What is the graph that corresponds to $Q'_8$ generalized quadrangle ? Could you please explain this in plain english? In this paper a table about large graphs with given degree and diameter graphs is shown: 

I would like to know what the adjacency list of the graph denoted by:
 
in the table above is. 
Could you please explain how I can get that adjacency list of $Q'_8$?
The paper says that :

and 
.
So $Q_8$ is the incidence graph of the regular generalised quadrangle and $Q'_8$ is the quotient of this. The problem is that I only understand the words "the", "so", "and", "is", "of" and "this" from the previous sentence.
What is the incidence graph of a regular generalized quadrangle ? 
Is this incidence graph bipartite ? If yes, why ?
What is the quotient of that ?
How can I get the actual graph based on the above sentence ?
Is Mathematica a good tool for this task?
 A: A Finite Generalized Quadrangle (FGQ) is an incidence structure $\mathcal{S}$ where we have points, lines (finite sets), and incidence relation satisfying:


*

*Any two points lie on at most one line

*Any two lines meet in at most one point

*If $\ell$ is a line and $P$ is a point not incident with $\ell$, then there is a unique line $m$ on $P$ intersecting $\ell$.


So this is a partial linear space containing no triangles.  The Wikipedia page here is actually super nice, Bill Cherowitzo works hard on the finite geometry pages.
Now the most studied examples of generalized quadrangles are the classical examples, these are defined by certain types of nondegenerate sesquilinear forms on a vector space over a finite field.  Your example, given as $Q_{8}$, is commonly known as $W(8)$, obtained from a nondegenerate alternating bilinear form on the projective space $\mathrm{PG}(3,8)$ (defined by a four dimensional vector space over $\mathbb{F}_{8}$).
Two important graphs related to these objects are the point collinearity graph (which are strongly regular) and the incidence graph, bipartite with parts corresponding to points and lines. The incidence graph of a FGQ is another important structure, it has diameter four and girth eight.
Okay, now for your important questions: I don't know what they are taking the quotient by, so I can't help there.  I would program it in Magma, but that's my personal preference for working with a bilinear form over a finite field.  Your best bet is probably to capitalize on the hard work of Eric Moorhouse, who has the data related to the quadrangle here (it is $W(8)$, having order $(8,8)$).
A: The most useful page I know of is degree-diameter table, which actually has pictures of many of the graphs. 
The requested graph is described in this paper --  C. Delorme and J. Gómez, Some New Large Compound Graphs, European Journal of Combinatorics, Volume 23, Issue 5, July 2002, Pages 539–547. 
But take a look at the first link.  Even when the construction is given, figuring it out can be a murky process.  The best describer of complex graphs that I know of is Brouwer.
