Table 10.

Edge partition of $T_{\mathcal{C}}(\sigma ,\tau )$

$({\kappa }_{\Gamma }(x),{\kappa }_{\Gamma }(y)),xy\in E(T_{\mathcal{C}}(\sigma ,\tau ))$	Frequency
$(\sigma ,2\sigma -1)$	$\sigma {(\sigma -1)}^{\tau -1}$
$(2\sigma -1,{\sigma }^2)$	$\sigma {(\sigma -1)}^{\tau -2}$
$({\sigma }^2,{\sigma }^2)$	$\sigma [\sum _{i=1}^{\tau }{(\sigma -1)}^{i-1}-\sigma {(\sigma -1)}^{\tau -2}]$