  The sine-Gordon equation on a metric graph with a structure represented by a
$\mathcal{Y}$-junction, is considered. The model is endowed with boundary
conditions at the graph-vertex of $\delta'$-interaction type, expressing
continuity of the derivatives of the wave functions plus a Kirchhoff-type rule
for the self-induced magnetic flux. It is shown that particular stationary,
kink and kink/anti-kink soliton profile solutions to the model are linearly
(and nonlinearly) unstable. To that end, a recently developed linear
instability criterion for evolution models on metric graphs by Angulo and
Cavalcante (2020), which provides the sufficient conditions on the linearized
operator around the wave to have a pair of real positive/negative eigenvalues,
is applied. This leads to the spectral study to the linearize operator and of
its Morse index. The analysis is based on analytic perturbation theory,
Sturm-Liouville oscillation results and the extension theory of symmetric
operators. The methods presented in this manuscript have prospect for the study
of the dynamic of solutions for the sine-Gordon model on metric graphs with
finite bounds or on metric tree graphs and/or loop graphs.
