User:Jason Kimberley/Moore bound

wikipedia:Moore_graph

$M(d,2k+1)=1+d\sum _{i=0}^{k-1}(d-1)^{i}=1+{\frac {d}{2}}M(d,2k).$

$M(d,2k)=2\sum _{i=0}^{k-1}(d-1)^{i}=1+(d-1)^{k-1}+d\sum _{i=0}^{k-2}(d-1)^{i}=M(d,2k-1)+(d-1)^{k-1}.$