91Ó°ÊÓ

Characterize the adjacency matrix of the complete graph \(K_{n}\).

Find the number of vertices in the bipartite graph \(K_{m, n}\).

Find the number of edges in the bipartite graph \(K_{m, n}\).

Identify the general form of the adjacency matrix for \(K_{m, n}\).

Let \(G\) be a graph with \(n\) vertices and \(e\) edges. Let \(M\) and \(m\) denote the maximum and minimum of the degrees of vertices in \(G,\) respectively. Prove that \(m \leq 2 e / n \leq M\).

A simple graph \(G\) is regular if every vertex has the same degree. If every vertex has degree \(r, G\) is \(r\) -regular with \(r\) the degree of the graph. Draw a regular graph with the given properties. \(r=1\) and two vertices.

A simple graph \(G\) is regular if every vertex has the same degree. If every vertex has degree \(r, G\) is \(r\) -regular with \(r\) the degree of the graph. Draw a regular graph with the given properties. \(r=2\) and three vertices.

Prove that a connected graph with \(n\) vertices has at least \(n-1\) edges. (Hint: Use induction.)

A simple graph \(G\) is regular if every vertex has the same degree. If every vertex has degree \(r, G\) is \(r\) -regular with \(r\) the degree of the graph. Draw a regular graph with the given properties. \(r=2\) and four vertices.

A simple graph \(G\) is regular if every vertex has the same degree. If every vertex has degree \(r, G\) is \(r\) -regular with \(r\) the degree of the graph. Draw a regular graph with the given properties. \(r=3\) and four vertices.