5.3.2. Construction of Sequence S*

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 this chunk, we introduce the concept of a degree sequence in graph theory. A degree sequence is simply a list that represents the degrees (number of edges connected) of all vertices in a graph, arranged from the largest to the smallest. This means that if we have a graph with some vertices, we start by identifying which vertex has the highest number of connections (the highest degree) and we place that degree first in our list. We then continue this process, listing each vertex's degree in descending order until we have covered all vertices.

Think of a sports team, where each player has a number of assists. To create a ranking of the players based on their assists, you would list the player with the most assists first, followed by the player with the second most assists, and so forth. This ranking acts like the degree sequence, showing the performance levels of the players in a clear order.

And we say a sequence of n values as 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 as a graphic sequence.

A graphic sequence is one that can represent a simple graph. A simple graph means it has no multiple edges between the same vertices and no loops (edges connected at both ends to the same vertex). For a sequence to be graphic, it must be possible to draw a corresponding simple graph. If it's impossible to create such a graph from the given degree sequence, it does not qualify as a graphic sequence. This distinction is crucial when analyzing graphs because it defines what degree sequences can realistically exist in graph structures.

Imagine a group's seating arrangement at a table. The degrees represent how many people are directly seated next to each person. If, based on the seating arrangement described by the degree sequence, you can't physically arrange people to match those relationships without sitting someone next to themselves or ignoring others, then that seating arrangement isn't possible – similar to how a non-graphic sequence cannot form a simple graph.

So here we are given the following you are given sequence S of n non negative integers in non increasing order and you have a reduced sequence S*. It is reduced in the sense it has n - 1 values whereas the sequence S has n values.

The process of constructing the reduced sequence S* starts with a sequence S that contains n non-negative integers ordered from largest to smallest. The reduced sequence S* is created by removing the first element, which is the highest degree, and then decrementing the next few elements. Specifically, we subtract one from the next d elements (where d is the degree we removed) in the original sequence. Therefore, S* contains one fewer element than S and represents a modified degree distribution from which we can derive further insights.

Consider a class of students who have scores in different subjects. If the highest scorer drops out, the class dynamic changes. To understand how the average score might change, you would remove that highest score and look at the remaining scores, applying slight adjustments if necessary to represent the class as accurately as possible.

So, that’s the way we obtained the sequence S*. And what Havel-Hakimi theorem says is the following it says that your sequence S is a graphic sequence if and only if the reduced sequence S* when arranged in a non increasing order is also a graphic sequence.

The Havel-Hakimi theorem provides a method to determine whether a given degree sequence represents a valid graph. According to this theorem, you can determine the validity of the original sequence S by examining the reduced sequence S*. If S* is graphic when arranged in non-increasing order, this implies that the original sequence S is also graphic. Thus, the behavior of S* provides essential insights into the characteristics of S, allowing us to verify whether S can indeed represent a graph.

This theorem can be compared to assembling a puzzle. If part of the puzzle can fit together neatly, it offers insights into how the whole puzzle may be completed. Similarly, if the reduced sequence S* is valid, we can confidently state that the original sequence S has potential to form a valid graph.

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