It is actually a very simple statement that just looks complicated because of the notation. The idea is this:
Recall that the degree of a vertex is the number of lines that end at that vertex.
How many line-endings will there be in the whole graph?
Well, two for every line, of course! One on each end!
Why the fancy notation?
The need for the notation has to do with wanting to say, "When you count them all up, no matter how many there are..." This sounds just fine in conversational English, but could potentially have some loopholes for those who wish to be utterly, mathematically precise. How many is "all of them"? How will you know you have counted them all? Are you sure it doesn't matter how many there are? Mathematicians don't want to have to go back later and find that they didn't take everything into consideration, so they try to find absolutely clear, waterproof, doubtproof ways to express things.
Study what the symbols mean, and experiment with different ways of expressing the same thing. Once you get used to the symbol, it doesn't seem so intimidating or hard. Statements that look like this one are very common in mathematics, so learning to read this will help make formal mathematical statements seem less like a foreign language to you.
Decoding the notation
The fancy E is a Greek letter called sigma. It is a symbol for a sum. It means that we are going to add something up. It doesn't matter how many numbers we will add up, the sum symbol tells us that we will add up as many as we end up with.
deg v is a shorthand way of saying the degree of a vertex in a graph.
The letter i which is a subscript of the v in deg v tells us which vertex. If the subscript was 1, it would mean the first vertex. If the subscript was 2, it would mean the second vertex, and so on. The fact that the subscript is i is related to the statement i=1 that appears beneath the sigma.
The statement i=1 that appears beneath the sigma means that when you start adding up the sums of vertices, you should start with the first one..
The letter p is the letter that graph theorists use to stand for the number of vertices in a graph. The number that p actually stands for at a given moment depends on the graph that you are thinking about. Notice that the letter p appears above the sigma
The p above the sigma is a partner to the i = 1. i = 1 says to begin with 1, p means to end with p. Since we are adding up degrees of vertices, "from i = 1 to p" means that we add up the degrees of all the vertices.
The q is the letter graph theorists use to stand for the number of edges in a graph. The number that q actually stands for, of course, will be different for different graphs.
2q means "2 times q".
How you would read it?
"The sum of the degrees of all the vertices in a graph is equal to twice the number of edges."
Theorem and Proof
The statement that is made by the notation in the picture is the theorem. It is a theorem because it has been proved. Until it is proved, it is just a statement, although sometime it is called a claim, or a conjecture (depending on how confident the person making the statement is that it will turn out to be true.)
The proof of the First Theorm of Graph Theory is not presented here, because you can write a proof yourself and demonstrate once and for all that writing proofs doesn't have to be scary. Here's all you have to do:
Make sure you understand the statement of the theorem and what the notation means.
Think about why that would be true. Try to imagine a case where it would be false.
Write down your ideas about why it would be true. Write down your questions, too, and the things that seem confusing about it. It's okay if it comes out in a jumble. Just keep trying to explain it to yourself.
Once you think you've got it, organize the jumbled information as well as you can.
Show what you have written to someone else and see if they can follow it.
Change or add to it until you are satisfied that it is clear.
Deduction and proof
A proof of the First Theorem of Graph Theory is likely to be a deductive proof, or a proof that relies on the technique called deduction (It wouldn't be very hard to think up an inductive proof, but deductive reasoning in this case is probably easier.)
At some point you are likely to say to yourself as you are thinking about this: Because each edge (or line) has two ends, I can be sure that... That is deductive reasoning. When the logical consequences of a true statement lead to other statements that must be true, the path of our thinking is deductive reasoning.
