Let $G$ be a finite group with $\left|G\right|\u2a7e4$ and $S$ be a subset of $G$ with $\left|S\right|=d$ such that the Cayley sum graph ${C}_{\Sigma}(G,S)$ is undirected and connected. We show that the non-trivial spectrum of the normalised adjacency operator of ${C}_{\Sigma}(G,S)$ 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 $\left(-1+\frac{{h}_{\Sigma}{\left(G\right)}^{4}}{\eta},1-\frac{{h}_{\Sigma}{\left(G\right)}^{2}}{2{d}^{2}}\right]$, where ${h}_{\Sigma}\left(G\right)$ denotes the vertex Cheeger constant of the $d$-regular graph ${C}_{\Sigma}(G,S)$ and $\eta ={2}^{9}{d}^{8}$. 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.

Revised:

Accepted:

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Keywords: Expander graphs, Cheeger inequality, Spectra of Cayley sum graphs.

^{1}; Saha, Jyoti Prakash

^{2}

@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 = {http://archive.numdam.org/articles/10.5802/alco.166/} }

TY - JOUR AU - Biswas, Arindam AU - Saha, Jyoti Prakash TI - A Cheeger type inequality in finite Cayley sum graphs JO - Algebraic Combinatorics PY - 2021 SP - 517 EP - 531 VL - 4 IS - 3 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.166/ DO - 10.5802/alco.166 LA - en ID - ALCO_2021__4_3_517_0 ER -

%0 Journal Article %A Biswas, Arindam %A Saha, Jyoti Prakash %T A Cheeger type inequality in finite Cayley sum graphs %J Algebraic Combinatorics %D 2021 %P 517-531 %V 4 %N 3 %I MathOA foundation %U http://archive.numdam.org/articles/10.5802/alco.166/ %R 10.5802/alco.166 %G en %F ALCO_2021__4_3_517_0

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. http://archive.numdam.org/articles/10.5802/alco.166/

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