Journal article
An Index for Graphs and Graph Groupoids
Axioms, Vol.11(2), p.47
02/01/2022
DOI: 10.3390/axioms11020047
Abstract
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 GG contains at least one loop-reduced finite path, then the order of G is infinity; hence, the canonical groupoid index G:K of the inclusion K & SUBE;G is either & INFIN; or 1 (under the definition and a natural axiomatization) for the graph groupoids K of all "parts " K of G. 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 G is guaranteed by the existence of loop-reduced finite paths in G. The non-semicircularity induced by graphs yields a new index-like notion called the graph-tree index & UGamma; of G. 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.
Details
- Title: Subtitle
- An Index for Graphs and Graph Groupoids
- Creators
- Ilwoo Cho - Saint Ambrose UniversityPalle Jorgensen - Univ Iowa, Dept Math, 14C McLean Hall, Iowa City, IA 52242 USA
- Resource Type
- Journal article
- Publication Details
- Axioms, Vol.11(2), p.47
- DOI
- 10.3390/axioms11020047
- eISSN
- 2075-1680
- Publisher
- MDPI
- Number of pages
- 55
- Language
- English
- Date published
- 02/01/2022
- Academic Unit
- Mathematics
- Record Identifier
- 9984241042002771
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