Dimer coverings and dimer-monomer model on the Tower of Hanoi graph

Abstract

We present the number of dimer coverings N_d(n) and the number of dimer-monomers M_d(n) on the Tower of Hanoi graph TH_d(n) at stage n with dimension d equal to two three four and ve for N_d(n) and d equal to three and four for M_d(n) When the number of vertices denoted as v(n) of the Tower of Hanoi graph is an even number Nd(n) is the number of close-packed dimers When the number of vertices is an odd number no close-packed con gurations are possible and we allow one of the outmost vertices uncovered The entropy of both S_TH_d and z_TH_d are respectively de ned as STHd = lim lnN_d(n)/v(n) and zTHd = lim lnM_d(n)/v We get the upper bounds and the lower bounds for S_TH_d and z_TH_d respectively As the di erence between these bounds converges to zero as the calculated stage increases with d = 3; 5 for dimer coverings and with d = 3; 4 for dimer-monomers the numerical value of both S_TH_d and z_TH_d can be evaluated with more than a hundred signi ficant fi gures accurate But the dimer covering with d = 4 is merely evaluated with more than six signifi cant fi gures accurate

Date of Award	2019
Original language	English
Supervisor	Shu-Chiuan Chang (Supervisor)