Basic Properties of a Graph - GeeksforGeeks

25 Oct, 2018
A Graph is a non-linear data structure consisting of nodes and edges. The nodes are sometimes also referred to as vertices and the edges are lines or arcs that connect any two nodes in the graph.
Properties of Graphs are basically used for characterization of graphs depending on their structures. We defined these properties in specific terms that pertain to the domain of graph theory. In this article, we are going to discuss some properties of Graphs these are as follows:
Distance between two VerticesIt is basically the number of edges that are available in the shortest path between vertex A and vertex B.If there is more than one edge which is used two connect two vertices then we basically considered the shortest path as the distance between these two vertices.Notation used :
d(A, B)
here function d is basically showing the distance between vertex A and vertex B.
Let us understand this using an example:In the above diagram, let’s try to find the distance between vertices b and d.d(b, c)
We can go from vertex b to vertex d in different ways such as
1.ba, af, fe, ed here the d(b, c) will be 4.
2.ba, af, fc, cd here the d(b, c) will be 4.
3.bc, cf, fe, ed here the d(b, c) will be 4.
4.bc, cd here the d(b, c) will be 2.
hence the minimum distance between vertex b and vertex d is 2.
Eccentricity of a VertexMaximum distance from a vertex to all other vertices is considered as the Eccentricity of that vertex.Notation used:
e(V)
here e(v) determines the eccentricity of vertex V.
Let us try to understand this using following example.From the above diagram lets try to find the eccentricity of vertex b.
e(b)
d(b, a)=1
d(b, c)=1
d(b, d)=2
d(b, e)=3
d(b, f)=2
d(b, g)=2
Hence the eccentricity of vertex b is 3
Radius of a Connected GraphThe minimum value of eccentricity from all vertices is basically considered as the radius of connected graph.Notation used:
r(G)
here G is the connected graph.
Let us try to understand this using following example.From the above diagram:
r(G) is 2.
Because the minimum value of eccentricity from all vertices is 2.
Diameter of A Connected GraphUnlike the radius of the connected graph here we basically used the maximum value of eccentricity from all vertices to determine the diameter of the graph.Notation used:
d(G)
where G is the connected graph.
Let us try to understand this using following example.From the above diagram:
d(G) is 3.
Because the maximum value of eccentricity from all vertices is 3.
Central Point and CentreThe vertex having minimum eccentricity is considered as the central point of the graph.And the sets of all central point is considered as the centre of Graph.if
e(V)=r(G)
then v is the central point.
Let us try to understand this using following example.In the above diagram the central point will be f.
because 
e(f)=r(G)=2
hence f is considered as the central point of graph.
Hence f is also the centre of the graph.

Distance between two VerticesIt is basically the number of edges that are available in the shortest path between vertex A and vertex B.If there is more than one edge which is used two connect two vertices then we basically considered the shortest path as the distance between these two vertices.Notation used :
d(A, B)
here function d is basically showing the distance between vertex A and vertex B.
Let us understand this using an example:In the above diagram, let’s try to find the distance between vertices b and d.d(b, c)
We can go from vertex b to vertex d in different ways such as
1.ba, af, fe, ed here the d(b, c) will be 4.
2.ba, af, fc, cd here the d(b, c) will be 4.
3.bc, cf, fe, ed here the d(b, c) will be 4.
4.bc, cd here the d(b, c) will be 2.
hence the minimum distance between vertex b and vertex d is 2.

Notation used :
d(A, B)
here function d is basically showing the distance between vertex A and vertex B.

Let us understand this using an example:In the above diagram, let’s try to find the distance between vertices b and d.
d(b, c)
We can go from vertex b to vertex d in different ways such as
1.ba, af, fe, ed here the d(b, c) will be 4.
2.ba, af, fc, cd here the d(b, c) will be 4.
3.bc, cf, fe, ed here the d(b, c) will be 4.
4.bc, cd here the d(b, c) will be 2.
hence the minimum distance between vertex b and vertex d is 2.

Eccentricity of a VertexMaximum distance from a vertex to all other vertices is considered as the Eccentricity of that vertex.Notation used:
e(V)
here e(v) determines the eccentricity of vertex V.
Let us try to understand this using following example.From the above diagram lets try to find the eccentricity of vertex b.
e(b)
d(b, a)=1
d(b, c)=1
d(b, d)=2
d(b, e)=3
d(b, f)=2
d(b, g)=2
Hence the eccentricity of vertex b is 3

Notation used:
e(V)
here e(v) determines the eccentricity of vertex V.

Let us try to understand this using following example.
From the above diagram lets try to find the eccentricity of vertex b.
e(b)
d(b, a)=1
d(b, c)=1
d(b, d)=2
d(b, e)=3
d(b, f)=2
d(b, g)=2
Hence the eccentricity of vertex b is 3

Radius of a Connected GraphThe minimum value of eccentricity from all vertices is basically considered as the radius of connected graph.Notation used:
r(G)
here G is the connected graph.
Let us try to understand this using following example.From the above diagram:
r(G) is 2.
Because the minimum value of eccentricity from all vertices is 2.

Notation used:
r(G)
here G is the connected graph.

Let us try to understand this using following example.
From the above diagram:
r(G) is 2.
Because the minimum value of eccentricity from all vertices is 2.

Diameter of A Connected GraphUnlike the radius of the connected graph here we basically used the maximum value of eccentricity from all vertices to determine the diameter of the graph.Notation used:
d(G)
where G is the connected graph.
Let us try to understand this using following example.From the above diagram:
d(G) is 3.
Because the maximum value of eccentricity from all vertices is 3.

Notation used:
d(G)
where G is the connected graph.

Let us try to understand this using following example.
From the above diagram:
d(G) is 3.
Because the maximum value of eccentricity from all vertices is 3.

Central Point and CentreThe vertex having minimum eccentricity is considered as the central point of the graph.And the sets of all central point is considered as the centre of Graph.if
e(V)=r(G)
then v is the central point.
Let us try to understand this using following example.In the above diagram the central point will be f.
because 
e(f)=r(G)=2
hence f is considered as the central point of graph.
Hence f is also the centre of the graph.

if
e(V)=r(G)
then v is the central point.

Let us try to understand this using following example.
In the above diagram the central point will be f.
because 
e(f)=r(G)=2
hence f is considered as the central point of graph.
Hence f is also the centre of the graph.

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