DEPARTAMENTO DE MATEMÁTICA – CFM / UFSC

Seminário Especial de Álgebra

RELATION MORPHISMS OF DIRECTED GRAPHS

Prof. Dr. Piotr Hajac (IMPAN-Varsaw)

Abstract: Associating graph algebras to directed graphs leads to both covariant and contravariant functors from suitable categories of graphs to the category k-Alg of algebras and algebra homomorphims. As both functors are often used at the same time, one needs a new category of graphs that allows a “common denominator” functor unifying the covariant and contravariant constructions. In this talk, I will show how to solve this problem by first introducing the relation category of graphs RG, and then determining the concept of admissible graph relations that yields a subcategory of RG admitting a contravariant functor to k-Alg simultaneously generalizing the aforementioned covariant and contravariant functors. I will illustrate relation morphisms of graphs by the naturally occurring example of including Cuntz algebras into matrices over Cuntz algebras. Based on joint work with Gilles G. de Castro and Francesco D’Andrea.

Data e Horário: Segunda feira, dia 18 de agosto de 2025, 16h

Local: Auditório Airton Silva, Departamento de Matemática – MTM /CFM

E. Krukoski

Colóquio de Matemática

Título: COUNTING PATHS IN DIRECTED GRAPHS

Palestrante: Prof. Dr. Piotr Hajac  (IMPAN, Warsaw, Poland)

Resumo: Graph theory is considered one of the oldest and most accessible branches of combinatorics and has numerous natural connections to other areas of mathematics. In particular, directed graphs, or quivers, are fundamental tools in representation theory as well as in noncommutative geometry and topology. In this talk, I will consider the class of directed graphs with N ≥ 1 edges and without loops shorter than k. Using the concept of a labelled graph, I will show how to determine graphs from this class that maximize the number of all paths of length k. To end with, I shall pose a related open problem concerning the maximal dimension of the path algebra of an acyclic graph with N ≥ 1 edges, and compute an upper bound for this dimension. Based on joint work with Oskar Stachowiak.

Data: Segunda-feira, 15 de agosto às 14h
Local: Auditório Airton Silva, Departamento de Matemática – MTM /CFM

E. Krukoski