Group signature formulas constructed from graphs

Given a finite undirected graph $\Gamma$ without loops, we define a sentence $\Phi(\Gamma)$ of group theory. A sequence of graphs $\Gamma_i$ is used to obtain a sequence of sentences $\Phi(\Gamma_i)$. These are employed to determine the $\Gamma$-dimension of a group and to study properties of the dimension. Under certain restrictions on a group, the known centralizer dimension is the $\Gamma$-dimension for some sequence of graphs. We mostly focus on dimensions defined by using linear graphs and cycles. Dimensions for a number of partially commutative metabelian groups are computed.