Fixed point indices and fixed words at infinity of selfmaps of graphs II

The index $\mathrm{ind}(\mathbf{F})$ of a fixed point class $\mathbf{F}$ is a classical invariant in the Nielsen fixed point theory. In the recent paper \cite{ZZ}, the authors introduced a new invariant $\mathrm{ichr}(\mathbf{F})$ called the improved characteristic, and proved that $\mathrm{ind}(\mathbf{F})\leq \mathrm{ichr}(\mathbf{F})$ for all fixed point classes of $\pi_1$-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig.