Problem 38 - Discrete Mathematics with Applications

Question

Prove that a connected graph with \(n\) vertices has at least \(n-1\) edges. (Hint: Use induction.)

Step-by-Step Solution

Answer

For a connected graph with \(n=1\) vertex, there are 0 edges. For a connected graph with \(n=2\) vertices, there must be at least one edge connecting the vertices. Now, we assume that for a connected graph with \(k\) vertices (\(k\geq2\)), it has at least \(k-1\) edges. To prove the inductive step, let G be a connected graph with \((k+1)\) vertices. After removing a vertex (v) and its associated edges, creating a new connected graph G' with \(k\) vertices, we know that G' must have at least \(k-1\) edges by the inductive hypothesis. Adding vertex v and its connecting edge back to G' results in a total of \(k\) edges, confirming that a connected graph with \((k+1)\) vertices has at least \(k\) edges. Thus, a connected graph with \(n\) vertices has at least \(n-1\) edges for all \(n\geq1\).

Step 1: Base Case: n = 1

For a connected graph with n = 1 vertex, there must be 0 edges since a single vertex does not have any other vertex to connect to. Thus, in this case, the number of edges is indeed n-1, or 1-1=0.

Step 2: Base Case: n = 2

For a connected graph with n = 2 vertices, there must be at least one edge connecting the two vertices, otherwise, the graph would not be connected. Thus, in this case, the number of edges is also n-1, or 2-1=1.

Step 3: Inductive Hypothesis

Assume that for a connected graph with k vertices (k≥2), it has at least k-1 edges.

Step 4: Inductive Step

Now, we want to show that a connected graph with (k+1) vertices has at least (k+1)-1=k edges. Let G be a connected graph with (k+1) vertices and let v be an arbitrary vertex in G. Since G is connected and contains (k+1) vertices, vertex v must have at least one edge connected to another vertex, say vertex u. Let's create a new graph G' by removing vertex v and its associated edge(s). Now, G' has k vertices, and since G was connected, G' must also be connected. By the inductive hypothesis, G' must have at least k-1 edges.

Step 5: Adding v Back to the Graph

Now, add vertex v and its associated edge back to G'. Since we already have k-1 edges in G', the act of adding a vertex v along with its connecting edge will result in a total of (k-1) + 1 = k edges. This confirms that a connected graph with (k+1) vertices has at least k edges. As we have shown the base cases and completed the induction process, we can conclude that a connected graph with n vertices has at least n-1 edges for all n>=1.

Key Concepts

Connected Graph

In graph theory, a connected graph is a type of graph where there is a path between every pair of vertices.
This implies that there are no isolated vertices. A graph can be visualized as a collection of points (vertices) with lines (edges) connecting them.

• In a connected graph, traveling from one vertex to any other should be possible.
• If it's impossible to reach one or more vertices from others, the graph is said to be disconnected.

Understanding connected graphs is crucial because many problems and theorems in graph theory revolve around this property.
They ensure the structure is unified rather than split into separate parts.

Inductive Proof

Inductive proof is a logical technique used to prove a statement for every integer above a certain number.
This method involves establishing a base case and showing the truth of a statement for an arbitrary case, assuming it's true for the previous one.

• First, check the base case: prove the statement for the smallest value (usually 1).
• Then, assume that the statement holds for some arbitrary natural number, like \(k\).
• Finally, prove that it holds for \(k+1\) under this assumption.

Inductive proofs can rigorously demonstrate propositions concerning sequences or structures like graphs.
When working with graphs, this often means confirming properties as vertices or edges are added.

Vertices and Edges

Vertices and edges are fundamental components of any graph. A vertex can be thought of as a point or a node,
while an edge is a connection between two vertices.

• They can represent numerous things depending on what the graph models: places, people, or states (vertices) and paths, relationships, or transitions (edges).
• The number of vertices in a graph is a critical aspect influencing its connectivity.
• Edges give structure to the graph, determining how vertices are connected.

In a connected graph, analyzing the vertices and edges helps in understanding the graph's structure and functionality.
They are foundational to any analysis or proof in graph theory, including determining connectivity and other properties.

Mathematical Induction

Mathematical induction is a powerful proof technique often used in graph theory and other mathematical areas.
It consists of proving a statement by confirming it for the initial case and then proving it for the next one,
based on the assumption that it holds for a given instance.

• Start by confirming the base case. This could be as simple as proving that a graph with one or two vertices behaves as expected.
• Make an inductive hypothesis that assumes the statement is true for a scenario with \(k\) elements, such as vertices in a graph.
• Show that if the statement holds for \(k\), it must also hold for \(k+1\).

Mathematical induction is particularly useful when proving things about sequences or connected structures,
like the minimum number of edges in a connected graph with a given number of vertices.