Communications in Mathematical Sciences

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

Murad Banaji and Gheorghe Craciun

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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.

Article information

Source
Commun. Math. Sci. Volume 7, Number 4 (2009), 867-900.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
http://projecteuclid.org/euclid.cms/1264434136

Mathematical Reviews number (MathSciNet)
MR2604624

Zentralblatt MATH identifier
1195.05038

Subjects
Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05C38: Paths and cycles [See also 90B10] 34C99: None of the above, but in this section 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]

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

Citation

Banaji, Murad; Craciun, Gheorghe. Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements. Commun. Math. Sci. 7 (2009), no. 4, 867--900. http://projecteuclid.org/euclid.cms/1264434136.


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