  The aim of this work is to establish a instability study for stationary kink
and antikink/kink profiles solutions for the sine-Gordon equation on a metric
graph with a structure represented by a Y-junction so-called a Josephson
tricrystal junction. By considering boundary conditions at the graph-vertex of
$\delta'$-interaction type, it is shown that these kink-soliton type stationary
profiles are linearly (and nonlinearly) unstable. The extension theory of
symmetric operators, Sturm-Liouville oscillation results and analytic
perturbation theory of operators are fundamental ingredients in the stability
analysis. The local well-posedness of the sine-Gordon model in $H^1(Y)\times
L^2(Y)$ is also established. The theory developed in this investigation has
prospects for the study of the (in)-stability of stationary wave solutions of
other configurations for kink-solitons profiles.
