Abstract: 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 talk, I will explain how to make combinatorial methods functorial, and show that collapsing an outsplitted 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, I will 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. I will demonstrate the new framework by applying it to various graphs that yield matrix algebras, Cuntz algebras, and matrices over Cuntz algebras. (Based on joint work with G. G. de Castro, M. Lowiel, E. Pacheco and M. Tobolski.) Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page