  A graph is an instrument which is extensively utilized to model various
problems in different fields. Up to date, many graphs have been developed to
represent algebraic structures, particularly rings in order to study their
properties. In this article, by focusing on commutative ring $ R $, we
introduce a new notion of unity product graph associated with $ R $ and its
complement. In addition, we prove that if the number of vertices of the unity
product graph is at least 2, then the graph is disconnected, while its
complement graph is connected. Furthermore, it is shown that there are some
commutative rings with such as Boolean ring and the Cartesian product of
Boolean rings in which their associated unity product graphs are trivial.
Consequently, some results are established to determine the number of isolated
vertices in unity product graph. We also characterize commutative rings with
unity in which their associated unity product and complement unity product
graphs are empty graph and complete graph, respectively. Finally, we prove some
results on the properties of the unity product graph and its complement in
terms of girth, diameter, radius, dominating number, chromatic number and
clique number as well as planarity and Hamiltonian.
