Gibbs measure for the HC-Blume-Capel model in the case of a "wand" type graph on a Cayley tree

In this paper, we investigate translation-invariant splitting Gibbs measures (TISGMs) for the HC-Blume-Capel model on a “wand” graph embedded in the Cayley tree of arbitrary order $k\ge 2$. It is known that there is the exact critical value ${\theta }_{cr}$ such that for $\theta \ge {\theta }_{cr}$, there exists a unique TISGM, whereas for $\theta <{\theta }_{cr}$, precisely three TISGMs exist in the case of a “wand” graph for the given model. In the present paper, we completely solve the (non-)extremality problem for one of these measures for any order $k$ of Cayley tree