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A263855 Number of connected graphs on n nodes up to isomorphism with a factor of (1+x) in their independence polynomial. 0
1, 6, 38, 277, 3056, 59768, 2376028, 195245762, 31700259751 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

LINKS

Table of n, a(n) for n=4..12.

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version b3ab217.

EXAMPLE

For n = 4, the a(4)=1 solution is the path of length 3.

PROG

(Sage)

from sage.graphs.independent_sets import IndependentSets

from math import factorial

from time import time

#Function to calculate a binomial coefficient (n choose r)

def choose(n, r):

    return factorial(n) / (factorial(r) * factorial(n - r))

#Function that checks if a polynomial has a certain root

def root_in_poly(poly, root):

    root_list = poly.roots()

    for tuple in root_list:

        for elt in tuple:

            if root == elt:

                return True

    return False

#Builds an independence polynomial for a graph

def build_ip(graph):

    number_of = [0] * graph.order()

    for set in IndependentSets(graph):

        number_of[len_set] += 1;

    poly = 0

    for index in range(0, len(number_of)):

        poly += (number_of[index]) * (x ** index)

    return poly

ip_list = []

R.<x> = QQ[]

root = -1

for v in range(4, 10):

    count = 0

    for g in graphs(v):

        if g.is_connected():

            ip = build_ip(g)

            if root_in_poly(ip, root):

                ip_list.append(ip)

                count += 1

    print(v, ": ", count)

CROSSREFS

Sequence in context: A026940 A082427 A192941 * A221283 A064309 A075197

Adjacent sequences:  A263852 A263853 A263854 * A263856 A263857 A263858

KEYWORD

nonn,more

AUTHOR

Ethan J. Brockmann, Nov 03 2015

EXTENSIONS

a(10)-a(12) added using tinygraph by Falk Hüffner, Jan 20 2016

STATUS

approved

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Last modified October 16 10:09 EDT 2018. Contains 316262 sequences. (Running on oeis4.)