91Ó°ÊÓ

Write the adjacency matrix of each graph. The complete bipartite graph \(K_{2,3}\)

Write the adjacency matrix of each graph. The complete graph on five vertices \(K_{5}\)

Tell whether the given path in the graph is (a) A simple path (b) A cycle (c) A simple cycle (b, c, d, a, b, e, d, c, b)

Write an algorithm that finds the lengths of the shortest paths from a given vertex to every other vertex in a connected, weighted graph \(G\).

Write an algorithm that finds the lengths of the shortest paths between all vertex pairs in a simple, connected, weighted graph having \(n\) vertices in time \(O\left(n^{3}\right)\).

A connected, planar graph has nine vertices having degrees \(2,2,2,3,3,3,4,4,\) and \(5 .\) How many edges are there? How many faces are there?

Give an example of a graph that has an Euler cycle that is also a Hamiltonian cycle.

Give an example of a graph that has an Euler cycle and a Hamiltonian cycle that are not identical.

Show that any graph having four or fewer vertices is planar.

Show that if \(n \geq 3,\) the complete graph on \(n\) vertices \(K_{n}\) contains a Hamiltonian cycle.