Communications in Mathematical Sciences

Volume 7 (2009)

Number 4

Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements

Pages: 867 – 900



Murad Banaji

Gheorghe Craciun


We extend previous work on injectivity in chemical reaction networks to general interaction networks. Matrix- and graph-theoretic conditions for injectivity of these systems are presented. A particular signed, directed, labelled, bipartite multigraph, termed the "DSR graph", is shown to be a useful representation of an interaction network when discussing questions of injectivity. A graph-theoretic condition, developed previously in the context of chemical reaction networks, is shown to be sufficient to guarantee injectivity for a large class of systems. The graph-theoretic condition is simple to state and often easy to check. Examples are presented to illustrate the wide applicability of the theory developed.


Interaction networks, chemical reactions, injectivity, SR graph, network structure, multiple equilibria

2010 Mathematics Subject Classification

05C38, 05C50, 15A15, 34C99

Published 1 January 2009