4.3.2. Edge Set Cardinality

In question number 2 you are given an unknown simple graph G, the graph G is not known to you known to you in the sense that you are just given that it has 6 vertices but exact cardinality of edge set is not given. But it is given to you that your graph G is such that if you delete the vertex v from the graph then you are left with 7 edges. If you delete vertex v from the graph you are left with 6 edges and so on.

In this chunk, we explore the idea of edge set cardinality in a graph. Given that a graph G has 6 vertices, we're interested in identifying how many edges (the edge set cardinality) are present initially. We have information about the number of edges remaining after removing specific vertices from the graph G. Essentially, when a vertex is removed from a graph, the edges connected to that vertex are also removed, which reduces the total number of edges in the graph. For instance, removing vertex v1 leaves 7 edges, meaning that vertex v1 was incident to a certain number of edges which we will determine.

To think about this, imagine a group of six friends connected in various ways – some are close friends and share mutual connections (edges). If one friend (a vertex) leaves the group, the connections they had (edges) are disrupted. By looking at how many connections remain when each friend leaves, we can deduce how well-connected our group is, akin to finding the edge set cardinality.

So, if that is the case the question asks you to find out the cardinality of the edge set of the original graph. Again do not try to do a brute force and try all possible graphs in your mind and then hit upon the answer because that will take enormous amount of time.

This chunk invites us to think critically and analytically rather than through brute-force trial and error. The cardinality of the edge set can be derived using known properties of graph theory. We won't have to list every possible graph to find the edge count; instead, we can use the relationships we’ve established while looking at edges left after removing vertices. This approach saves time and increases efficiency by relying on mathematical logic and insights from the structure of the graph.

Imagine you're trying to guess how many cards are in a deck without counting them directly. Instead, you know that when you remove a known group of cards, you can determine how many were there originally based on what’s left. This is similar to how we assess the edges in a graph.

If you have the vertex v here and if you remove the vertex v from the graph the cardinality of the edge set gets decremented by the degree of v because the deletion of the vertex v will delete how many edges from the graph? All the edges which are incident with the vertex v namely degree of v number of edges from my edge set will be removed.

Here, we discuss the impact of removing a vertex on the edge set of a graph. The degree of a vertex indicates how many edges are connected to that vertex. When a vertex is removed, we lose those edges connected to it, thereby reducing the total number of edges in the graph. So, if we have several equations based on different vertices and their respective degrees, we can deduce the original total edge cardinality.

Think of a network of water pipes where each junction (vertex) has a certain number of pipes (edges) connected. If you remove a junction, all the pipes connected to it are also removed, leading to less flow in the entire network. This analogy helps to visualize the concept of edge removals based on vertex degrees.

Now if I sum all these 6 equations I get that 6 times the cardinality of E minus the summation of the degree of 6 vertices in the graph is 36.

We gather information from the previously established relationships about vertex removals. By summing all the equations derived from removing a vertex and utilizing their degrees, we can find a single equation that relates directly to the original edge count. This consolidation of information allows us to simplify the problem and focus on just one key equation that leads us to the answer.

If you were to tally the number of friends removed from a gathering and how many friendships they represented, summing those counts can help you figure out just how connected the entire group was before anyone left. It's a generalization of many smaller observations into one clear picture.

And now I can apply the handshaking theorem which says that if you take the summation of the degrees of all the vertices in your original graph it is the same as twice the number of edges.

The Handshaking Theorem forms a pivotal concept in understanding the relationships between the vertices' degrees and edges in a graph. It tells us that if we sum the degrees of all the vertices in a graph, we will end up with twice the number of edges, given that each edge connects two vertices and counts towards both of their degrees. This relationship is essential for solving for edge cardinality in our earlier equations.

Picture a social gathering: each handshake between two people creates a connection. If we list every person and count how many people they've shaken hands with, we actually count each handshake between two friends twice. Thus, summing all individual counts gives us double the actual number of handshakes, just as the theorem suggests for edges in a graph.

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