a module theoretic approach to‎ ‎zero-divisor graph with respect to (first) dual

let $m$ be an $r$-module and $0 neq fin m^*={rm hom}(m,r)$. we associate an undirected graph $gf$ to $m$ in which non-zero elements $x$ and $y$ of $m$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. weobserve that over a commutative ring $r$, $gf$ is connected anddiam$(gf)leq 3$. moreover, if $gamma (m)$ contains a cycle,then $mbox{gr}(gf)leq 4$. furthermore if $|gf|geq 1$, then$gf$ is finite if and only if $m$ is finite. also if $gf=emptyset$, then $f$ ismonomorphism (the converse is true if $r$ is a domain). if $m$ iseither a free module with ${rm rank}(m)geq 2$ or anon-finitely generated projective module there exists $fin m^*$with ${rm rad}(gf)=1$ and ${rm diam}(gf)leq 2$. we prove thatfor a domain $r$ the chromatic number and the clique number of $gf$ are equal.