| Summary: | In this paper, we consider certain quantities that arise in the images of the so-called graph-tree indexes of graph groupoids. In text, the graph groupoids are induced by connected finite-directed graphs with more than one vertex. If a graph groupoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula><i>G</i> contains at least one loop-reduced finite path, then the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula> is infinity; hence, the canonical groupoid index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi mathvariant="double-struck">G</mi><mo>:</mo><mi mathvariant="double-struck">K</mi></mfenced></semantics></math></inline-formula> of the inclusion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">K</mi><mo>⊆</mo><mi mathvariant="double-struck">G</mi></mrow></semantics></math></inline-formula> is either <i>∞</i> or 1 (under the definition and a natural axiomatization) for the graph groupoids <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">K</mi></semantics></math></inline-formula> of all “parts” <i>K</i> of <i>G</i>. A loop-reduced finite path generates a semicircular element in graph groupoid algebra. Thus, the existence of semicircular systems acting on the free-probabilistic structure of a given graph <i>G</i> is guaranteed by the existence of loop-reduced finite paths in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula>. The non-semicircularity induced by graphs yields a new index-like notion called the graph-tree index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula>. We study the connections between our graph-tree index and non-semicircular cases. Hence, non-semicircularity also yields the classification of our graphs in terms of a certain type of trees. As an application, we construct towers of graph-groupoid-inclusions which preserve the graph-tree index. We further show that such classification applies to monoidal operads.
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