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 $R$. 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 $\langlea\rangle\langle b \rangle\hspace{-1mm}=0$ 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 $\widilde{\Gamma}(R)$ 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.