Let
R be a commutative ring with non-zero identity. The cozero-divisor graph of
R, denoted by
, is a graph with vertices in
, which is the set of all non-zero and non-unit elements of
R, and two distinct vertices
a and
b in
are adjacent if and only if
and
. In this paper, we investigate some combinatorial properties of the cozero-divisor graphs
and
such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.