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