  Let R be a commutative ring with unity (not necessarily finite). The ring R
consider as a simple graph whose vertices are the elements of R with two
distinct vertices x and y are adjacent if xy=0 in R, where 0 is the zero
element of R. In 1988 [8], Beck raised the conjecture that the chromatic number
and clique number are same in a graph associated to any commutative ring with
unity. In 1993 [2], Anderson and Naseer disproved the conjecture by giving a
counter example of the conjecture (Note that, till date this is a only one
counter example). In this paper, we find the clique number and bounds for
chromatic number of a graph associated to any finite product of commutative
rings with unity in terms of its factors (In particular, if R is finite which
is not a local ring, we obtain the clique number and bounds for chromatic
number of a graph associated to R in terms of its local rings). As a
consequence of our results, we construct infinitely many counter examples of
the above conjecture. Also, some of the main results, proved by Beck in 1988
[8] and Anderson and Naseer in 1993 [2] are consequences of our results.
