Graph Theory - Radius and Center

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Radius of Graph

The radius of a graph is defined as the minimum eccentricity among all vertices. The eccentricity of a vertex is the greatest distance between that vertex and any other vertex in the graph.

The radius, therefore, represents the smallest maximum distance from any vertex to the farthest vertex in the graph.

Formally, the radius r of a graph G is given by −

r = min(ecc(v)) for all v in G

Where ecc(v) represents the eccentricity of vertex v.

Eccentricity

The eccentricity of a vertex v in a graph is the greatest distance from v to any other vertex in the graph. Eccentricity provides a measure of how far a vertex is from the farthest vertex. The concept of eccentricity is important because it helps in determining both the radius and center of a graph.

For example, in a tree graph, the eccentricity of a leaf node is equal to the length of the longest path from that leaf node to the farthest vertex.

Calculating the Radius

To compute the radius of a graph, follow these steps −

• For each vertex in the graph, calculate its eccentricity.
• Determine the vertex with the minimum eccentricity. This value is the radius.

Example: Simple Graph Radius

Consider the following graph with 5 vertices −

Let us calculate the eccentricity of each vertex −

Vertex 1:

• Distance to 2: 1 (direct edge)
• Distance to 3: 1 (direct edge)
• Distance to 4: 2 (through vertex 2)
• Distance to 5: 2 (through vertex 3)
• Maximum distance: 2 (between vertices 1 and 4 or 1 and 5)

Eccentricity of Vertex 1: 2

Vertex 2:

• Distance to 1: 1 (direct edge)
• Distance to 3: 2 (through vertex 1)
• Distance to 4: 1 (direct edge)
• Distance to 5: 3 (through vertices 1 and 3)
• Maximum distance: 3 (between vertices 2 and 5)

Eccentricity of Vertex 2: 3

Vertex 3:

• Distance to 1: 1 (direct edge)
• Distance to 2: 2 (through vertex 1)
• Distance to 4: 3 (through vertices 1 and 2)
• Distance to 5: 1 (direct edge)
• Maximum distance: 3 (between vertices 3 and 4)

Eccentricity of Vertex 3: 3

Vertex 4:

• Distance to 1: 2 (through vertex 2)
• Distance to 2: 1 (direct edge)
• Distance to 3: 3 (through vertices 2 and 1)
• Distance to 5: 4 (through vertices 2, 1, and 3)
• Maximum distance: 4 (between vertices 4 and 5)

Eccentricity of Vertex 4: 4

Vertex 5:

• Distance to 1: 2 (through vertex 3)
• Distance to 2: 3 (through vertices 3 and 1)
• Distance to 3: 1 (direct edge)
• Distance to 4: 4 (through vertices 3, 1, and 2)
• Maximum distance: 4 (between vertices 5 and 4)

Eccentricity of Vertex 5: 4

Therefore, the radius of the graph is 2, as the minimum eccentricity is 2 for vertex 1.

Radius in Trees

In a tree, the radius is the minimum distance from any vertex to the farthest vertex, and the center is the set of vertices that have the smallest eccentricity. A tree always has a center, and it is always either one or two vertices. The center can be identified by finding the central vertex or vertices of the tree.

For example, in a path graph (a simple tree), the center is the middle vertex (or two middle vertices if the number of vertices is even). The radius is the distance from the middle vertex to the farthest vertex.

Eccentricity of each vertex −

• Vertex 1: 2
• Vertex 2: 3
• Vertex 3: 3
• Vertex 4: 4
• Vertex 5: 4
• Vertex 6: 4

Radius of the tree: 2
Center of the tree: [1]

Radius of Cyclic Graphs

In cyclic graphs, the radius is the minimum eccentricity, and the center is usually one or two vertices located at the center of the cycle.

The eccentricity in a cyclic graph is typically equal for all vertices, and the center can be identified by selecting the middle vertex (or two middle vertices for an even number of vertices).

Complete Graphs

In a complete graph, the radius and the center are easily determined. Since every vertex is connected to every other vertex, the eccentricity of all vertices is 1. Therefore, the radius is 1.

Grid Graphs

In grid graphs, the radius is the minimum distance from any vertex to the farthest vertex.

For an m x n grid graph, the radius is typically ⌊(m + n) / 2⌋ , where m is the number of rows, and n is the number of columns. This represents the minimum distance between vertices.

Diameter vs. Radius: Key Differences

The diameter of a graph is the longest shortest path between any two vertices, while the radius measures the smallest maximum distance from any vertex to others. The diameter is always greater than or equal to the radius.

For example, in a complete graph, the diameter is 1, while the radius is also 1, indicating all vertices are equally connected.

Center of Graph

The center of a graph is defined as the set of vertices that have the minimum eccentricity. These vertices are located in the most central position, with the smallest maximum distance to any other vertex in the graph.

Formally, the center C of a graph G is defined as −

C = { v ∈ G | ecc(v) = r }

Where r is the radius of the graph, and ecc(v) is the eccentricity of vertex v.

