### Abstract

The minimum semidefinite rank (Formula presented.) of a graph is defined to be the minimum rank among all Hermitian positive semidefinite matrices associated to the graph. A problem of interest is to find upper and lower bounds for (Formula presented.) of a graph using known graph parameters such as the independence number and the minimum degree of the graph. We provide a sufficient condition for (Formula presented.) of a bipartite graph to equal its independence number. The delta conjecture gives an upper bound for (Formula presented.) of a graph in terms of its minimum degree. We present classes of graphs for which the delta conjecture holds.

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

Pages (from-to) | 774-787 |

Number of pages | 14 |

Journal | Linear and Multilinear Algebra |

Volume | 63 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 3 2015 |

### Fingerprint

### Keywords

- connectivity of a graph
- delta conjecture
- independence number
- matrix of a graph
- minimum semidefinite rank

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Linear and Multilinear Algebra*,

*63*(4), 774-787. https://doi.org/10.1080/03081087.2014.898763

**Bounds on minimum semidefinite rank of graphs.** / Narayan, Sivaram K.; Sharawi, Yousra.

Research output: Contribution to journal › Article

*Linear and Multilinear Algebra*, vol. 63, no. 4, pp. 774-787. https://doi.org/10.1080/03081087.2014.898763

}

TY - JOUR

T1 - Bounds on minimum semidefinite rank of graphs

AU - Narayan, Sivaram K.

AU - Sharawi, Yousra

PY - 2015/4/3

Y1 - 2015/4/3

N2 - The minimum semidefinite rank (Formula presented.) of a graph is defined to be the minimum rank among all Hermitian positive semidefinite matrices associated to the graph. A problem of interest is to find upper and lower bounds for (Formula presented.) of a graph using known graph parameters such as the independence number and the minimum degree of the graph. We provide a sufficient condition for (Formula presented.) of a bipartite graph to equal its independence number. The delta conjecture gives an upper bound for (Formula presented.) of a graph in terms of its minimum degree. We present classes of graphs for which the delta conjecture holds.

AB - The minimum semidefinite rank (Formula presented.) of a graph is defined to be the minimum rank among all Hermitian positive semidefinite matrices associated to the graph. A problem of interest is to find upper and lower bounds for (Formula presented.) of a graph using known graph parameters such as the independence number and the minimum degree of the graph. We provide a sufficient condition for (Formula presented.) of a bipartite graph to equal its independence number. The delta conjecture gives an upper bound for (Formula presented.) of a graph in terms of its minimum degree. We present classes of graphs for which the delta conjecture holds.

KW - connectivity of a graph

KW - delta conjecture

KW - independence number

KW - matrix of a graph

KW - minimum semidefinite rank

UR - http://www.scopus.com/inward/record.url?scp=84919873548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919873548&partnerID=8YFLogxK

U2 - 10.1080/03081087.2014.898763

DO - 10.1080/03081087.2014.898763

M3 - Article

VL - 63

SP - 774

EP - 787

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 4

ER -