Graph Theory: what are the two vertices and how to draw them? 
Define a graph $G$ such that $V(G) = \{2,3,4,5,11,12,13,14\}$ and two
  vertices $s$ and $t$ are adjacent if and only if $\gcd\{s,t\} = 1$. Draw a
  diagram of $G$ and find its size $e(G)$.

I can understand V(G) = {2,3,4,5,11,12,13,14} but what are "two vertices $s$ and $t$ are adjacent if and only if $\gcd\{s,t\} = 1$"?
 A: You have $8$ vertices, labelled $2$, $3$, and so on.   Now we need to determine the edges.
Look for example at the vertices $2$ and $3$. Are they joined by an edge?  They are to be joined precisely if $\gcd(2,3)=1$.  The greatest common divisor of $2$ and $3$ is indeed $1$, so draw an edge joining $2$ and $3$.
Are vertices $2$ and $4$ joined by an edge? Well, $\gcd(2,4)=2\ne 1$, so no edge.
Are vertices $2$ and $5$ joined by an edge? Yes, because $\gcd(2,5)=1$. Continue.
Vertex $2$ will be joined to $3$, $5$, $11$, $13$.  Now we have produced  all the edges that involve $2$.
In addition, $3$ is joined to $4$, $5$, $11$, $13$, $14$.
In addition, $4$ is joined to $5$, $11$, and $13$.
In addition, $5$ is joined to $11$, $12$, $13$, $14$.
In addition, $11$ is joined to $12$, $13$, $14$.
In addition, $12$ is joined to $13$.
And finally, $13$ is joined to $14$. 
A: A graph is a bunch of dots on your paper, called "vertices".  Each dot has a label, which is its name.  In your example there are eight dots, named 2, 3, 4, 5, 11, 12, 13, and 14.
Sometimes two vertices are connected by a line, and sometimes they aren't.  When two dots are connected, we say they are "adjacent". The line is called an "edge".
This question is asking about a graph where two vertices are connected whenever their labels (which are numbers) have no common divisor bigger than one.  For example, vertices 4 and 14 are not connected because the numbers 4 and 14 are both divisible by 2.  But vertices 5 and 14 are connected because there is no  number bigger than 1 that divides both 5 and 14.  We write $\gcd(4, 14)$ for the greatest number, 2, that divides both 4 and 14, and $\gcd(5,14)$ for the greatest number, 1, that divides both 5 and 14. 
Your job is to draw all the connections between the dots.  You should decide if 2 and 3 are connected, and then draw an edge between them if so.  Then decide if 2 and 4 are connected, and so on.
$e(G)$ is just the total number of edges in the graph.
A: $G=G(V,E)$ is a graph, where $V$ is the set of vertices - in your
case the vertices "have names'' or correspond to natural numbers.
Now $E$ is the set of edges, in your case $E:=\{(u,v):gcd(u,v)=1\}$
and by $gcd(u,v)$ I mean the gcd of the natural numbers corresponding
to $u,v$.
For example there is an edge between the vertices corresponds
to the numbers $1$ and $2$ i.e. $(1,2)\in E$ since $gcd(1,2)=1$
but $(2,4)\not\in E$ since $gcd(2,4)=2\neq1$.
Note:$gcd$ - is the greatest common divisor 
A: Here's a drawing of the graph in question.

The $7$ dashed lines represent non-edges, and the $\binom{7}{2}-7=14$ solid lines are the present edges.
