Betti numbers of split graphs

Abstract: A split graph is a graph where the vertices are a disjoint union of a complete part $C={x_i,\ldots,x_n}$ and a stable part $S={y_1,\ldots,y_m}$. We will determine the Betti numbers of the edge ring of all split graphs, in particular show that the only nonzero Betti numbers are $β{0,0}$ and $β{i,i+1}$, $i>0$. 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 $i=1,\ldots,\min{m,n}$). We also determine which split graphs that have Cohen-Macaulay edge ring.