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Properties of a Graph
Graphs come with various properties which are used for characterization of graphs depending on their structures. These properties are defined in specific terms pertaining to the domain of graph theory. In this chapter, we will discuss a few basic properties that are common in all graphs.
Radius of a Connected Graph
The minimum eccentricity from all the vertices is considered as the radius of the Graph G. The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the Graph G.
Notation − r(G)
From all the eccentricities of the vertices in a graph, the radius of the connected graph is the minimum of all those eccentricities.
Example − In the above graph r(G) = 2, which is the minimum eccentricity for ‘d’.
Diameter of a Graph
The maximum eccentricity from all the vertices is considered as the diameter of the Graph G. The maximum among all the distances between a vertex to all other vertices is considered as the diameter of the Graph G.
Notation − d(G)
From all the eccentricities of the vertices in a graph, the diameter of the connected graph is the maximum of all those eccentricities.
Example − In the above graph, d(G) = 3; which is the maximum eccentricity.
Central Point
If the eccentricity of a graph is equal to its radius, then it is known as the central point of the graph. If
e(V) = r(V),
then ‘V’ is the central point of the Graph ’G’.
Example − In the example graph, ‘d’ is the central point of the graph.
e(d) = r(d) = 2
Centre
The set of all central points of ‘G’ is called the centre of the Graph.
Example − In the example graph, {‘d’} is the centre of the Graph.
Circumference
The number of edges in the longest cycle of ‘G’ is called as the circumference of ‘G’.
Example − In the example graph, the circumference is 6, which we derived from the longest cycle a-c-f-g-e-b-a or a-c-f-d-e-b-a.
Girth
The number of edges in the shortest cycle of ‘G’ is called its Girth.
Notation − g(G).
Example − In the example graph, the Girth of the graph is 4, which we derived from the shortest cycle a-c-f-d-a or d-f-g-e-d or a-b-e-d-a.
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