Elizabeth A. Pacheco | Searcharxiv

arXiv2026

The maximal dimensions of path and graph algebras

We consider the class of acyclic connected directed graphs with $N\geq 1$. In this paper we find the optimal upper bound for the number of paths amongst acyclic, connected graphs with $N$ edges. We prove that it is in fact optimal by finding an acyclic, connected graph with $N$ edges that realizes this bound. We then adapt these methods to find an optimal bound for Leavitt path algebras over a finite, acyclic, connected graph with $N$ edges.

math.CO↗

arXiv2026

The functoriality of moves on graphs and the extended covariant functoriality of graph algebras

Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this paper, we make these combinatorial methods functorial, and show that collapsing an out-split graph to the original graph and transforming a graph to a shifted graph can be implemented by admissible graph homomorphisms and admissible path homomorphisms, respectively. To include the inverses of such isomorphisms, we introduce a new category of graphs where morphisms are given as regular homomorphisms of graph inverse semigroups. This new category admits a covariant functor to the category of C*-algebras and $*$-homomorphisms which extends the known covariant functor from the category of graphs and admissible path homomorphisms.

math.OA↗