In a graph, a path is a sequence of vertices such that each vertex is connected to the next by an edge. If the last vertex is connected to the first vertex, then the path is called a circuit.

Which of the following is a path in the graph below?

A -- B -- C -- D -- E

Which of the following is a circuit in the graph below?

A -- B -- C -- D -- E F -- G -- H

In a graph with $n$ vertices and $m$ edges, the maximum number of paths between two vertices is:

In a graph with $n$ vertices and $m$ edges, the maximum number of circuits is:

A graph is called a connected graph if there is a path between every pair of vertices. Which of the following graphs is connected?

A -- B -- C D -- E -- F

A graph is called a tree if it is connected and has no cycles. Which of the following graphs is a tree?

A -- B -- C D -- E -- F

In a graph, a spanning tree is a tree that includes all the vertices of the graph. Which of the following graphs is a spanning tree of the graph below?

A -- B -- C D -- E -- F

In a graph, a Hamiltonian path is a path that visits every vertex exactly once. Which of the following graphs has a Hamiltonian path?

A -- B -- C D -- E -- F

In a graph, a Eulerian circuit is a circuit that visits every edge exactly once. Which of the following graphs has an Eulerian circuit?

A -- B -- C D -- E -- F

Which of the following statements is true about a graph with $n$ vertices and $m$ edges?

A. If $m > n$, then the graph must have a circuit. B. If $m < n$, then the graph must be a tree. C. If $m = n$, then the graph must be a connected graph.

Which of the following statements is true about a tree with $n$ vertices?

A. A tree with $n$ vertices has $n-1$ edges. B. A tree with $n$ vertices has $n+1$ edges. C. A tree with $n$ vertices has $2n-2$ edges.

Which of the following statements is true about a graph with $n$ vertices and $m$ edges?

A. If $m < n-1$, then the graph must be a tree. B. If $m > n-1$, then the graph must have a circuit. C. If $m = n-1$, then the graph must be a connected graph.

Which of the following statements is true about a Hamiltonian path in a graph?

A. A Hamiltonian path must visit every vertex exactly once. B. A Hamiltonian path must visit every edge exactly once. C. A Hamiltonian path must start and end at the same vertex.

Which of the following statements is true about an Eulerian circuit in a graph?

A. An Eulerian circuit must visit every vertex exactly once. B. An Eulerian circuit must visit every edge exactly once. C. An Eulerian circuit must start and end at the same vertex.