### Abstract

Our former study Tóbiás and Tasi (J Math Chem 54:85, 2016) is continued, where a simple algebraic solution was given to the kinetic problem of triangle, quadrangle and pentangle reactions. In the present work, after defining chemical reaction networks and their connectedness, first-order chemical reaction networks (FCRNs) are studied on the basis of the results achieved by Chellaboina et al. (Control Syst 29:60, 2009). First, it is proved that an FCRN is disconnected iff its coefficient matrix is block diagonalizable. Furthermore, mass incompatibility is used to interpret the reducibility of subconservative networks. For conservative FCRNs, the so-called marker network is introduced, which is linearly conjugate to the original one, to describe the zero eigenvalue associated to the coefficient matrix of an FCRN. Instead of using graph-theoretical concepts, simple algebraic tools are applied to present and solve these problems. As an illustration, an industrially important ten-component (formal) FCRN is presented which has algebraically exact solution.

Original language | English |
---|---|

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Journal of Mathematical Chemistry |

DOIs | |

Publication status | Accepted/In press - Jun 14 2016 |

### Fingerprint

### Keywords

- Algebraic model
- First-order reaction network
- Marker network
- Mass incompatibility
- Multiplicity of the zero eigenvalue
- Network decomposition

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*, 1-16. https://doi.org/10.1007/s10910-016-0655-2

**First-order chemical reaction networks I : theoretical considerations.** / Tóbiás, Roland; Stacho, László L.; Tasi, Gyula.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, pp. 1-16. https://doi.org/10.1007/s10910-016-0655-2

}

TY - JOUR

T1 - First-order chemical reaction networks I

T2 - theoretical considerations

AU - Tóbiás, Roland

AU - Stacho, László L.

AU - Tasi, Gyula

PY - 2016/6/14

Y1 - 2016/6/14

N2 - Our former study Tóbiás and Tasi (J Math Chem 54:85, 2016) is continued, where a simple algebraic solution was given to the kinetic problem of triangle, quadrangle and pentangle reactions. In the present work, after defining chemical reaction networks and their connectedness, first-order chemical reaction networks (FCRNs) are studied on the basis of the results achieved by Chellaboina et al. (Control Syst 29:60, 2009). First, it is proved that an FCRN is disconnected iff its coefficient matrix is block diagonalizable. Furthermore, mass incompatibility is used to interpret the reducibility of subconservative networks. For conservative FCRNs, the so-called marker network is introduced, which is linearly conjugate to the original one, to describe the zero eigenvalue associated to the coefficient matrix of an FCRN. Instead of using graph-theoretical concepts, simple algebraic tools are applied to present and solve these problems. As an illustration, an industrially important ten-component (formal) FCRN is presented which has algebraically exact solution.

AB - Our former study Tóbiás and Tasi (J Math Chem 54:85, 2016) is continued, where a simple algebraic solution was given to the kinetic problem of triangle, quadrangle and pentangle reactions. In the present work, after defining chemical reaction networks and their connectedness, first-order chemical reaction networks (FCRNs) are studied on the basis of the results achieved by Chellaboina et al. (Control Syst 29:60, 2009). First, it is proved that an FCRN is disconnected iff its coefficient matrix is block diagonalizable. Furthermore, mass incompatibility is used to interpret the reducibility of subconservative networks. For conservative FCRNs, the so-called marker network is introduced, which is linearly conjugate to the original one, to describe the zero eigenvalue associated to the coefficient matrix of an FCRN. Instead of using graph-theoretical concepts, simple algebraic tools are applied to present and solve these problems. As an illustration, an industrially important ten-component (formal) FCRN is presented which has algebraically exact solution.

KW - Algebraic model

KW - First-order reaction network

KW - Marker network

KW - Mass incompatibility

KW - Multiplicity of the zero eigenvalue

KW - Network decomposition

UR - http://www.scopus.com/inward/record.url?scp=84974806686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84974806686&partnerID=8YFLogxK

U2 - 10.1007/s10910-016-0655-2

DO - 10.1007/s10910-016-0655-2

M3 - Article

SP - 1

EP - 16

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

ER -