The semicircular law of free probability as noncommutative multivariable operator theory

In this paper, we study semicircular elements induced by connected finite directed graphs. It is shown that if the graph groupoid G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}$$\end{document} of a given graph G contains at least one loop finite path, then it induces a semicircular element under suitable representations of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}$$\end{document}. As application, if a graph G is fractal (or, satisfies the fractal property) in a certain sense, then it automatically generates infinitely many semicircular elements.