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]


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