Question

Let \(G=(V, E)\) be a connected graph. The distance between two vertices \(v, w \in V\). denoted \(D(v, w),\) is the length of a shortest path joining \(v\) to \(w\). A vertex \(u\) of \(G\) is in the center of \(G\) if \(\max _{v \in V(G)} d(u, v)\) is as small as possible. Prove that a tree has either one or two vertices in its center.

Short answer

A tree has either one or two vertices in its center, based on its diameter.

Step-by-step solution

Understand Tree Properties

In graph theory, a tree is a connected graph without cycles, meaning it has no closed loops. Trees have the property that there is exactly one path between any two vertices. This uniqueness will play an important role in determining the structure of the center of the tree.

Define the Eccentricity of a Vertex

The eccentricity of a vertex \(u\), denoted \(e(u)\), is defined as the maximum distance from \(u\) to any other vertex, i.e., \(e(u) = \max_{v \in V} D(u, v)\). A vertex is in the center of the graph if it has the minimum eccentricity among all vertices.

Consider the Consequences of a Tree's Structure

Given the unique path property, the farthest vertices from any vertex in a tree are its leaves (vertices with degree 1). The longest path in a tree, called a diameter, has a path that specifically connects two leaves.

Find the Midpoint of the Diameter

The center of the tree relates closely to the diameter of the tree. The midpoint (or midpoints if even-numbered) of the longest path (diameter) from one end to the other will minimize the maximum distance to other vertices in the tree.

Analyze the Possible Center Conditions

If the number of vertices in the diameter is odd, there is a single central vertex. If the number of vertices is even, there are two adjacent central vertices because both contribute equally to reducing the maximum path length to any leaf in the tree.

Conclusion from Diameter Analysis

Thus, through considering the properties of the tree's diameter and the definition of eccentricity, we see that the center must consist of either the single midpoint vertex (odd-length diameter) or two adjacent vertices (even-length diameter). Therefore, a tree has exactly one or two vertices in its center.

Key concepts

Tree Properties

In graph theory, a tree is a special kind of graph that is connected and acyclic. This means there are no closed loops or cycles anywhere within the tree. One of the noteworthy properties of trees is that there is exactly one path between any two vertices. This is a fundamental characteristic, providing a unique structure that simplifies many graph computations.
Trees are incredibly useful in various applications such as data structures (binary trees), organizational hierarchies, and spanning trees in network design. Understanding these properties can help in visualizing many types of problems where hierarchical representation is needed.

Eccentricity in Graphs

Eccentricity is a concept that helps measure how far a vertex is from the "farthest" point in a graph. For a vertex \(u\), its eccentricity, denoted \(e(u)\), is the greatest distance you can find to another vertex in the graph. Mathematically, it's defined as \(e(u) = \max_{v \in V} D(u, v)\). So, the eccentricity tells you how "central" or "peripheral" a vertex is.
The lower the eccentricity, the more central the vertex is considered to be. In this way, eccentricity helps identify key vertices that could play significant roles, such as in optimizing network designs or in determining the most efficient communication paths.
In a tree, evaluating eccentricity is straightforward because each vertex's connections are directly determined by the tree's unique path property.

Graph Center

The center of a graph is a pivotal concept focused on identifying the most central points—where the maximum eccentricity is minimal across the graph. In mathematical terms, a vertex is in the graph center if it possesses the smallest possible maximum distance to all other vertices.
For trees, determining the center can be particularly insightful. A tree can have one central vertex if its diameter (the longest path between two vertices) is odd, or two adjacent vertices if the diameter is even. These central points are crucial for several purposes, such as minimizing maximum travel time in transport networks or balancing loads in parallel computing systems.

Diameter of a Graph

In any connected graph, the diameter is the length of the longest shortest path between any two vertices. This represents the "widest" part of the graph. For trees, this notion is particularly simplified since there's only one path between any two vertices.
The significance of the diameter in trees helps ascertain the center. The central vertex or vertices of a tree lie on this longest path, precisely to ensure the minimal maximum distance to all other nodes. If a tree's diameter path is made up of an odd number of vertices, there's a clear central vertex. If it's even, there are two central vertices, providing stable midpoints for network considerations and helping in understanding the depth and reach of the graph.