5.1. Discrete Mathematics

The degree sequence of a graph is basically the sequence of degrees of the vertices in non-increasing order. So, you list down the highest degree vertex or the degree of the highest vertex first followed by the next highest degree, followed by the next highest degree and so on.

A degree sequence represents how many edges connect to each vertex of a graph, arranged from highest to lowest. For example, if you have three vertices with degrees 3, 2, and 1, their degree sequence would be (3, 2, 1). This arrangement makes it easier to analyze and prove properties about the graph.

Imagine a classroom where students can only interact with each other. Each student has a certain number of friends. If one student has three friends, another has two, and the last one has one, we arrange their friendships (or degrees) from most to least. This arrangement helps us understand the social interactions in the classroom more clearly.

We say a sequence of n values is a graphic sequence if you can construct a simple graph whose degree sequence is the given sequence. If you cannot draw any simple graph whose degree sequence is a given sequence, then that sequence will not be called a graphic sequence.

A graphic sequence must allow for the construction of a simple graph, which is a graph without loops or multiple edges between any pair of vertices. Not all numerical sequences meet this requirement. For instance, the sequence (2, 2, 2) can represent a simple triangle (three vertices connected to each other). However, a sequence that implies more edges than available vertices cannot be graphic.

Think of it like trying to set up a team project where each team member needs to communicate with a specific number of others. If one team member needs to talk to five others, but there are only four available, it’s impossible to satisfy that requirement. Hence, the setup is not graphic.

If you take the sum of the values in the sequence, it should be an even quantity, since the sum of the degrees of all the vertices is twice the number of edges, which is always an even number.

This property stems from the Handshaking Lemma in graph theory, which states that every edge contributes to the degree of two vertices. Therefore, if you sum up all the vertex degrees, the result must be even. If a sequence's sum is odd, it cannot represent a graphic sequence because you cannot create a graph with an odd number of total edges balanced between two vertices.

Imagine each handshake in a room counts as two: one for each participant. If everyone shakes hands and you count the total handshakes, it should always yield an even number because each handshake involves two people. If the handshake count is odd, something went wrong in the accounting.

If you are given a sequence with n values, how can you verify whether that sequence is a graphic sequence or not? Here, the Havel-Hakimi theorem provides a necessary and sufficient condition.

The Havel-Hakimi theorem offers a systematic way to verify if a degree sequence can form a simple graph. You take a sequence, remove the largest element, and decrement the next largest elements accordingly. By repeating this process, if you end with a valid graphic sequence, the original sequence is also graphic.

Consider this like planning a series of meals for a group. You start with the biggest meal (largest number in the sequence) and then reduce plans for the next big meals based on how many people can cover it. If you can still serve everyone with remaining portions, your original plan is feasible.

The theorem states that a sequence S is graphic if and only if the reduced sequence S*, when arranged in a non-increasing order, is also a graphic sequence.

Proving the theorem involves showing two implications. First, if S* is graphic, then S must also be graphic. This is based on the ability to construct a graph using vertices of degrees in S*. Then you have to prove the reverse. If S is graphic, then S* must also show the properties of a graphic sequence.

Think of the process like verifying if a team can always produce a product. If smaller indicators of team performance (S*) are strong, it shows the overall team setup (S) is robust. Conversely, if the whole team is proven effective, so must the sub-group indicators be.

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