Given the graphic sequence of a simple graph, how to construct the adjacency matrix? Suppose I have the graphic sequence (7,6,6,6,5,4,4,2), how do I get the adjacency matrix out of it?
 A: You can construct one (there may be many) of the realizations of a graphic sequence using the Havel-Hakimi Theorem:

Theorem (Havel, Hakimi, reference on Wikipedia)
  Let $s = s_{0}, \ldots , s_{d}, s_{d+1}, \ldots, s_{n}$ be a non-increasing sequence of non-negative integers with $s_{0} = d$. The sequence $s$ is graphic if and only if the sequence $s' = s_{1}-1, s_{2}-1,\ldots,s_{d}-1,s_{d-1}\ldots,s_{n}$ is graphic.

What this amounts to algorithmically is that we can take such a sequence and construct a graph (and hence the adjacency matrix, if that's the way you want to do it) by starting with a an empty graph on $n$ vertices (i.e. with no edges), labelled with the degree requirements (so vertex $i$ has initial requirement $s_{i}$), the repeatedly take the vertex with the largest remaining degree requirement $s_{j}$ adding making it adjacent to the $s_{j}$ vertices with the next highest remaining degree requirements.
In terms of the adjacency matrix, we start with a disconnected graph - all entries in the matrix are zero - then fill them in according to the scheme above.
A: The graph has $8$ vertices, and $\tfrac{1}{2}(7+6+6+6+5+4+4+2)=20$ edges by the Handshaking Lemma.  We can exhaustively generate the non-isomorphic $8$-vertex $20$-edge graphs with vertex degrees between $2$ and $7$ using geng which comes with nauty using:
geng 8 20:20 -d2 -D7

I wrote a script that filters out the ones that don't have degree sequence (7,6,6,6,5,4,4,2) leaving two remaining.  The two graphs are as follows:

I've marked the vertices with their degrees, and their corresponding adjacency matrices are given below.
$\begin{array}{|cccccccc|} \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 
1 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 
1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\ \hline \end{array}
 \quad\quad\quad
\begin{array}{|cccccccc|} \hline 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 
1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 
1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\ \hline \end{array}$
