We study a shift invariant space on an undirected graphs $G$ having $N$ vertices. We obtain a characterization theorem for a system of generalized translates $\{T_{i}g : 1\leq i\leq N\}$, for $g\in C^N$, to form an orthonormal basis. Moreover, we find a necessary and sufficient condition for the system $\{T_{i}g : 1\leq i\leq m\}$, $m\leq N$, to form a linearly independent set and an orthonormal set. Further, we obtain a characterization result for a system of generalized translates which is generated by multiple generators $g_{1},...,g_{M}$ to form a frame for $C^N$. In particular, we deduce similar results for the system $\{T_{i}M_{s}g : 1\leq i,s\leq N\}$ with modulation $M_{s}$ and the spectral graph wavelet system. We also provide an illustration for the spectral graph wavelet system.