Table 9.

Vertex partition of $T_{\mathcal{C}}(\sigma ,\tau )$ based on neighborhood degree sum

${\kappa }_{\Gamma }(x),x\in V(T_{\mathcal{C}}(\sigma ,\tau ))$	Frequency
$\sigma$	$\sigma {(\sigma -1)}^{\tau -1}$
${\sigma }^2$	$\sigma \sum _{i=1}^{\tau -2}{(\sigma -1)}^{i-1}+1$
$2\sigma -1$	$\sigma {(\sigma -1)}^{t-2}$