Alpha-matrix and graph-generated grammars

In this paper we consider the extension of graph-generated grammars based on their matrix representations. We study two classes of graph-generated grammars associated with the vertex and edge marking of graphs. We define alpha-matrices over a semiring of languages specified by finite alphabet $\mathcal{A}$ and then define the corresponding matrix algebras. These concepts are then used for constructive representation of graph-generated languages and research of their equivalence. We define a matrix-generated grammars as a natural superclass of graph-generated grammars. All the proofs are illustrated by examples.