Math

A graph that has an edge between each pair of its vertices is called a ______?

Complete Graph

A method that determines the solution to the traveling salesperson problem involves listing all Hamilton circuits and selecting the circuit with the minimum sum of weights.

Brute Force method

A tree that is created from a weighted graph and that has the smallest possible weight is called the_______?

Minimum spanning tree

Is every path is a circuit?

No, because not every path ends at the same vertex where it starts.

A graph whose edges have numbers attached to them is called a ______?

Weighted graph.

Is every circuit is a path?

Yes, because a circuit is a path that begins and ends at the same vertex.

Is every Euler circuit is an Euler path?

Yes, because both an Euler circuit and an Euler path are paths that travel through every edge of a graph once and only once. An Euler circuit also begins and ends on the same vertex.

Can a graph can be drawn in many equivalent ways?

Yes, because the placement of vertices and the shapes of the edges are unimportant.

If the graph has exactly two odd​ vertices/vertex then ____?

A connected graph has at least one Euler​ path, but no Euler​ circuit.

If the graph has more than two odd vertices _____?

A connected graph has no Euler paths and no Euler circuits.

A circuit that travels through every edge of a graph exactly once is called​ a/an _____?

Euler circuit.

A path that passes through each edge of a graph exactly one time?

Euler path.

A path that begins and ends at the same vertex and passes through all other vertices exactly once is called a ________?

Hamilton circuit.

A path that passes through each vertex of a graph exactly once is called a_____?

Hamilton path.

A procedure that yields the minimum spanning tree of a graph is called________?

Kruskal's Algorithm, always pick the edge with the smallest available weight, but avoid creating any circuits.

Does The Nearest Neighbor Method provides exact solutions to the traveling salesperson problems?

No, because The Nearest Neighbor Method is meant only to approximate the optimal solution.

Does every complete graph that has a Hamilton circuit has at least one Euler circuit?

No, because there are complete​ graphs, which have Hamilton​ circuits, that have vertices with odd​ degree, so they cannot have Euler circuits.

A tree that is created from another connected graph and that contains all of the connected​ graph's vertices, is​ connected, and contains no circuits is called​ a/an________?

Spanning tree

A tree is a complete graph?

The statement is false. A tree does not have an edge between each pair of its vertices.

Most connected graphs have many possible spanning trees?

The statement is true. Each spanning tree is a skeleton holding together the connected​ graph, and for most graphs there are many such skeletons.

If a graph has no odd vertices​ (all even​ vertices)?

Then it has at least one Euler circuit​.

A graph that is connected and has no circuits is called​ a/an ________?

Tree, every edge is​ a/an bridge.