TY - JOUR

T1 - Asymptotic enumeration of independent sets on the Sierpinski gasket

AU - Chang, Shu Chiuan

AU - Chen, Lung Chi

AU - Yan, Weigen

PY - 2013

Y1 - 2013

N2 - The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets md,b(n) on the generalized Sierpinski gasket SGd,b(n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three for d = 2. Upper and lower bounds for the asymptotic growth constant, defined as zSGd,b = limv→∞ lnmd,b(n)/v where v is the number of vertices, on these Sierpinski gaskets are derived in terms of the numbers at a certain stage. The numerical values of these zSGd,b are evaluated with more than a hundred significant figures accurate. We also conjecture upper and lower bounds for the asymptotic growth constant zSGd,2 with general d, and an approximation of zSGd,2 when d is large.

AB - The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets md,b(n) on the generalized Sierpinski gasket SGd,b(n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three for d = 2. Upper and lower bounds for the asymptotic growth constant, defined as zSGd,b = limv→∞ lnmd,b(n)/v where v is the number of vertices, on these Sierpinski gaskets are derived in terms of the numbers at a certain stage. The numerical values of these zSGd,b are evaluated with more than a hundred significant figures accurate. We also conjecture upper and lower bounds for the asymptotic growth constant zSGd,2 with general d, and an approximation of zSGd,2 when d is large.

UR - http://www.scopus.com/inward/record.url?scp=84880142600&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880142600&partnerID=8YFLogxK

U2 - 10.2298/FIL1301023C

DO - 10.2298/FIL1301023C

M3 - Article

AN - SCOPUS:84880142600

SN - 0354-5180

VL - 27

SP - 23

EP - 40

JO - Filomat

JF - Filomat

IS - 1

ER -