An extended duality theorem and multiplication of several matrices

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.