  Degree distribution, or equivalently called degree sequence, has been
commonly used to be one of most significant measures for studying a large
number of complex networks with which some well-known results have been
obtained. By contrast, in this paper, we report a fact that two arbitrarily
chosen networks with identical degree distribution can have completely
different other topological structure, such as diameter, spanning trees number,
pearson correlation coefficient, and so forth. Besides that, for a given degree
distribution (as power-law distribution with exponent $\gamma=3$ discussed
here), it is reasonable to ask how many network models with such a constraint
we can have. To this end, we generate an ensemble of this kind of random graphs
with $P(k)\sim k^{-\gamma}$ ($\gamma=3$), denoted as graph space
$\mathcal{N}(p,q,t)$ where probability parameters $p$ and $q$ hold on $p+q=1$,
and indirectly show the cardinality of $\mathcal{N}(p,q,t)$ seems to be large
enough in the thermodynamics limit, i.e., $N\rightarrow\infty$, by varying
values of $p$ and $q$. From the theoretical point of view, given an ultrasmall
constant $p_{c}$, perhaps only graph model $N(1,0,t)$ is small-world and other
are not in terms of diameter. And then, we study spanning trees number on two
deterministic graph models and obtain both upper bound and lower bound for
other members. Meanwhile, for arbitrary $p(\neq1)$, we prove that graph model
$N(p,q,t)$ does go through two phase transitions over time, i.e., starting by
non-assortative pattern and then suddenly going into disassortative region, and
gradually converging to initial place (non-assortative point). Among of them,
one "null" graph model is built.
