## Publication: Grafos de divisores irreducibles ##### Authors
Ortiz-Albino, Reyes M.
##### College
College of Arts and Sciences - Art
##### Department
Department of Mathematics
M.S.
2020-04-21
##### Abstract
La teor\'ia de grafos de $\tn$-divisores $\tn$-irreducibles, conocidos tambi\'en como $\tn$-grafos, surge como una posible aplicaci\'on a la teor\'ia de $\tn$-factorizaciones. La teor\'ia de $\tn$-grafos se desarroll\'o conectando la teor\'ia de factorizaciones generalizadas y grafos de divisores irreducibles. En el 2013, Mooney public\'o algunos resultados que conectan las propiedades de los dominios con aquellas de los grafos, tal y como lo hizo Coykendall. Por ejemplo, un dominio con integridad $D$ es un $\tau$-UFD si y solo si el $\tau$-grafo simple de cualquier elemento distinto de cero y no unidad es un grafo completo. Nuestra definici\'on es m\'as estricta, y en algunos casos coincide con la definici\'on de Mooney. En nuestro caso, hay ocasiones en las que un entero puede tener un grafo completo de divisores irreducibles, pero su $\tn$-grafo no es un grafo completo. Presentamos algunas propiedades y caracter\'isticas generales de los $\tdos$-grafos. Exploramos propiedades de subgrafos e isomorfismos de grafos entre $\tdos$-grafos utilizando las propiedades de las factorizaciones estudiadas. En adici\'on, investigamos el problema del clique'' m\'aximo y el problema del isomorfismo de subgrafos entre los $\tdos$-grafos. Adem\'as, presentamos condiciones necesarias para que un grafo simple conexo sea un $\tdos$-grafo reducido y algunas implicaciones de este resultado. Finalmente, generalizamos los resultados obtenidos para $\tdos$-grafos para valores de $n \in \{3,4,6\}$ y valores primos para elementos en la clase de equivalencia $[n]_{\tn}$.

The theory of $\tn$-irreducible $\tn$-divisor graphs, also known as $\tn$-graphs, arises as a possible application to the theory of $\tn$-factorizations. The theory of $\tn$-graphs was developed by connecting the generalized factorizations theory and a concept of an irreducible divisor graph on domains. In 2013, Mooney published some results that connected the properties of domains with those of graphs, just as Coykendall did in the past. For example, an integral domain $D$ is an $\tau$-UFD if and only if the simple $\tau$-graph of any nonzero nonunit element is a complete graph. Our definition is more strict, and in some cases it coincides with Mooney's definition. In our case, there are instances in which an integer may have a complete irreducible divisor graph, but its $\tn$-graph need not be a complete graph. We presented some properties and general characteristics of the $\tdos$-graphs. We explored some properties of subgraphs and graph isomorphisms between $\tdos$-graphs using some properties of the factorizations we studied. In addition, we investigated the maximum clique problem and the subgraph isomorphism problem between $\tdos$-graphs. Also, we presented some necessary conditions for a simple graph to be a reduced $\tdos$-graph and some implications of these results. Finally, we generalized the results obtained for $\tdos$-graphs for values of $n \in \{3,4,6\}$ and prime values for integers in the equivalence class $[n]_{\tn}$.
grafos,
divisores,
graphs,
divisors,
irreducibles
##### Usage Rights
Except where otherwise noted, this item’s license is described as CC0 1.0 Universal