Let $p$ and $q$ be distinct primes. The \textit{semiprime divisor function graph} denoted by $G_{D(pq)}$, is the graph with vertex set $V(G_{D(pq)})=\{1,p,q,pq\}$ and edge set $E(G_{D(pq)})=\{\{1,p\}, \{1,q\},\{1,pq\},\{p,pq\},\{q,pq\}\}$. The semiprime divisor function graph is a special type of divisor function graph $G_{D(n)}$ in which $n=pq$. Recently, the energy and some indices of semiprime divisor function graph have been determined. In this paper, we introduce a natural extension to the semiprime divisor function graph which we call the \textit{$k$-dprime divisor function graph}. Moreover, we present results on some distance-based and degree-based topological indices of $k$-dprime divisor function graph. We end the paper by giving some open problems.