Graph measurements: length, distance, diameter, eccentricity, radius, center



Prerequisite – Graph Theory Basics – Set 1, Set 2
A graph is defined as set of points known as ‘Vertices’ and line joining these points is known as ‘Edges’. It is a set consisting of where ‘V’ is vertices and ‘E’ is edge.

Vertices: {A, B, C, D, E, F}
Edges: {{A, B}, {A, D}, {A, E}, {B, C}, {C, E}, {C, F}, {D, E}, {E, F}} 

Graph Measurements: There are few graph measurement methods available:

1. Length –
Length of the graph is defined as the number of edges contained in the graph.



Length of the graph: 8
AB, BC, CD, DE, EF, FA, AC, CE 

2. Distance between two Vertices –
The distance between two vertices in a graph is the number of edges in a shortest or minimal path.It gives available minimum distance between two edges.There can exist more than one shortest path between two vertices.

Shortest Distance between 1-5 is 2
1->2->5 

3. Diameter of graph –
Diameter of graph is maximum distance between the pair of vertices.It can also be defined as maximal distance between the pair of vertices. Way to solve it is find all the paths and then find maximum of all.

Diameter: 3
BC->CF->FG  

4. Radius of graph –
Radius of graph exists only if it has diameter.The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the Graph G.It is denoted as r(G).

Radius: 2
All available minimum radius: 
BC->CF,
BC->CE,
BC->CD,
BC->CA

5. Center of graph –
It consists of all the vertices whose eccentricity is minimum. Here the eccentricity is equal to the radius.For example, if school is at the center of town it will reduces the distance buses has to travel.

Center: A  

6. Eccentricity of graph –
It is defined as maximum distance of one vertex from other vertex.The maximum distance between a vertex to all other vertices is considered as the eccentricity of vertex. It is denoted by e(V).

Eccentricity from:
(A, A) = 0
(A, B) = 1
(A, C) = 2
(A, D) = 1
Maximum value  is 2, So Eccentricity is 2


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