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 \(\Delta(G).\)
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.
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.
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 \(H\)).
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 \(V_H\text{.}\)
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 \(G.\)
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 \(\Omega(G).\)
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.