Let $G = (V, E)$ be a graph. The first Zagreb index and the forgotten topological index of a graph $G$ are defined respectively as $\sum_{u \in V} d^2(u)$ and $\sum_{u \in V} d^3(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. If the minimum degree of $G$ is at least one, the inverse degree of $G$ is defined as $\sum_{u \in V} \frac{1}{d(u)}$. In this paper, we, for a graph with minimum degree at least one, present an upper bound for the first Zagreb index of the graph and lower bounds for the forgotten topological index and the inverse degree of the graph. We also present sufficient conditions involving the first Zagreb index, the forgotten topological index, or the inverse degree for some Hamiltonian properties of a graph.