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

Indices of fixed point classes play a central role in Nielsen theory. Jiang-Wang-Zhang proved that for selfmaps graphs and surfaces, the index any class has an upper bound called its characteristic. In this paper, we study difference between characteristic graphs. First, free groups, extend notion attracting words at infinity automorphisms into injective endomorphisms. Then, by using relative train track technique, show mentioned above is quite likely to be number equivalence endomorphism induced on fundamental group. This gives new algebraic approach estimating indices graph selfmaps. As consequence, obtain endomorphisms generalizing one due Gaboriau-Jaeger-Levitt-Lustig. Furthermore, give simple roughly detecting whether exist or not.