  The aim of this work is to establish a linear instability criterium of
stationary solutions for the Korteweg-de Vries model on a star graph with a
structure represented by a finite collections of semi-infinite edges. By
considering a boundary condition of $\delta$-type interaction at the
graph-vertex, we show that the continuous tail and bump profiles are linearly
unstable in a balanced star graph. The use of the analytic perturbation theory
of operators and the extension theory of symmetric operators is a piece
fundamental in our stability analysis.
  The arguments presented in this investigation has prospects for the study of
the instability of stationary waves solutions of other nonlinear evolution
equations on star graphs.
