The $\mathcal{F}$-co-index (forgetten topological co-index) for a simple connected graph ${\mathcal{\mathcal{G}}}$ is defined as the sum of the terms $\zeta_{\mathcal{\mathcal{\mathcal{G}}}}^2(y)+\zeta_{\mathcal{\mathcal{\mathcal{G}}}}^2(x)$ over all non-adjacent vertex pairs $(x,y)$ of $\mathcal{\mathcal{\mathcal{G}}}$, where $\zeta_{\mathcal{\mathcal{\mathcal{G}}}}(y)$ and $\zeta_{\mathcal{\mathcal{\mathcal{G}}}}(x)$ are the degrees of the vertices $y$ and $x$ in ${\mathcal{\mathcal{G}}}$, respectively. The $\mathcal{F}$-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total $\pi$-electron energy. Therefore, considering the importance of the $\mathcal{F}$-index and $\mathcal{F}$-co-index, in this paper we study this indices and we present new bounds for the $\mathcal{F}$-index and $\mathcal{F}$-co-index.
