Fractals on Graphs

Recently graphs have been studied and applied in various math and science fileds. In this monograph we consider graphs with fractal property. Starting with graphs (combinatorial objects) we construct the corresponding groupoids (algebraic objects). The fractal property of graphs and groupoids is detected by the automata labelings (automata-theoretic objects). The groupoids with fractal property will be called graph fractaloids. By defining suitable representations of groupoids we establish von Neumann algebras (operator-algebraic objects). As elements of the von Neumann algebras we define the labeling operators (operator-theoretic objects) of graph fractaloids. In Part 1 by computing the free moments (free-probabilistic data) of the operators we verify how the graph fractaloids act in the von Neumann algebras. Also based on such computations we can classify the graph fractaloids in Part 2. Our classification shows the richness of graph fractaloids which are not fractal groups in general. In Part 3 we show that for any finite graph there always exists a finite fractal graph containing it as its part.