Let G be a (molecular) graph with n vertices, and $d_i$ be the degree of its i-th vertex. Then, the inverse sum indeg matrix of G is the $n\times n$ matrix $\mathcal{C}\left(G\right)$ with entries $c_{ij}=\frac{d_i{d}_j}{d_i+d_j}$, if the i-th and the j-th vertices are adjacent and 0 otherwise. Let ${\mu }_1\ge {\mu }_2\ge \dots \ge {\mu }_n$ be the eigenvalues of $\mathcal{C}$ arranged in order. The inverse sum indeg energy of G, ${\epsilon }_{isi}\left(G\right)$ can be represented as ${\sum }_{j=1}^n\left|{\mu }_i\right|$. In this paper, we establish several novel upper and lower sharp bounds on ${\mu }_1$ and ${\epsilon }_{isi}\left(G\right)$ via some other graph parameters, and describe the structures of the extremal graphs.