Bounds on the sum of minimum semidefinite rank of a graph and its complement

Sivaram K. Narayan; Yousra Sharawi

doi:10.13001/1081-3810.3539

Bounds on the sum of minimum semidefinite rank of a graph and its complement

Sivaram K. Narayan, Yousra Sharawi^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.

Original language	English
Article number	31
Pages (from-to)	399-406
Number of pages	8
Journal	Electronic Journal of Linear Algebra
Volume	34
DOIs	https://doi.org/10.13001/1081-3810.3539
Publication status	Published - 2018
Externally published	Yes

Keywords

Graph complement conjecture
Independence number
Matrix of a graph
Minimum semidefinite rank

ASJC Scopus subject areas

Algebra and Number Theory

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abstract = "The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.",

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AB - The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.

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Bounds on the sum of minimum semidefinite rank of a graph and its complement

Abstract

Keywords

ASJC Scopus subject areas

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