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An element $a$ in a ring $R$ is called a strong
zero-divisor if, either $\langle a \rangle\langle b \rangle=0$ or $\langle b \rangle\langle a \rangle=0$,
for some $0\neq b\in R$ ($\langle x \rangle$ is the ideal generated by $x\in R$).
Let $S(R)$ denote the set of all strong zero-divisors of $R$. This notion of
strong zero-divisor has been extensively studied by these authors in [8]. In this paper, for any ring $R$, we
associate an undirected graph $\widetilde{\Gamma}(R)$ with vertices
$S(R)^*\hspace{-2mm}=\hspace{-1mm}S(R)\hspace{-1mm}\setminus\{0\}$,
where distinct vertices $a$ and $b$ are adjacent if and only if either
$\langle a \rangle\langle b \rangle\hspace{-1mm}=0$ or
$\langle b \rangle\langle a \rangle\hspace{-1mm}=0$. We investigate the interplay
between the ring-theoretic properties of $R$ and the graph-theoretic
properties of $\widetilde{\Gamma}(R)$. It is shown that for every ring
$R$, every two vertices in $\widetilde{\Gamma}(R)$ are connected
by a path of length at most 3, and if $\widetilde{\Gamma}(R)$
contains a cycle, then the length of the shortest cycle in
$\widetilde{\Gamma}(R)$, is at most 4. Also we characterize all
rings $R$ whose $\widetilde{\Gamma}(R)$ is a complete
graph or a star graph. Also, the interplay of between the ring-theoretic properties of a ring
$R$ and the graph-theoretic properties of $\widetilde{\Gamma}(M_n(R))$, are fully investigated.
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