As usual, let \(C(X)\) denote the ring of all real-valued continuous functions on a Tychonoff space \(X\). By the zero-divisor graph \(\Gamma (C(X))\) of \(C(X)\) we mean the graph with vertices nonzero zero-divisors of \(C(X)\) such that there is an edge between vertices \(f\), \(g\) if and only if \(f\neq g\) and \(fg=0\). The goal of the authors is to analyze the relationship among ring properties of \(C(X)\), graph properties of \(\Gamma (C(X))\) and topological properties of the base space \(X\). The results presented in this article are in the flavor of the following: (1) \(C(X)\) is an almost regular ring if and only if for every pair of vertices \(g\) and \(h\) of \(\Gamma (C(X))\) and every nonunit \(f\in C(X)\) , \(c(fg,fh)\leq 4\), where \(c(f,g)\) stands for the length of the smallest cycle containing \(f\) and \(g\); \(c(f,g)=\infty\) if there no cycle containing \(f\) and \(g\); (2) \(C(X)\) is a regular ring if and only if \(C(X)\) is an almost regular ring and for every vertex \(f\) of \(\Gamma (C(X))\), there exists a vertex \(g\) of \(\Gamma (C(X))\) adjacent to \(f\) such that \(c(f,g)=4\); (3) the smallest cardinal number \(\alpha\) such that every complete subgraph of \(\Gamma (C(X))\) has cardinality \(\leq \alpha\) (which is denoted by \(\omega\Gamma (C(X))\), the cellularity of \(X\) and the Goldie dimension of \(C(X)\) coincide. In particular, \(\omega\Gamma (C(X))\) and the cardinality of \(X\) coincide for every discrete space \(X\); (4) \(d(X)\leq dt\, \Gamma (C(X))\leq \omega (X)\). In particular, whenever \(d(X)=\omega (X)\), then \(dt \,\Gamma (C(X))=\omega (X)\) (here \(d(X)\) stands for the density of \(X\), \(\omega (X)\) for the weight of \(X\) and \(dt(X)\) for the dominating number of \(\Gamma (C(X))\): the smallest cardinal number of the form \(|A|\) where \(A\) is a dominating set of vertices of \(\Gamma (C(X))\), i.e., a set \(A\) such that every vertex outside \(A\) is adjacent to at least one vertex in \(A\)); (5) if the character \(\chi (X)\) of \(X\) is less or equal to the density \(d(X)\) of \(X\), then \(dt \Gamma (C(X))=d(X)\), and (6) the following statements are equivalent: (i) \(\Gamma (C(X))\) is not triangulated and the set of centers of \(\Gamma (C(X))\) is a dominating set, (ii) the set of isolated points of \(X\) is dense in \(X\), and (iii) the socle of \(C(X)\) is an essential ideal.