8 - GRAPHS OF TYPE 1Z AND GIRTH 4 THAT ARE CAYLEY GRAPHS OF GROUPS Z(m,2,k)

Abstract

This chapter focuses on Cayley graphs of the groups Z(m,2,k) of order 4m, where n = 2. The defining relations in the case are R2 = S4 = (RS2)m = E, S–1 RS3 = (RS2)k, where k2 = -1(mod m). It follows from the equation that the inverse of k (mod m), k′ = m - k. Thus, two values of k whose sum equals m give rise to isomorphic Cayley graphs. It is easier to check (SR)4 = E in the Cayley diagram of the groups, which can be drawn as a tessellation. The chapter describes the Cayley graphs of the groups Z(m,2,k) by the LCF code and discusses the cases of odd and even values of m.