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Section 5.2 Properties of Graphs

Subsection 5.2.1 Definitions of Graph Properties

Definition 5.2.1. Adjacency.

Two vertices, \(v_i\) and \(v_j,\) in a graph \(G\) are adjacent if and only if \(\{v_i,v_j\}\) is an edge in \(G.\)

Definition 5.2.3. Incidence.

Two edges in a graph \(G\) are incident if and only if they share a vertex.

Definition 5.2.5. Vertex/Edge Incidence.

A vertex \(v\) and an edge \(e=\{v_i,v_j\}\) in a graph \(G\) are incident if and only if \(v \in e.\)

Example 5.2.6. Vertex Incident with Edge.

Vertex A is incident with edge \(\{A,B\}\) in the graph in FigureΒ 5.2.11, that is, A is in the edge.

Definition 5.2.7. Degree.

The degree of a vertex \(v\) is the number of edges incident with \(v.\)

Example 5.2.8. Vertex Degree.

Vertex A in the graph in FigureΒ 5.2.11 has degree 4, because \(\{A,B\}\text{,}\) \(\{A,L\}\text{,}\) \(\{A,K\}\text{,}\) and \(\{A,F\}\) are edges incident with it.
The minimum degree of all vertices in a graph \(G\) is denoted \(\delta(G)\) and the maximum degree of all vertices in a graph \(G\) is denoted \(\Delta(G).\)

Definition 5.2.9. Regular.

A graph \(G\) is regular if and only if the degree of all vertices are the same.
Figure 5.2.11. Regular Graph

Definition 5.2.12. Complete Graph.

A graph \(G\) is a complete graph, denoted \(K_n,\) if and only if \(\{v_i,v_j\} \in E\) for all \(i \ne j.\)
Figure 5.2.14. Complete Graph \(K_5\)
Note the size of a graph or subgraph is the number of vertices. Thus \(K_5\) has size 5.

Definition 5.2.15. Independent Set.

A subset \(A\) of vertices in a graph are independent if and only if no pair of vertices in \(A\) are adjacent.

Example 5.2.16. Vertex Independence Set.

\(A,C,E\) is an independent set for the graph in FigureΒ 5.2.11. Note no edge contains any two of these vertices.

Definition 5.2.17. Bipartite Graph.

A graph \(G\) is bipartite if and only if the vertices can be partitioned into two sets such that no two vertices in the same partition are adjacent.

Example 5.2.18. Understanding the bipartite definition.

This video shows how to demonstrate a graph is bipartite.
This video shows how to determine if a graph is bipartite.
This video shows how to demonstrate a graph is not bipartite.

Definition 5.2.19. Complete Biparite.

If each vertex in any partition of a bipartite graph is adjacent to all vertices in the other partition, the graph is called complete bipartite and is denoted \(K_{n,m}\) where \(n,m\) are the sizes of the partitions.

Definition 5.2.21. Subgraph.

A graph \(H=(V_H,E_H)\) is a subgraph of a graph \(G=(V_G,E_G)\) if and only if \(V_H \subseteq V_G\) and \(E_H \subseteq E_G.\)

Example 5.2.22. Subgraph.

The graph with vertex set \(V_H=\{A,B,C,G,L\}\) and edge set \(E=\{\{A,B\}, \{A,L\}, \{L,G\}, \{B,C\}, \{C,G\} \}\) is a subgraph of the graph in FigureΒ 5.2.11.

Definition 5.2.23. Induced Subgraph.

A graph \(H=(V_H,E_H)\) is an induced subgraph of a graph \(G=(V_G,E_G)\) if and only if \(V_H \subseteq V_G\) and \(E_H=\{(v_1,v_2) \in E_G : v_1,v_2 \in V_H\}\) (the set of all edges from \(G\) using only vertices in \(H\)).

Example 5.2.24. Subgraph.

The graph with vertex set \(V_H=\{A,B,C,G,L\}\) and edge set \(E=\{\{A,B\}, \{A,L\}, \{B,L\}, \{B,G\}, \{L,G\}, \{B,C\}, \{C,G\} \}\) is the subgraph of the graph in FigureΒ 5.2.11 induced by \(V_H\text{.}\)

Definition 5.2.25. Graph Complement.

The complement of a graph \(G=(V,E)\) is the graph \(H=(V,E_2)\) such that \(v_1,v_2\) are adjacent in \(H\) if and only if they are not adjacent in \(G.\)

Example 5.2.26. A Graph and its Complement.

Definition 5.2.27. Graph Dual.

The dual of a graph \(G=(V,E)\) is the graph \(H=(E,E_2)\) such that for two vertices (edges of \(G\)) are adjacent if they were incident in \(G.\)

Example 5.2.28. A Graph and its Dual.

Subsection 5.2.2 Practice

Complete graphs are also known as cliques. The complete graph on five vertices, \(K_5,\) is shown in FigureΒ 5.2.14. The size of the largest clique that is a subgraph of a graph \(G\) is called the clique number, denoted \(\Omega(G).\)

Checkpoint 5.2.33.

Draw a graph with at least five vertices. Calculate the degree of each vertex. Add these degrees. Count the number of edges. Compare the sum of the degrees to the number of edges. Add an edge. Repeat the experiment. Conjecture a relationship.

Checkpoint 5.2.34.

After the class confirms the result above prove that the number of vertices of odd degree is even.
The size of the maximum independent set in a graph \(G\) is denoted \(\alpha(G).\)

Checkpoint 5.2.36.

Re-write the definition of independent set exchanging vertices for edges. Note this is called a matching.
Figure 5.2.43. Graph Set 1
Figure 5.2.44. Graph Set 2