In this paper, we determine precise formulas for the diameters, the number of perfect matchings, and the Tutte polynomials for an infinite family of finite graphs, namely the Schreier graphs of tree automaton groups, also called tree graph automata. This enables us to easily find the number of spanning trees, spanning forests, and an explicit form for the chromatic polynomials. In the second part of the paper, we provide the precise values for the Wiener and Szeged index of any tree graph automaton.