A Cheeger type inequality in finite Cayley sum graphs
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 517-531.

Let G be a finite group with |G|4 and S be a subset of G with |S|=d such that the Cayley sum graph C Σ (G,S) is undirected and connected. We show that the non-trivial spectrum of the normalised adjacency operator of C Σ (G,S) is controlled by its Cheeger constant and its degree. We establish an explicit lower bound for the non-trivial spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval - 1 + h Σ (G) 4 η , 1 - h Σ (G) 2 2d 2 , where h Σ (G) denotes the vertex Cheeger constant of the d-regular graph C Σ (G,S) and η=2 9 d 8 . Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the Cayley graph of finite groups.

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DOI: https://doi.org/10.5802/alco.166
Classification: 05C25,  05C50,  05C75
Keywords: Expander graphs, Cheeger inequality, Spectra of Cayley sum graphs.
@article{ALCO_2021__4_3_517_0,
     author = {Biswas, Arindam and Saha, Jyoti Prakash},
     title = {A {Cheeger} type inequality in finite {Cayley} sum graphs},
     journal = {Algebraic Combinatorics},
     pages = {517--531},
     publisher = {MathOA foundation},
     volume = {4},
     number = {3},
     year = {2021},
     doi = {10.5802/alco.166},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.166/}
}
Biswas, Arindam; Saha, Jyoti Prakash. A Cheeger type inequality in finite Cayley sum graphs. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 517-531. doi : 10.5802/alco.166. https://alco.centre-mersenne.org/articles/10.5802/alco.166/

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