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Unified bounds for the independence number of graphs

Part of: Graph theory Basic linear algebra Communication, information

Published online by Cambridge University Press:  11 December 2023

Jiang Zhou
Jiang Zhou*
Affiliation:
College of Mathematical Sciences, Harbin Engineering University, Harbin, PR China
*
e-mail: zhoujiang@hrbeu.edu.cn
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Abstract

The Hoffman ratio bound, Lovász theta function, and Schrijver theta function are classical upper bounds for the independence number of graphs, which are useful in graph theory, extremal combinatorics, and information theory. By using generalized inverses and eigenvalues of graph matrices, we give bounds for independence sets and the independence number of graphs. Our bounds unify the Lovász theta function, Schrijver theta function, and Hoffman-type bounds, and we obtain the necessary and sufficient conditions of graphs attaining these bounds. Our work leads to some simple structural and spectral conditions for determining a maximum independent set, the independence number, the Shannon capacity, and the Lovász theta function of a graph.

Keywords

Independence numbergraph matrixLovász theta functionShannon capacitygeneralized inverseeigenvalue

MSC classification

Primary: 05C69: Dominating sets, independent sets, cliques 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 94A15: Information theory, general 15A09: Matrix inversion, generalized inverses
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant No. 12071097) and the Natural Science Foundation for The Excellent Youth Scholars of the Heilongjiang Province (Grant No. YQ2022A002).

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