A191595 Order of smallest n-regular graph of girth 5.

5, 10, 19, 30, 40, 50

Offset: 2

Author

N. J. A. Sloane, Jun 07 2011

Keywords

• nonn
• more
• hard

Comments

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012

Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

Links

• M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
• Andries E. Brouwer, Cages
• Geoff Exoo, Regular graphs of given degree and girth
• G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
• G. Royle, Cages of higher valency

Crossrefs

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).

Moore lower bound on the orders of (k,g) cages: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306(k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10),A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 02 2011

Formula

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

Extensions

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013