On domination polynomials of some graphs

The article investigates the domination polynomial of generalized friendship graphs. The domination polynomial captures the number of dominating sets of each cardinality in a graph and is known to be NP-complete to compute for general graphs. We establish the log-concave and unimodal properties of these polynomials, and determine their peaks. Furthermore, we analyze the distribution of the zeros of aforesaid polynomial and identify their region in the complex plane. Several open problems are proposed for future exploration.