Diameters and Eigenvalues
نویسندگان
چکیده
We derive a new upper bound for the diameter of a k-regular graphG as a function of the eigenvalues of the adjacency matrix. Namely, supposethe adjacency matrix of G has eigenvalues AI , A2 .••.• An with lAd:::: IA21 ::::... :::: IAnl where AI = k, A = IA21. Then the diameter D(G) must satisfy D(G) :::; rlog(n 1)f1og(k/A)l. We wilJ consider families of graphs whose eigenvalues can be explicitly de-termined. These graphs are determined by sums or differences of vertex labels.Namely, the pair {i. j} being an edge depends only on the value i + j (or i jfor directed graphs). We will show that these graphs are expander graphs withsmall diameters by using an inequality on character sums, which was recentlyproved by N. M. Katz. BELL COMMUNICATIONS RESEARCH, 445 SOUTH STREET, MORRISTOWN, NEW JERSEY 07960 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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