Algebraic shifting of finite graphs
  In the present paper, exterior algebraic shifting and symmetric algebraic shifting of bipartite graphs and chordal graphs are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that, for $a>3$ and $b>3$, the exterior algebraic shifted graph of the complete bipartite graph $K_{a,b}$ of size $a,b$ is different from the symmetric algebraic shifted graph of $K_{a,b}$. Second, we will show that the exterior algebraic shifted graph of any chordal graph $G$ is coincident with the symmetric algebraic shifted graph of $G$. 
Satoshi Murai