Let G be a finite group. The generating graph of subgroups 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 if and only if \(G=\langle H, K\rangle \) . In this paper, we obtain some sufficient and necessary conditions for \(\Gamma (G)\) being planar when it is connected, and characterize the structure of the finite group G where \(\Gamma (G)\) has a universal vertex. We define a new graph \(\Gamma ^{*}(G)\) , which is obtained by removing isolated vertices from \(\Gamma (G)\) . We also characterize finite groups G when \(\Gamma ^{*}(G)\) is a star graph, a complete graph, a complete k-partite graph, or has a universal vertex.
