What graph is this? For my game I am trying to implement a continues world by interconnecting the nodes like below
I beg your pardon for my bad drawings

I don't know how to explain it but its NOT DENSE GRAPH
It is representation of 3x3 nodes
Where every node is connected to adjacent node vertically or diagonally (edges in turquoise color)
Ex:
1-2, 1-4
2-1,2-3,2-5
5-2, 5-6, 5-4, 5-8
Now there are some edges (colored in blue and purple)
1-7, 1-3
4-6
2-8
I need edges like this for creating endless/continues world for my game
My world is actually lot bigger than this but I made 3x3 for the sake of drawing.
Is there any name for this type of graph?
 A: @Johannes Kloos is right. The graph in the picture is just the cartesian product of $C_3$ with itself. I think in general you are interested in the cartesian product of $C_k$ and $C_l$.
A: 
"Where every node is connected to adjacent node vertically or diagonally"

Judging from the picture, it should say horizontally instead of diagonally.  And if the vertices are connected to every vertex vertically and horizontally, fidbc's answer is correct for the $3 \times 3$ case, but not in general.
If the above is correct, this graph is known as the $3 \times 3$ rook's graph.  Generalising this to the $k \times n$ case, it is the Cartesian product of $K_k$ and $K_n$, and the line graph of the complete bipartite graph $K_{k,n}$.
If $n \geq k$, then the number of $n$-colourings of of the $k \times n$ rook's graph is the number of Latin rectangles of order $n$.  In particular, the number of $n$-colourings of of the $n \times n$ rook's graph is the number of Latin squares of order $n$.  Further, the chromatic polynomial of this graph counts a generalisation of Latin rectangles.
I discuss this graph in my survey paper The Many Formulae for the Number of Latin Rectangles.
