Multidegrees of binomial edge ideals

Let $G$ be a simple graph with binomial edge ideal $J_G$. We prove how to calculate the multidegree of $J_G$ based on combinatorial properties of $G$. In particular, we study the set $S_{\min }(G)$ defined as the collection of subsets of vertices whose prime ideals have minimum codimension. We provide results which assist in determining $S_{\min }(G)$, then calculate $S_{\min }(G)$ for star, horned complete, barbell, cycle, wheel, and friendship graphs, and use the main result of the paper to obtain the multidegrees of their binomial edge ideals.