Cycle Indices of Cyclic and Dihedral Groups Acting on Diagonals of a Regular N-Gon and R-Element Subsets of the Set X = * +.

Abstract

Polya (1937) used the cycle index in counting configurations. Several authors have computed cycle index formula of several group actions which have been used globally in chemistry and mathematics, in counting of chemical compounds and graphs. However dihedral group Dn acting on the set of diagonals of a regular n - gon, 4-elements and r-elements subsets of the set X = ( 1,2,…n) has not been done. The project shall come up with the formulae for Z. / and Z. /= [Σ ( ) ]. The project seek to extend Muthoka’s findings on the formula for Z( , -) and Z( , -) to derive the formulae for Z( , -) and Z( , -) = [Σ ( ) ( ) ( ) . ( ) ( )/]. This shall be done by re-visiting some of the results by Harary and Palmer (1973) and also some of the contributions by Kamuti and Njuguna (2004) in the study shall be used.