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