The neighbourhood polynomial of some families of dendrimers

The neighbourhood polynomial N(G, x) is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph and it is defined as (G, x) = Sigma(U is an element of N(G)) x(vertical bar U vertical bar), where N(G) is neighbourhood complex of a graph, whose vertices of the graph and faces are subsets of vertices that have a common neighbour. A dendrimers is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, we compute this polynomial for some families of dendrimer.