Betti numbers of split graphs

A split graph is a graph where the vertices are a disjoint union of a complete part $C=\{x_i,\dots ,x_n\}$ and a stable part $S=\{y_1,\dots ,y_m\}$. We will determine the Betti numbers of the edge ring of all split graphs, in particular show that the only nonzero Betti numbers are ${\beta }_{0,0}$ and ${\beta }_{i,i+1}$, $i>0$. The Betti numbers only depend on the multiset of the number of neighbors in $S$ the $x_i$’s have. Singh and Verma have earlier determined the Betti numbers for complete split graphs (where all $y_i$ are neighbors to all $x_j$), and for “nearly complete” split graphs (where all $y_i$ are neighbors to all $x_j$, except that $y_i$ is not a neighbor to $x_i$ for $i=1,\dots ,\min \{m,n\}$). We also determine which split graphs that have Cohen-Macaulay edge ring.