Let
k
≥
1
k \geq 1
be an odd integer,
t
=
⌊
k
+
2
4
⌋
{t = \left \lfloor {\tfrac {{k + 2}}{4}} \right \rfloor }
, and q be a prime power. We construct a bipartite, q-regular, edge-transitive graph
C
D
(
k
,
q
)
CD(k,q)
of order
υ
≤
2
q
k
−
t
+
1
\upsilon \leq 2{q^{k - t + 1}}
and girth
g
≥
k
+
5
g \geq k + 5
. If e is the the number of edges of
C
D
(
k
,
q
)
CD(k,q)
, then
e
=
Ω
(
υ
1
+
1
k
−
t
+
1
)
e = \Omega ({{\upsilon ^{1 + \frac {1}{{k - t + 1}}}}})
. These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order
υ
\upsilon
and girth at least g,
g
≥
5
g \geq 5
,
g
≠
11
g \ne 11
, 12. For
g
≥
24
g \geq 24
, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for
5
≤
g
≤
23
5 \leq g \leq 23
,
g
≠
11
g \ne 11
, 12, it improves on or ties existing bounds.