On freely generated semigraph $C^{\ast }$-algebras

For special universal $C^{\ast }$-algebras associated to $k$-semigraphs we present the universal representations of these algebras, prove a Cuntz–Krieger uniqueness theorem, and compute the $K$-theory. These $C^{\ast }$-algebras seem to be the most universal Cuntz–Krieger like algebras naturally associated to $k$-semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient of such an algebra.