In this work we obtain new lower and upper optimal bounds for general (exponential) indices of a graph. In the same direction, we show new inequalities involving some well-known topological indices like the generalized atom-bound connectivity index $ ABC_\alpha $ and the generalized second Zagreb index $ M_2^\alpha $. Moreover, we solve some extremal problems for their corresponding exponential indices ($ e^{ABC_\alpha} $ and $ e^{M_2^{\alpha}} $).
Keywords: exponential indices; general sum connectivity index; generalized second Zagreb index; topological indices.