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# User:Jason Kimberley/C girth ge 4

 girth C D E ${\displaystyle \geq }$ Cge Dge Ege ${\displaystyle =}$ Ceq Deq Eeq

## A186714:

Triangular array ${\displaystyle C(n,k)=}$ the number of connected ${\displaystyle k}$-regular graphs, having girth at least 4, with ${\displaystyle n}$ nodes, for ${\displaystyle 0\leq k\leq n/2}$.

The ${\displaystyle (k,4)}$-cage is ${\displaystyle K_{k,k}}$.

These counts are the output from Markus Meringer's GENREG. The italicised values are from my running of GENREG at The University of Newcastle High Performance Computing Facility for the durations described in the column sequences.

Girth at least: 3 4 5 6 7 8

${\displaystyle n}$ ${\displaystyle \forall }$ ${\displaystyle k}$ 0 1 2 3 4 5 6 7 8 A186724 A185114 A014371 A033886 A058275 A058276 A181153 A181154 A181170 1 1 1 0 1 0 0 0 1 0 0 1 1 0 0 1 2 0 0 1 1 1 0 0 1 0 4 0 0 1 2 1 1 0 0 1 0 0 10 0 0 1 6 2 1 3 0 0 1 0 2 0 37 0 0 1 22 12 1 1 32 0 0 1 0 31 0 0 340 0 0 1 110 220 7 1 1 1608 0 0 1 0 1606 0 1 0 18020 0 0 1 792 16828 388 9 1 1 193907 0 0 1 0 193900 0 6 0 0 2867725 0 0 1 7805 2452818 406824 267 8 1 1 32674058 0 0 1 0 32670330 0 3727 0 0 0 1581632114 0 0 1 97546 456028474 1125022325 483012 741 13 1 1 6705889824 0 0 1 0 6636066099 0 69823723 0 1 0 0 0 0 1 1435720 100135577747 3813549359274 14836130862 2887493 14 1 0 0 1 0 1582718912968 0 0 0 1 23780814 0 0 1 0 0 0 1 432757568 0 0 1 0 0 0 1 8542471494 0 0 1 0 0 0 1 181492137812 0 0 1 0 0 0 1 4127077143862
${\displaystyle C(23,10)=0}$ was found using GENREG taking 33.3 processor days on 18th May 2011.