## Basic Properties of Graph TheoryProperties of graph theory are basically used for characterization of graphs depending on the structures of the graph. Following are some basic properties of graph theory: ## 1 Distance between two vertices
## ExampleSuppose, we want to find the distance between vertex B and D, then first of all we have to find the shortest path between vertex B and D. There are many paths from vertex B to vertex D: - B -> C -> A -> D, length = 3
- B -> D, length = 1(Shortest Path)
- B -> A -> D, length = 2
- B -> C -> D, length = 2
- B -> C -> A -> D, length = 3
Hence, the minimum ## 2. Eccentricity of a vertex
To count the eccentricity of vertex, we have to find the distance from a vertex to all other vertices and ## ExampleIn the above example, if we want to find the maximum eccentricity of vertex 'a' then: - The distance from vertex a to b is 1 (i.e. ab)
- The distance from vertex a to c is 1 (i.e. ac)
- The distance from vertex a to f is 2 (i.e. ac -> cf or ad -> df)
- The distance from vertex a to d is 1 (i.e. ad)
- The distance from vertex a to e is 2 (i.e. ab -> be or ad -> de)
- The distance from vertex a to g is 3 (i.e. ab -> be -> eg or ac -> cf -> fg etc.)
Hence, the Similarly, maximum eccentricities of other vertices of the given graph are: - e(b) = 3
- e(c) = 3
- e(d) = 2
- e(e) = 3
- e(f) = 3
- e(g) = 3
## 3. Radius of connected GraphThe ## ExampleFrom the example of 5.2, r(G) = 2, which is the minimum eccentricity for the vertex 'd'. ## 4. Diameter of a Graph
## ExampleFrom the above example, if we see all the eccentricities of the vertices in a graph, we will see that the diameter of the graph is the maximum of all those eccentricities. Diameter of graph ## 5. Central pointIf the eccentricity of the graph is equal to its radius, then it is known as Or If ## ExampleFrom the above example, 'd' is the central point of the graph. i.e. ## 6. CentreThe set of all the central point of the graph is known as centre of the graph. ## ExampleFrom the example of 5.2, { ## 7. CircumferenceThe total number of edges in the longest cycle of graph G is known as the circumference of G. ## ExampleIn the above example, the ## 8. GirthThe total number of edges in the shortest cycle of graph G is known as ## ExampleIn the above example, the ## 9. Sum of degrees of vertices TheoremFor non-directed graph G = (V,E) where, Vertex set V = {V1, V2, .... Vn} then, In other words, for any graph, the sum of degrees of vertices equals twice the number of edges.
For directed graph G = (V, E) where, Vertex Set V = {V1, V2, ... Vn} then,
The number of vertices in any non- directed graph with odd degree is even. ## ExampleIt is impossible to make a graph with v (number of vertices) = 6 where the vertices have degrees 1, 2, 2, 3, 3, 4. This is because the sum of the degrees deg(V) is,
In an non-directed graph, if the degree of each vertex is k, then
If the degree of each vertex in a non-directed graph is at least k, then
If the degree of each vertex in a non- directed graph is at most k, then Next TopicGraph Representations |