Shellability of Polyhedral Joins of Simplicial Complexes and its Application to Graph Theory

We investigate the shellability of the polyhedral join Z(M)* (K, L) of simplicial complexes K, M and a subcomplex L subset of K. We give sufficient conditions and necessary conditions on (K, L) for Z(M)* (K, L) being shellable. In particular, we show that for some pairs (K, L), Z(M)* (K, L) becomes shellable regardless of whether M is shellable or not. Polyhedral joins can be applied to graph theory as the independence complex of a certain generalized version of lexicographic products of graphs which we define in this paper. The graph obtained from two graphs G, H by attaching one copy of H to each vertex of G is a special case of this generalized lexicographic product and we give a result on the shellability of the independence complex of this graph by applying the above results.