TY - GEN
T1 - ON THE SECOND EIGENVALUE OF RANDOM REGULAR GRAPHS.
AU - Broder, Andrei
AU - Shamir, Eli
PY - 1987
Y1 - 1987
N2 - It is known that random d regular graphs are very efficient expanders, almost surely. However, checking whether a particular graph is a good expander is co-NP-complete. It is shown that the second eigenvalue of d-regular graphs, lambda //2, is concentrated in an interval of width O( ROOT d) around its mean, and that its mean is O(d**3**/**4). The result holds under various models for random d-regular graphs. As a consequence, a random d-regular graph on n vertices is, with high probability, a certifiable efficient expander for n sufficiently large. The bound on the width of the interval is derived from martingale theory, and the bound on E( lambda //2) is obtained by exploring the properties of random walks in random graphs.
AB - It is known that random d regular graphs are very efficient expanders, almost surely. However, checking whether a particular graph is a good expander is co-NP-complete. It is shown that the second eigenvalue of d-regular graphs, lambda //2, is concentrated in an interval of width O( ROOT d) around its mean, and that its mean is O(d**3**/**4). The result holds under various models for random d-regular graphs. As a consequence, a random d-regular graph on n vertices is, with high probability, a certifiable efficient expander for n sufficiently large. The bound on the width of the interval is derived from martingale theory, and the bound on E( lambda //2) is obtained by exploring the properties of random walks in random graphs.
UR - http://www.scopus.com/inward/record.url?scp=0023535763&partnerID=8YFLogxK
U2 - 10.1109/sfcs.1987.45
DO - 10.1109/sfcs.1987.45
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AN - SCOPUS:0023535763
SN - 0818608072
SN - 9780818608070
T3 - Annual Symposium on Foundations of Computer Science (Proceedings)
SP - 286
EP - 294
BT - Annual Symposium on Foundations of Computer Science (Proceedings)
PB - IEEE
ER -