  This paper concerns asymptotic stability, instability, and bifurcation of
constant steady state solutions of the parabolic-parabolic and
parabolic-elliptic chemotaxis models on metric graphs. We determine a threshold
value $\chi^*>0$ of the chemotaxis sensitivity parameter that separates the
regimes of local asymptotic stability and instability, and, in addition,
determine the parameter intervals that facilitate global asymptotic convergence
of solutions with positive initial data to constant steady states. Moreover, we
provide a sequence of bifurcation points for the chemotaxis sensitivity
parameter that yields non-constant steady state solutions. In particular, we
show that the first bifurcation point coincides with threshold value $\chi^*$
for a generic compact metric graph. Finally, we supply numerical computation of
bifurcation points for several graphs.
