Free multiderivations of connected subgraph arrangements

Cuntz and K\"uhne introduced the class of connected subgraph arrangements $A_G$, depending on a graph $G$, and classified all graphs $G$ such that the corresponding arrangement $A_G$ is free. We extend their result to the multiarrangement case and classify all graphs $G$ for which the corresponding arrangement $A_G$ supports some multiplicity $\mu$ such that the multiarrangement $(A_G,\mu)$ is free.