Reaction networks (RNs) comprise a set $X$ of species and a set $\RR$ of reactions $Y\to Y'$, each converting a multiset of educts $Y\subseteq X$ into a multiset $Y'\subseteq X$ of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix S have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for $\SM$ that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively.

Chemically plausible RNs allow neither a perpetuum mobile, i.e., a ``futile cycle'' of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix $\SM$. The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and -- in a stricter sense -- futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of ``atoms''. In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless $\SM$ implies that they are ``obligatory isomers''. In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN.

``Chemical'' RNs are directed hypergraphs with a stoichiometric matrix $\SM$ whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain $\SM$ by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes. Chemical reactions can be understood as transformations of molecular

multi-graphs that preserve vertex labels (atoms) and degrees (sum of bonding and non-bonding electrons), thereby implying the atom-atom map of a reaction. The corresponding reaction mechanism is often described by an electron pushing diagram that explains the transformation by consecutive local relocations of invidudal edges (electron pairs). Here we show that every degree-preserving map between multigraphs, and thus every atom-atom map, can be generated by cyclic electron pushing. Moreover, it is always possible to decompose such an explanation into electron pushing diagrams involving only four electron pairs. This is turn implies that every reaction can be decomposed into a sequence of elementary reactions that involve at most two educt molecules and two product molecules. The requirement of a mechanistic explantion in terms of electron pushing and small imaginary transition states therefore does not impose a combinatorial constraint on the feasibility of hypothetical chemical reactions.