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|Title:||Nonexistence of a circulant expander family|
|Authors:||Leung, K.H. |
|Source:||Leung, K.H., Nguyen, V., So, W. (2011-02). Nonexistence of a circulant expander family. Bulletin of the Australian Mathematical Society 83 (1) : 87-95. ScholarBank@NUS Repository. https://doi.org/10.1017/S0004972710001644|
|Abstract:||The expansion constant of a simple graph G of order n is defined as h(G) = min0 ∈ for some fixed ∈ > 0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger's inequalities d-λ2(G)/2 ≤ h(G) ≤ √2d(d - λ2(G)) where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed. © Copyright Australian Mathematical Publishing Association Inc. 2010.|
|Source Title:||Bulletin of the Australian Mathematical Society|
|Appears in Collections:||Staff Publications|
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