4.3. Question 2

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 the vertex v from the graph you are left with 6 edges and so on.

In this part of the question, we have a graph G with 6 vertices, but we don't know how many edges it has. We do know something interesting about the edges: if we remove a vertex from the graph, the number of edges changes according to the degree of that vertex. Specifically, the graph loses edges equivalent to the number of edges connected to the removed vertex. This indicates we need to use the properties of graph theory to deduce the total number of edges in G.

Imagine a community of 6 friends where each friend has a certain number of connections (friendships) with each other. If one friend (vertex) leaves the community, the remaining friendships (edges) decrease. Just as we can count the total number of friendships knowing how many each friend has, we can find the total edges in graph G by considering how many edges disappear when each friend leaves.

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. Instead we will try to apply some rules of logic and properties of graph here.

Instead of guessing and checking, we will systematically analyze the problem using the relationship between vertex deletion and edge count. The principle states that removing a vertex from a graph decreases the total edge count by the degree of that vertex. Therefore, we can set up an equation based on the given conditions of the graph after removing individual vertices, allowing us to calculate the total number of edges in G without testing every possible graph.

Imagine you’re organizing a group event and need to calculate how many connections (friendships) are left if certain friends (vertices) are not attending. Rather than randomly guessing connections for everyone, you can make a chart of who is connected to whom. This would allow you to easily determine how many friends remain and their interactions without unnecessary guesswork.

So that means what I can say is that my graph G is such that the edge set cardinality minus the degree of v is 7 because it is given that after deleting vertex v in the graph you are left with 7 edges. That means in the leftover graph which I obtain after deleting v if I would have added the edges which were incident with vertex v and how many such edges would have been there? : degree of v number of edges that would have given me the cardinality of the edge set.

Based on the properties outlined, we create equations for each vertex in G. For each vertex v_i removed, the resulting edge count is related to its degree. For instance, if removing vertex v_1 leaves us with 7 edges, we can express this mathematically: (total edges in G) - (degree of v_1) = 7. This process is repeated for all vertices with their respective edge counts after removal.

Think of this as needing to account for how many friends are left after each invited friend declines the invitation to a party. By counting who has said they will attend after each decline, you can amend your final guest count more accurately without needing to recalculate the entire group each time.

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. 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 same as twice the number of edges.

After establishing the equations, we sum them together to form a compact representation of edge counts, leading to a single central equation. We observe that the sum of vertex degrees equals twice the total number of edges (known as the handshaking theorem). Thus, we can resolve for the total edge count from this finetuned equation.

Similar to creating a budget for a project involving team members. As you account for each member’s contributions, you gather all their input into one sheet, allowing you to calculate the total effort put in. Just as the budget figures can be combined, here we can combine equations to determine the total edges.

So, now if I take the summation of the degrees of all the vertices in the original graph it is same as twice the number of edges. So, now I have 1 equation just involving the unknown which is my cardinality of the edge set. So, I get my edge set’s cardinality to be 9.

Applying the handshaking theorem, we relate our summation of degrees to twice the edge count, yielding a conclusive equation exclusively in terms of the total edge number. Solving this final equation provides the edge set's cardinality, establishing a clear result: the graph has 9 edges.

This final step is akin to reaching the conclusion of your budget calculations, leading you to the final project cost. By analyzing each contributor's sum of efforts, you can coherently determine the total required resource, finishing your calculations just as we finalize the edge count.

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