  The aim of this work is to establish a linear instability result of
stationary, kink and kink/anti-kink soliton profile solutions for the
sine-Gordon equation on a metric graph with a structure represented by a
$\mathcal Y$-junction. The model considers boundary conditions at the
graph-vertex of $\delta$-interaction type, or in other words, continuity of the
wave functions at the vertex plus a law of Kirchhoff-type for the flux. It is
shown that kink and kink/anti-kink soliton type stationary profiles are
linearly (and nonlinearly) unstable. For that purpose, a linear instability
criterion that provides the sufficient conditions on the linearized operator
around the wave to have a pair of real positive/negative eigenvalues, is
established. As a result, the linear stability analysis depends upon of the
spectral study of this linear operator and of its Morse index. The extension
theory of symmetric operators, Sturm-Liouville oscillation results and analytic
perturbation theory of operators are fundamental ingredients in the stability
analysis. A comprehensive study of the local well-posedness of the sine-Gordon
model in $\mathcal E(\mathcal Y) \times L^2(\mathcal{Y})$ where $\mathcal
E(\mathcal Y) \subset H^1(\mathcal{Y})$ is an appropriate energy space, is also
established. The theory developed in this investigation has prospects for the
study of the instability of stationary wave solutions of other nonlinear
evolution equations on metric graphs.
