Math 1029 Test 3 Ledet

A connected graph has 30 even vertices and no odd vertices. Determine whether the graph has an Euler path​ (but not an Euler​ circuit), an Euler​ circuit, or neither an Euler path nor an Euler​ circuit, and explain why.

An Euler circuit because it has no odd vertices

A Hamilton circuit must begin and end at the same edge.

False

A​ z-score describes how many standard deviations a data item in a normal distribution lies above or below the​ _____.

Mean

T/F: A Hamilton path must contain every vertex in the graph exactly once.

True

T/F: An Euler path can start and end at the same vertex.

True

T/F: The​ mean, median, and mode of a normal distribution are all equal.

True

T/F: An edge can be a part of a path only once.

True because a path is a sequence of adjacent vertices and the edges connecting them.

In a normal​ distribution, approximately​ __________% of the data items fall within 1 standard deviation of the​ mean, approximately​ __________% of the data items fall within 2 standard deviations of the​ mean, and approximately​ __________% of the data items fall within 3 standard deviations of the mean.

68, 95, 99.7

If an edge is removed from a connected graph and leaves behind a disconnected​ graph, such an edge is called a​ _____.

Bridge

If​ n% of the items in a distribution are less than particular data​ item, we say the data item is in the nth​ __________ of the distribution.

Percentile

A finite set of points connected by line segments or curves is called​ ____. The points are called​ ____. The line segments or curves are called​ ____. Such a line segment or curve that starts and ends at the same point is called a​ ____.

A graph, vertices, edges, loop

If there is at least one edge connecting two vertices in a​ graph, the vertices are called​ _____. A sequence of such vertices and the edges connecting them is called a​ _____. If this sequence of vertices and connecting edges begins and ends at the same​ vertex, it is called a​ _____.

Adjacent, path, circuit

A connected graph has 38 even vertices and two odd vertices. Determine whether the graph has an Euler path​ (but not an Euler​ circuit), an Euler​ circuit, or neither an Euler path nor an Euler​ circuit, and explain why.

An Euler path because it has exactly two odd vertices

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. This method is called the​ _______ Method.

Brute Force

A graph that has an edge between each pair of its vertices is called​ a/an ________ graph. If such a graph has n​ vertices, the number of Hamilton circuits in the graph is given by the factorial expression​ ______.

Complete, (n-1)!

The number of edges that connect to a vertex is called the​ _____ of the vertex.

Degree

A connected graph has even vertices​ A, B, and​ C, and odd​ vertices, D and E. Each Euler path must begin at vertex D and end at vertex​ _______, or begin at vertex​ _______ and end at vertex​ _______.

E, E, D

Two graphs that have the same number of vertices connected to each other in the same way are called​ _____.

Equivalent

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

Euler

A path that passes through each edge of a graph exactly one time is called​ a(n) ______ path.

Euler

A connected graph with 116 even vertices and no odd vertices.

Euler circuit

A path that crosses each border once is an ______________. The graph has ____________ vertices.​

Euler path, exactly two odd

T/F: A connected graph with exactly one odd vertex has at least one Euler circuit.

False

A path that passes through each vertex of a graph exactly once is called​ a/an _______ path. Such a path that begins and ends at the same vertex and passes through all other vertices exactly once is called​ a/an _______ circuit.

Hamilton, Hamilton

A method that approximates the solution to the salesperson problem is called the​ ______ Method. This method involves continually choosing an edge with the smallest​ ______.

Nearest Neighbor, weight

A connected graph with 25 even vertices and three odd vertices.

Neither

A connected graph has 46 even vertices and four odd vertices. Determine whether the graph has an Euler path​ (but not an Euler​ circuit), an Euler​ circuit, or neither an Euler path nor an Euler​ circuit, and explain why.

Neither because it has more than two odd vertices

A​ city's sanitation department is reviewing their routes. By labeling each intersection as a​ vertex, they discover they have 43 vertices in a certain area of the city. Does that area have an Euler path if six of those intersections have 3 roads meeting at that​ point?

No

A connected graph has at least one Euler​ circuit, which, by​ definition, is also an Euler​ path, if the graph has​ _______ odd vertices. An Euler circuit can start and end at​ _______ vertex.

No, any

A connected graph has no Euler paths and no Euler circuits if the graph has more than two​ _______ vertices.

Odd

What does a​ z-score measure?

The number of standard deviations above or below the mean a specified data item is.

T/F: In a normal​ distribution, the​ z-score for the mean is 0.

True because a​ z-score describes how many standard deviations a data item in a normal distribution lies above or below the mean.

A connected graph has at least one Euler​ path, but no Euler​ circuit, if the graph has exactly​ _______ odd​ vertices/vertex.

Two

A graph whose edges have numbers attached to them is called​ a/an ______ graph. The numbers shown along the edges of such a graph are called the​ ______ of the edges. The problem of finding a Hamilton circuit for which the sum of these numbers is a minimum is called the​ ______ salesperson problem. Such a Hamilton circuit is called the​ ______ solution.

Weighted, weights, traveling, optimal