I found a Graph that defies the answer to this question... I have this problem from graph theory which I am not entirely convinced with...
I am asked if there is a graph with six vertices with degrees {2,2,2,4,5,5}.
Answer, is apparently NO.
My question is...is there a rule to the order in which the degrees are presented?
like say, if the graph is presented to have degree {a,b,c} then does vertices with degree a and degree b have to be adjacent because they are presented next to each other in the set?? 
Because if not....I found a counter example and my answer to the question is YES there is.
I am not asked for a simple graph or anything so the multiple edges from one vertex to another is allowed, yes?
I cannot find a reason to negate my example as a proof of existence for such a graph. The answer reasons with the sum of degrees being 20 which is twice the number of edges i.e. there are 10 edges, and if we let $v_1,v_2$ have degree 5 each, we come to the conclusion that it is impossible to construct such a graph.
Am I misunderstanding the notation of {2,2,2,4,5,5} here?? I don't see the rules...it would be great if someone can tell me why my example is not valid. Thanks!
 A: It is quite common to just say "graph" and implicitly assume that the graph has no multiple edges. This usage goes together with the word "multigraph" to describe a graph that may have multiple edges.
The terminology "simple graph" is also available to specify this, but is more often used in context where it is already established that multiple edges are a relevant possibilities.
Apparently, the source you have for the claim that $(2,2,2,4,5,5)$ is not a possible degree sequence for a graph must be using the "multigraph/graph" terminology rather than the "graph/simple graph" terminology. That's all there is to it -- merely a difference in terminology.
When multiple edges are allowed, every finite sequence of nonnegative integers whose sum is even (and which doesn't contain a number that is larger than half the sum, unless you also allow loops) is a possible degree sequence. Since this makes the problem almost trivial, it is usually considered in the context where multiple edges are not allowed.
A: The notation {2, 2, 2, 4, 5, 5} is the degree sequence of the graph and is often written in descending order like : [5, 5, 4, 2, 2, 2]. The Havel-Hakimi theorem gives an easy way to check if a sequence is 'graphical' (makes a graph). Conceptually, you take the highest degree d in the sequence and take one from the d next highest degrees. This is done repeatedly until either a) you get a sequence of zeros, b) you start getting negative values or c) you don't have enough to connect to.
So in your case, we have 5 as the largest value, so we take one from the next 5 degrees, leaving us with [0, 4, 3, 1, 1, 1]. We repeat with 4 to get [0, 0, 2, 0, 0, 0] - which is where we stop. Note that this procedure does not guarantee a connected graph, nor does it give you all possible graphs with the supplied degree sequence.
