvetkovic <function cvetkovic at 0x7fb5f5614d70> caro_wei <function caro_wei at 0x7fb5f59f1230> inverse_degree <function inverse_degree at 0x7fb5fd044de8> eulerian_faces <function eulerian_faces at 0x7fb5f56141b8> min_degree <function min_degree at 0x7fb5fd132cf8> connected_components_number <unbound method Graph.connected_components_number> cycle_space_dimension <function cycle_space_dimension at 0x7fb5f54a76e0> card_positive_eigenvalues <function card_positive_eigenvalues at 0x7fb5fd044e60> gutman_energy <function gutman_energy at 0x7fb5f59c3230> card_center <function card_center at 0x7fb5f54a78c0> kirchhoff_index <function kirchhoff_index at 0x7fb5f54a7500> max_even_minus_even_horizontal <function max_even_minus_even_horizontal at 0x7fb5f59bd488> lovasz_theta <unbound method Graph.lovasz_theta> welsh_powell <function welsh_powell at 0x7fb5f54a7320> laplacian_energy <function laplacian_energy at 0x7fb5f54a7c80> spanning_trees_count <unbound method Graph.spanning_trees_count> max_common_neighbors <function max_common_neighbors at 0x7fb5fd132410> sigma_2 <function sigma_2 at 0x7fb5f59f1398> wiener_index <unbound method Graph.wiener_index> wilf <function wilf at 0x7fb5f54a7cf8> order <unbound method Graph.order> edge_con <function edge_con at 0x7fb5f59f1320> girth <unbound method Graph.girth> independence_number(x) <= annihilation_number(x) independence_number(x) <= size(x) independence_number(x) <= fractional_alpha(x) independence_number(x) <= cvetkovic(x) independence_number(x) <= barrus_bound(x) independence_number(x) <= floor(fractional_alpha(x)) independence_number(x) <= ceil(lovasz_theta(x)) independence_number(x) <= 10^order_automorphism_group(x) independence_number(x) <= maximum(residue(x), floor(lovasz_theta(x))) independence_number(x) <= card_positive_eigenvalues(x) + max_even_minus_even_horizontal(x) independence_number(x) <= maximum(max_degree(x), floor(lovasz_theta(x))) independence_number(x) <= card_negative_eigenvalues(x) + max_even_minus_even_horizontal(x) independence_number(x) <= e^(cosh(max_degree(x) - 1)) independence_number(x) <= (2*diameter(x))^order_automorphism_group(x) independence_number(x) <= card_center(x) + max_even_minus_even_horizontal(x) + 1 independence_number(x) <= brinkmann_steffen(x) + max_even_minus_even_horizontal(x) + 1

for g in graph_objects:
    if independence_number(g) > floor(g.lovasz_theta()):
        print g.graph6_string()

N~~eeQoiCoM?Y?U?F??

g = Graph("N~~eeQoiCoM?Y?U?F??")

show(g)

independence_number(g)

10

g.lovasz_theta()

9.9999999

g._name_

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> AttributeError: 'Graph' object has no attribute '_name_'

for f in invariants:
    print f, f(Graph(1))

<function average_distance at 0x7fb5f59f1f50>

Error in lines 1-2 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 2, in <module> File "<string>", line 536, in average_distance File "sage/rings/integer.pyx", line 1858, in sage.rings.integer.Integer.__div__ (/projects/sage/sage-7.5/src/build/cythonized/sage/rings/integer.c:13046) return coercion_model.bin_op(left, right, operator.div) File "sage/structure/coerce.pyx", line 1055, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (/projects/sage/sage-7.5/src/build/cythonized/sage/structure/coerce.c:9336) return PyObject_CallObject(op, xy) File "sage/rings/integer.pyx", line 1843, in sage.rings.integer.Integer.__div__ (/projects/sage/sage-7.5/src/build/cythonized/sage/rings/integer.c:12880) raise ZeroDivisionError("rational division by zero") ZeroDivisionError: rational division by zero

