### Eigenvalues and expansion of regular graphsEigenvalues and expansion of regular graphs

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 Author Kahale, Nabil Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1995 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Ramanujan graphs ♦ Eigenvalues ♦ Expander graphs ♦ Induced subgraphs ♦ Load balancing ♦ Random walks ♦ Selection networks Abstract The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best-known explicit expanders. The spectral method yielded a lower bound of $\textit{k}/4$ on the expansion of linear-sized subsets of $\textit{k}-regular$ Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately $\textit{k}/2.$ Moreover, we construct a family of $\textit{k}-regular$ graphs with asymptotically optimal second eigenvalue and linear expansion equal to $\textit{k}/2.$ This shows that $\textit{k}/2$ is the best bound one can obtain using the second eigenvalue method. We also show an upper bound of roughly 1+k-1 . As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (respectively, extrovert graphs) of smaller size (respectively, degree) than was previously known. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1995-09-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 42 Issue Number 5 Page Count 16 Starting Page 1091 Ending Page 1106

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Source: ACM Digital Library