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We study the ideal triangulation graph $T\left(S\right)$ of an oriented punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T\left(S\right)$ is an isomorphism. We also show that the graph $T\left(S\right)$ of such a surface $S$, equipped with its natural simplicial metric is not Gromov hyperbolic. We also show that if the triangulation graph of two oriented punctured...