### 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 |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Narayan, S. K., & Sharawi, Y. (2015). Bounds on minimum semidefinite rank of graphs.

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