  We show how to calculate the operator algebra and the operator Lie algebra of
a stochastic labelled-graph grammar. More specifically, we carry out a generic
calculation of the product (and therefore the commutator) of time-evolution
operators for any two labelled-graph grammar rewrite rules, where the operator
corresponding to each rule is defined in terms of elementary two-state
creation/annihilation operators. The resulting graph grammar algebra has the
following properties: (1) The product and commutator of two such operators is a
sum of such operators with integer coefficients. Thus, the algebra and the Lie
algebra occurs entirely at the structural (or graph-combinatorial) level of
graph grammar rules, lifted from the level of elementary creation/annihilation
operators (an improvement over [1], Propositions 1 and 2). (2) The product of
the off-diagonal (state-changing) parts of two such graph rule operators is a
sum of off-diagonal graph rule operators with non-negative integer
coefficients. (3) These results apply whether the semantics of a graph grammar
rule leaves behind hanging edges (Theorem 1), or removes hanging edges (Theorem
2). (4) The algebra is constructive in terms of elementary two-state
creation/annihilation operators (Corollaries 3 and 8). These results are useful
because dynamical transformations of labelled graphs comprise a general
modeling framework, and algebraic commutators of time-evolution operators have
many analytic uses including designing simulation algorithms and estimating
their errors.
