Question

If the sequence \(d_{1}, d_{2}, \ldots, d_{n}\) of nonnegative integers represents the degrees of the vertices of a tree with \(n\) vertices, then \(\sum_{i=1}^{n} d_{i}=2(n-1)\). Show that the converse is false.

Short answer

The sequence \(0, 0, 3, 3\) sums to \(6\) but cannot represent a tree.

Step-by-step solution

Understanding the Problem

We need to show that having a sequence of nonnegative integers where the sum is \(2(n-1)\) does not necessarily imply the sequence represents the degrees of the vertices of a tree.

Exploring Tree Degree Properties

Recall that a tree with \(n\) vertices has exactly \(n-1\) edges, so the sum of the degrees of its vertices must be \(2(n-1)\). This is because each edge contributes 2 to the sum of degrees, one for each incident vertex.

Finding a Counterexample

To show the converse is false, we need a sequence of nonnegative integers that sums to \(2(n-1)\) but cannot represent the degree sequence of any tree. Let's consider \(n=4\), which means the degrees must sum to \(6\). A sequence like \(1, 1, 2, 2\) works as it can represent a tree, but let's check another.

Constructing a False Sequence

Consider the sequence \(0, 0, 3, 3\) for \(n = 4\). The sum is \(6\) (i.e., \(2(n-1)\)), but a tree with 4 vertices cannot have a degree sequence with vertices of degree 3, because then they wouldn't connect all vertices, and trees are connected.

Conclusion: Verifying the False Sequnce

Thus, the sequence \(0, 0, 3, 3\) sums to \(6\) but cannot form a tree because there are not enough connections to all vertices. Hence, not every sequence summing to \(2(n-1)\) represents a tree.

Key concepts

Tree Properties

A tree is a special type of graph in graph theory that has some unique and interesting properties. Trees are connected graphs with no cycles. This means all the vertices in a tree are connected by edges in such a way that there are no loops or circuits.

In a tree with \(n\) vertices, there are always exactly \(n-1\) edges. This is true regardless of the tree's configuration or structure. The sum of the degrees of all vertices in a tree equals twice the number of edges because each edge in a tree connects two vertices and thus counts towards two degrees.

Understanding these properties is essential for determining if a given sequence of nonnegative integers can represent a tree.

Degree Sequence

The degree sequence of a graph is a list of the degrees of its vertices. For a tree with \(n\) vertices, this would be a sequence of nonnegative integers where the sum is exactly \(2(n-1)\). This sequence accounts for each connection between vertices because each connection or edge contributes 2 to the total degree count.

However, merely having a sum that equals \(2(n-1)\) does not mean the sequence will always form a valid tree. This point introduces the notion of the degree sequence requiring more conditions beyond just the sum. Properly forming a tree involves ensuring that these degrees can be distributed in a way that maintains the tree's properties of connectivity and acyclicity.

Nonnegative Integers

A sequence of nonnegative integers includes whole numbers that are zero or positive. In the context of a tree's degree sequence, these integers represent how many connections, or edges, each vertex has.

When dealing with trees, the nonnegative integers in a sequence must sum up correctly but also be distributed in a manner to ensure all vertices connect appropriately, achieving a single connected structure with no cycles.

The sequence \(0, 0, 3, 3\) mentioned in the problem exemplifies the challenge. Each number is nonnegative, and they sum appropriately, but they cannot form a tree due to connection issues.

Counterexample

A counterexample serves to disprove a statement in mathematics by showing a specific case where the statement fails. In this context, we have a claim regarding sequences summing to \(2(n-1)\) and representing a tree’s degree sequence.

The counterexample \(0, 0, 3, 3\) illustrates that although the numbers sum to the correct total, they do not fulfill the other requirements of a tree. The sequence fails to connect all vertices in a single structure without cycles, thereby highlighting the falsehood of the converse statement that any sequence with the sum \(2(n-1)\) forms a tree.

Understanding such counterexamples is crucial in graph theory, as it helps identify limitations and refine concepts.