load("conjecturing-theory.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: lovasz_theta(g)

objects = graph_objects

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, fractional_thm, kwok_thm, cvetkovic_thm, cvetkovic_thm, min_degree_thm, hansen_thm, borg_thm, matching_number_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file different_degrees <function different_degrees at 0x7fb5f54d3488> diameter <unbound method Graph.diameter> card_KE_part <function card_KE_part at 0x7fb5f5a27500> largest_singular_value <function largest_singular_value at 0x7fb5f540cf50> vertex_con <function vertex_con at 0x7fb5f54d3e60> card_negative_eigenvalues <function card_negative_eigenvalues at 0x7fb5f54d3c08> randic <function randic at 0x7fb5f54d3668> fiedler <function fiedler at 0x7fb5f54d3848> independence_number <function independence_number at 0x7fb5f54d3050> average_distance <function average_distance at 0x7fb5f54d3d70> graph_rank <function graph_rank at 0x7fb5f54d3a28> radius <unbound method Graph.radius> card_independence_irreducible_part <function card_independence_irreducible_part at 0x7fb5f5410398> average_vertex_temperature <function average_vertex_temperature at 0x7fb5f54d3578> critical_independence_number <function critical_independence_number at 0x7fb5f54a7230> fractional_alpha <function fractional_alpha at 0x7fb5f540cb90> degree_sum <function degree_sum at 0x7fb5f5410140> size <unbound method Graph.size> residue <function residue at 0x7fb5f540caa0> mean_common_neighbors <function mean_common_neighbors at 0x7fb5f540c758> matching_number <function matching_number at 0x7fb5f540ca28> card_bridges <function card_bridges at 0x7fb5f54d3f50> density <unbound method Graph.density> card_pendants <function card_pendants at 0x7fb5f54d3de8> barrus_bound <function barrus_bound at 0x7fb5f540c410> min_common_neighbors <function min_common_neighbors at 0x7fb5f540c6e0> degree_variance <function degree_variance at 0x7fb5f54d3938> max_degree <function max_degree at 0x7fb5f540c8c0> barrus_q <function barrus_q at 0x7fb5f540c9b0> median_degree <function median_degree at 0x7fb5f59c3b90> card_cut_vertices <function card_cut_vertices at 0x7fb5f54d3c80> brinkmann_steffen <function brinkmann_steffen at 0x7fb5f54102a8> average_degree <function average_degree at 0x7fb5f54d38c0> order_automorphism_group <function order_automorphism_group at 0x7fb5f5410320> number_of_triangles <function number_of_triangles at 0x7fb5f54d39b0> fractional_covering <function fractional_covering at 0x7fb5f540cc08> szekeres_wilf <function szekeres_wilf at 0x7fb5f54d3500> card_periphery <function card_periphery at 0x7fb5f540ccf8> distinct_degrees <function distinct_degrees at 0x7fb5f540c5f0> maximum_average_degree <unbound method Graph.maximum_average_degree> max_eigenvalue <function max_eigenvalue at 0x7fb5f540ce60> sum_temperatures <function sum_temperatures at 0x7fb5f54d35f0> annihilation_number <function annihilation_number at 0x7fb5f540cb18> clustering_average <unbound method Graph.clustering_average> alon_spencer <function alon_spencer at 0x7fb5f5410050> szeged_index <unbound method Graph.szeged_index> brooks <function brooks at 0x7fb5f54d32a8> card_zero_eigenvalues <function card_zero_eigenvalues at 0x7fb5f54d3b90> cvetkovic <function cvetkovic at 0x7fb5f540cc80> caro_wei <function caro_wei at 0x7fb5f54100c8> inverse_degree <function inverse_degree at 0x7fb5f54d3aa0> eulerian_faces <function eulerian_faces at 0x7fb5f540c938> min_degree <function min_degree at 0x7fb5f540c848> connected_components_number <unbound method Graph.connected_components_number> cycle_space_dimension <function cycle_space_dimension at 0x7fb5f540cde8> card_positive_eigenvalues <function card_positive_eigenvalues at 0x7fb5f54d3b18> gutman_energy <function gutman_energy at 0x7fb5f54d37d0> card_center <function card_center at 0x7fb5f540cd70> kirchhoff_index <function kirchhoff_index at 0x7fb5f54a79b0> max_even_minus_even_horizontal <function max_even_minus_even_horizontal at 0x7fb5f59c31b8> lovasz_theta <unbound method Graph.lovasz_theta> welsh_powell <function welsh_powell at 0x7fb5f54d3230> laplacian_energy <function laplacian_energy at 0x7fb5f54d3410> spanning_trees_count <unbound method Graph.spanning_trees_count> max_common_neighbors <function max_common_neighbors at 0x7fb5fd132d70> sigma_2 <function sigma_2 at 0x7fb5f54101b8> wiener_index <unbound method Graph.wiener_index> wilf <function wilf at 0x7fb5f54d3320> order <unbound method Graph.order> edge_con <function edge_con at 0x7fb5f54d3ed8> girth <unbound method Graph.girth> independence_number(x) <= annihilation_number(x) independence_number(x) <= size(x) independence_number(x) <= fractional_alpha(x) independence_number(x) <= cvetkovic(x) independence_number(x) <= barrus_bound(x) independence_number(x) <= floor(fractional_alpha(x)) independence_number(x) <= ceil(lovasz_theta(x)) independence_number(x) <= 10^order_automorphism_group(x) independence_number(x) <= maximum(residue(x), floor(lovasz_theta(x))) independence_number(x) <= card_positive_eigenvalues(x) + max_even_minus_even_horizontal(x) independence_number(x) <= maximum(max_degree(x), floor(lovasz_theta(x))) independence_number(x) <= card_negative_eigenvalues(x) + max_even_minus_even_horizontal(x) independence_number(x) <= e^(cosh(max_degree(x) - 1)) independence_number(x) <= (2*diameter(x))^order_automorphism_group(x) independence_number(x) <= card_center(x) + max_even_minus_even_horizontal(x) + 1 independence_number(x) <= brinkmann_steffen(x) + max_even_minus_even_horizontal(x) + 1

