  With the extended logarithmic flow equations and some extended Vertex
operators in generalized Hirota bilinear equations, extended bigraded Toda
hierarchy(EBTH) was proved to govern the Gromov-Witten theory of orbiford
$c_{NM}$ in literature. The generating function of these Gromov-Witten
invariants is one special solution of the EBTH. In this paper, the multi-fold
Darboux transformations and their determinant representations of the EBTH are
given with two different gauge transformation operators. The two Darboux
transformations in different directions are used to generate new solutions from
known solutions which include soliton solutions of $(N,N)$-EBTH, i.e. the EBTH
when $N=M$. From the generation of new solutions, one can find the big
difference between the EBTH and the extended Toda hierarchy(ETH). Meanwhile we
plotted the soliton graphs of the $(N,N)$-EBTH from which some approximation
analysis will be given. From the analysis on velocities of soliton solutions,
the difference between the extended flows and other flows are shown. The two
different Darboux transformations constructed by us might be useful in
Gromov-Witten theory of orbiford $c_{NM}$.