The center is closely related to the radius of a graph, as it contains the vertices whose eccentricity equals the radius.

Properties of the Center

Following are the properties of the center of a graph −

• Uniqueness of the Center: The center of a graph may consist of one or two vertices, but it is always well-defined and can be identified using eccentricity calculations.
• Relation to Radius: The center is always a vertex or vertices with eccentricity equal to the radius, representing the most centrally located points in the graph.
• Geodesic Distance: The center is essential in minimizing geodesic distances, ensuring the most efficient connections in terms of shortest paths.

Identifying the Center

To identify the center of a graph, we follow these steps −

• For each vertex, calculate its eccentricity.
• Determine the set of vertices whose eccentricity equals the radius. These vertices form the center of the graph.

Example: Simple Graph

Consider a graph with 6 vertices −

Let us calculate the eccentricity for each vertex −

Eccentricity of Vertex 1

Vertex 1 is connected to vertices 2 and 3. The distances are −

• Distance to Vertex 2: 1
• Distance to Vertex 3: 1
• Distance to Vertex 4: 2 (via Vertex 2)
• Distance to Vertex 5: 2 (via Vertex 2)
• Distance to Vertex 6: 2 (via Vertex 2)

Maximum distance (eccentricity) from Vertex 1 is 2.

Eccentricity of Vertex 2

Vertex 2 is connected to vertices 1, 4, 5, and 6. The distances are −

• Distance to Vertex 1: 1
• Distance to Vertex 3: 2 (via Vertex 1)
• Distance to Vertex 4: 1
• Distance to Vertex 5: 1
• Distance to Vertex 6: 1

Maximum distance (eccentricity) from Vertex 2 is 2.

Eccentricity of Vertex 3

Vertex 3 is connected to Vertex 1. The distances are −

• Distance to Vertex 1: 1
• Distance to Vertex 2: 2 (via Vertex 1)
• Distance to Vertex 4: 3 (via Vertex 1 and Vertex 2)
• Distance to Vertex 5: 3 (via Vertex 1 and Vertex 2)
• Distance to Vertex 6: 3 (via Vertex 1 and Vertex 2)

Maximum distance (eccentricity) from Vertex 3 is 3.

Eccentricity of Vertex 4

Vertex 4 is connected to Vertex 2. The distances are −

• Distance to Vertex 1: 2 (via Vertex 2)
• Distance to Vertex 2: 1
• Distance to Vertex 3: 3 (via Vertex 2 and Vertex 1)
• Distance to Vertex 5: 2 (via Vertex 2)
• Distance to Vertex 6: 2 (via Vertex 2)

Maximum distance (eccentricity) from Vertex 4 is 3.

Eccentricity of Vertex 5

Vertex 5 is connected to Vertex 2. The distances are −

• Distance to Vertex 1: 2 (via Vertex 2)
• Distance to Vertex 2: 1
• Distance to Vertex 3: 3 (via Vertex 2 and Vertex 1)
• Distance to Vertex 4: 2 (via Vertex 2)
• Distance to Vertex 6: 2 (via Vertex 2)

Maximum distance (eccentricity) from Vertex 5 is 3.

Eccentricity of Vertex 6

Vertex 6 is connected to Vertex 2. The distances are −

• Distance to Vertex 1: 2 (via Vertex 2)
• Distance to Vertex 2: 1
• Distance to Vertex 3: 3 (via Vertex 2 and Vertex 1)
• Distance to Vertex 4: 2 (via Vertex 2)
• Distance to Vertex 5: 2 (via Vertex 2)

Maximum distance (eccentricity) from Vertex 6 is 3.

From the above calculations, we see that both vertices 1 and 2 have the smallest eccentricity of 2. Since vertex 2 connects directly to more vertices, it is considered more central.

Thus, vertex 2 is chosen as the center, and the radius is 2.

Center in Trees

In a tree, the center is either one vertex or two vertices located at the middle of the longest path in the tree. This central location ensures that the maximum distance from the center to any other vertex is minimized.

Center in Cyclic Graphs

In cyclic graphs, the center is either one or two vertices located at the center of the cycle. The eccentricity of all vertices in a cycle is usually equal, so the center can be identified by selecting the middle vertex (or two middle vertices for an even number of vertices).

Example

Consider a cyclic graph with 6 vertices connected in a cycle −

In this cyclic graph, all vertices have the same eccentricity because each vertex is the same distance from the farthest vertex in the cycle. The eccentricity of each vertex is 3, as it takes a maximum of 3 steps to reach any other vertex in the cycle.

Identifying the Center

Since the graph has an even number of vertices (6), the center is the pair of vertices that are equidistant from the ends of the cycle. In this case, the vertices at the center of the cycle are −

• Vertex 0
• Vertex 3

These vertices are the center of the cyclic graph because they are positioned at equal distances from the ends of the cycle.