Let G be a finite group. The generating graph of G, denoted by \(\Gamma (G)\) , is a graph whose vertices are non-trivial proper subgroups of G and two distinct vertices H and K are adjacent in \(\Gamma (G)\) if and only if \(G=\langle H, K\rangle \) . In this paper, we study the connectivity, diameter, girth, clique number, and vertex chromatic number of \(\Gamma (G)\) . For instance, we show that if \(\Gamma (G)\) has no isolated vertex, then \(\Gamma (G)\) is connected with diameter at most 3. Moreover, we characterize the structure of the finite groups whose maximal subgroups have degree 1, 2 or 3, respectively, and prove that there is no finite group G such that the degrees of its maximal subgroups are equal to 4. Furthermore, we show that clique number, and chromatic number and number of maximal subgroups of graphs of finite groups are equal.
