5.1.2. Tutorial 9: Part II

Hello, everyone, welcome to the second part of tutorial 9. So, let us start with question number 7. So, here we first define what we call as the degree sequence of a graph and 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.

In graph theory, the degree of a vertex is simply the number of edges connected to it. The degree sequence is a list that arranges all the vertex degrees of a graph in a specific order—usually from the highest degree to the lowest. This ordering helps in understanding the structure of the graph itself.

Imagine you have a basketball team and each player can score different amounts of points in a game. If you list out the players by their scores from highest to lowest, that's similar to what a degree sequence does for a graph. The player with the highest score (or degree) is listed first, just like the vertex with the highest degree is at the top of the degree sequence.

A sequence of n values is termed 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 the given sequence will not be called a graphic sequence.

A graphic sequence allows for the construction of a graph that matches the degrees specified in that sequence. For it to be graphic, there must be a way to connect the vertices such that all vertices match the specified degrees without contradiction. If a degree sequence fails to allow for such a graph, it is deemed non-graphic.

Think of constructing a house where each room (vertex) has a specific number of doorways (edges) to other rooms. If someone gives you a list of doorways needed for each room but it is impossible to arrange them without any dead ends or contradictions, then that list is not feasible—similar to a non-graphic sequence.

The first few parts of question 7 basically asks you to prove or disprove which of the given sequences is a graphic sequence. So let us take the first sequence 5, 4, 3, 2, 1, 0.

You are often challenged to verify whether certain sequences are graphic. In this example, the sequence includes degrees from 5 down to 0. To explore whether this sequence is graphic, we must check if it’s possible to create a simple graph with these degrees assigned to its vertices.

Imagine a group of friends, where the first friend (vertex) can connect to five other friends, the second can connect to four, and so forth down to one who has no connections. You need to analyze whether you can actually arrange them so these friendships are valid without leaving anyone isolated contrary to their defined friendships.

In this case, we have to verify whether we can draw a simple graph with 6 nodes where the highest degree is 5 and the smallest degree is 0. It is easy to see that this sequence is not a graphic sequence.

A sequence like 5, 4, 3, 2, 1, 0 is non-graphic because the vertex with degree 5 needs to connect to five other vertices, which are only 5 degrees in total. However, one of the vertices must have a degree of 0, meaning it cannot connect to anyone. This creates a contradiction, making the sequence non-graphic.

Suppose you have six people in a network where one must be completely isolated (degree 0) but one person claims to know everyone in the group (degree 5). This discrepancy is like having too many connections for one person and not enough for others, resulting in an impossible social network.

One simple way to refute that this sequence is not a graphic sequence is that if you take the sum of the values that are given in this sequence is not an even quantity.

In graph theory, a basic requirement for a degree sequence to be graphic is that the sum of all vertex degrees must be even, as each edge in a graph contributes to the degree of two vertices. If the total sum is odd, it is impossible to form a graph since each edge must connect two vertices.

Think of inviting pairs of friends to a party. For every pair of friends, there has to be a handshake (an edge) between them. If you have an odd total of handshakes, one will always be left out without a match—similar to an odd sum of degrees in a graphic sequence.

So now in question 8, we want to characterise that, we want to find out a characterization for graphic sequences. A necessary and sufficient condition is given by what we call as Havel-Hakimi theorem.

The Havel-Hakimi theorem provides a method to determine if a sequence is a graphic sequence by transforming it into a reduced form and checking its properties. Essentially, you repeatedly remove degrees and adjust the sequence until you can easily verify if it’s graphic.

Consider this as a game. You’re given a certain number of tokens representing friendships. The game involves checking if you can break down these friendships (removing degrees) while ensuring that new connections still uphold the new friendship balance—similar to ensuring a sequence remains valid through transformations.

So, that is a way I can apply the Havel-Hakimi theorem to verify whether a given sequence of S values is a graphic sequence or not.

By applying the Havel-Hakimi theorem, you can systematically reduce a degree sequence to determine if it can eventually lead back to a simple graph. If you reach a point where you can verify that a reduced sequence is not graphic, then the original sequence is also declared non-graphic.

Imagine a team trying to adjust their roles to meet a shared goal. They keep adjusting until they connect with only roles that actually work together. If they can't correctly align their tasks, the original team structure isn't effective either, much like how the original sequence can't be graphic if they fail in reduced formats.

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