The network of chemical reactions: graph grammars, curvature, order and categories

With access to more than 40 million chemical reactions from 1779 up to the present, which account for more than 20 million substances; historical and fundamental questions of chemistry can be addressed in a mathematical framework. The first question is on the nature of the network $N$ resulting from linking chemical reactions and on its structure and characterisation. Here some results on early explorations of parts of $N$ will be discussed, which address the issue of representing reactions in terms of graphs by highlighting the importance of hypergraphs and of directed bipartite graphs. A novel network curvature approach and its possible application and adaptation to analyse $N$ will be discussed. Results on how to mathematically approach the reduction of $N$ and its associated graph $G$ to a network with graph $G'$ relating classes of similar substances will be shown, where elements of category theory and partially ordered sets in the form of contexts of Formal Concept Analysis are highlighted.

Delving into the molecular description level for substances, a reaction is the transformation of educts into products, which is understood as the graph rewrite rule (graph grammar) of the graph of educts to the corresponding one of products. It will be shown how graph grammars work and it will be discussed an approach to look for new rules in $N$. The partially ordering of rules will be discussed as an approach to find central grammars for chemistry.

Finally, the historical aspect of $N$ and its temporal change will be stressed as a mathematical and computational approach to the history of chemical reactivity.