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The bipartite graph property means that a walk starting in $V_0$ ends there precisely when its length is even. Thus the even-length walk generating function on a bipartite graph is $Z_G^0(x)={\sum }_{k\ge 0}{\sigma }_{2k}{x}^{2k}$. Deleting the last edge of an odd-length self-avoiding walk leaves an even-length one; each such prefix has at most $\Delta$ possible last-edge choices. Hence ${\sigma }_{2k+1}\le \Delta {\sigma }_{2k}$, and for $x\ge 0$,These inequalities also hold with infinite values. Their finite positive prefactor shows that the two power series have the same radius of convergence $1/\mu$; alternatively apply the root formula to the even subsequence.