What is intuitive perception of $T_{\alpha_1} \circ T_{\alpha_2} \circ … \circ T_{\alpha_M}$ in graph domain?

if $$G(V,E,W)$$ be a weighted graph and $$\vert V \vert =N$$. for any vertex $$i \in \lbrace 1,2,...,,N \rbrace$$ define a generalized translation operator $$T_i:\mathbb{R}^N \to \mathbb{R}^N$$via generalized convolution with a delta centered at vertex i:

$$(T_{i}f)(n) := \sqrt{N}(f*\delta_{i})(n) = \sqrt{N}\sum_{\ell=0}^{N-1}\hat{f}(\lambda_{\ell}) \chi^{*}_{\ell}(i)\chi_{\ell}(n) .$$ [link]

the set of translation operators $${T_i} \in \lbrace 1,2,...,N\rbrace$$ do not form a mathematical group; i.e., $$T_iT_j \neq T_{i+j}.$$

unlike the classical translation operator, since The translation defined above is a kernelized operator. I don't have the intuitive perception of $$T_iT_j$$ in graph domain. And also In generally: $$T_{\alpha_1} \circ T_{\alpha_2} \circ ... \circ T_{\alpha_M}$$ [link]