Even-length walk generating function on a bipartite graph
= Even-length walk generating function on a bipartite graph
{title2=$Z_G^0(x)=\sum_{k\geq0}\sigma_{2k}x^{2k}$}
In a <bipartite graph>, a <self-avoiding walk> ends in its starting class exactly when it has even length. If degrees are bounded by $\Delta$, deleting the last <edge> gives $\sigma_{2k+1}\leq\Delta\sigma_{2k}$ and hence $Z_G^0(x)\leq Z_G(x)\leq(1+\Delta x)Z_G^0(x)$ for $x\geq0$. Thus these <generating functions> have the same <radius of convergence>.