nb1 = Graph('HLOO@d@')

nb1.lovasz_theta()

5.9999999

independence_number(nb1)

6

nb2 = Graph('K@??BGk@h@oc')

nb2.lovasz_theta()

6.9999999

independence_number(nb2)

7

load("conjecturing.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: g.lovasz_theta()

objects = graph_objects[:10]

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c

loaded graph dc1024 loaded graph dc2048 loaded graph properties data file loaded graph invariants data file independence_number(x) <= floor(lovasz_theta(x))

for g in graph_objects:
    if independence_number(g) > min(t(g) for t in theorems):
        print g.name()

starfish

for g in objects:
    print g.name(), independence_number(g), annihilation_thm(g), lovasz_theta_thm(g)

paw 2 2 2.0

load("conjecturing.py")
load("gt.sage")

annihilation_thm = lambda g: annihilation_number(g)
fractional_thm = lambda g: fractional_alpha(g)
kwok_thm = lambda g: order(g) - (size(g)/max_degree(g))
cvetkovic_thm = lambda g: cvetkovic(g)
trivial_thm = lambda g: order(g)
min_degree_thm = lambda g: order(g) - min_degree(g)
hansen_thm = lambda g: floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
borg_thm = lambda g: order(g) - ceil((order(g) - 1) / max_degree(g))
matching_number_thm = lambda g: order(g) - matching_number(g)
lovasz_theta_thm = lambda g: g.lovasz_theta()

objects = graph_objects[:10]

invariants = efficiently_computable_invariants + [independence_number]

theorems = [annihilation_thm, lovasz_theta_thm]

conjs = conjecture(objects, invariants, invariants.index(independence_number), theory = theorems)
for c in conjs:
    print c