Contents Online

# 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

DOI: http://dx.doi.org/10.4310/CMS.2009.v7.n4.a4

#### Authors

#### Abstract

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.

#### Keywords

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

#### 2010 Mathematics Subject Classification

05C38, 05C50, 15A15, 34C99