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Math
Applied Math
Graph Theory
Contemporary Math Chapter 6 & 7
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Gravity
Terms in this set (49)
Graph, Vertex, & Edges
A graph is a collection of one or more points, called vertices, and the paths connecting them, called edges.
Equivalent Graphs
Two graphs are considered the same if they have the same number of vertices connected in the same way, even if they look different.
Adjacent Verticies
Two vertices are called adjacent vertices if there is an edge connecting them.
Degree
The degree of a vertex is the total number of edges at the vertex.
d=2e theorem
If d is the sum of all the degrees of the vertices in a graph and e is the number of edges in the graph, then d = 2e.
Path
A path is a route in a graph.
Each edge in the graph is used at most
one time.
Any vertex can be used more than once.
Circuit
A circuit is a path that begins and ends at the same vertex.
Euler Paths
Euler Paths
• A path that uses every edge of a graph exactly once is called an Euler path.
Euler Circuit
Euler Circuits
• A circuit that uses every edge of a graph exactly once is called an Euler circuit.
Connected Graph
A graph is said to be connected if, for every pair of vertices, there is a path that contains them.
Bridge
An edge is called a bridge if its removal from the graph would change the graph from connected to disconnected.
Component
A component of a graph is a maximal connected piece of the graph.
A component is connected.
Every graph has one or more components.
Degree Theorem
Degree Theorem
• Any graph must have an even number of vertices with odd degree.
the sum of all the degrees in a graph is always an even number, so:
• Vertices with even degrees come in pairs. • Vertices with odd degrees come in pairs
If graph has no Vertices of odd degree then...
then it has at least one Euler circuit.
If graph has two vertices of odd degree then...
• It does not have an Euler circuit.
• It has at least one Euler path.
• Any Euler path starts at one of the two vertices of odd degree and ends at the other.
If the graph has more than two
vertices of odd degree, then...
• It does not have an Euler circuit.
• It does not have an Euler path.
Eulerization:
Making non Eulerian Path/circuits into Euler path/circuit
Weighted Graphs
• A graph in which each edge has a number associated with it is called a weighted graph.
The number corresponding to an edge is called the weight of the edge.
Redundant edges
are edges that can be removed while still leaving a path between the two vertices.
Subgraph
of a given graph is a set of vertices and edges chosen from among those of the original graph.
Tree
A connected graph that has no circuits is called a tree
A spanning tree is
a subgraph that:
Contains all the original vertices of the
graph. Is connected. Contains no circuits.
Minimal Spanning Tree
A spanning tree with the smallest possible total weight is called a minimal spanning tree
Kruskal's algorithm
begins with all the vertices of a graph and adds edges one by one, using the idea of acceptable and unacceptable edges.
Acceptable edges are
Edges that do not share a vertex with any edges already chosen.
Unacceptable edges are
Edges that add to a component of the subgraph, but do not add a new vertex.
Hamiltonian Path
A path that visits each vertex in a graph only once is called a Hamiltonian path.
Hamiltonian Circuit
If a Hamiltonian path begins and ends at the same vertex, it is called a Hamiltonian circuit
Hamiltonian Path or Circuit must...
Note: A Hamiltonian path or circuit must pass through every vertex of the graph, but it does not have to use every edge.
Complete Graph
A complete graph is a graph in which every pair of vertices is connected by exactly one edge.
Complete Graph Theorem
• The number of edges in a complete graph always follows a certain pattern.
• A complete graph with 𝑛 vertices
always has n (n-1)/2 edges.
N!
The symbol n! is read "n factorial".
Complete weighted graph
When costs are assigned to each edge in a complete graph, the graph is called a complete weighted graph.
Nearest-Neighbor Algorithm
1) Specify a starting vertex.
2) If unvisited vertices remain, go from the current vertex to the unused vertex with the lowest-cost connecting edge.
3) If no unvisited vertices remain, return to the starting vertex to finish forming the circuit.
Cheapest Link
Choose vertex according to the cheapest ones until a link is found.
Independent tasks
Tasks that may be done in any order
are called independent tasks
Isolated tasks
A task that does not have any
precedence relation, is called an "isolated" task.
An isolated task is an independent task.
Source
Source
A task that has no incoming arrows
Sink
A task that has no out going tasks.
Isolated task a source or sink?
An isolated task is a source as well as a
sink.
Finishing Time Is....
The weight of a path
The total weight of a selected path.
=total finishing time for the path
Maximal Path
Any path that begins at a source and
ends at a sink
Critical Path:
The highest weighted maximal path
Critical Time:
The total weight of the critical path.
A Project cannot be finished shorter than the...
A project cannot be finished shorter than the critical time
(1) no matter how many more people are involved
Project Finishing Time depends on
The project finishing time depends on
1) How many people or machines are available to do the work. ( processors)
2) How the tasks are assigned.(priority)
Priority List
A priority list is an ordered list of all the
tasks in a project.
• When tasks are performed, precedence
relations override a priority list.
• A priority list tells you which task to do first among those you can do at a certain point in time.
Increasing vs. Decreasing Time Priority List
...
An optimal schedule
is a schedule assigning processors to tasks in such
a way that it results in the shortest
possible finishing time for that project
with that number of processors.
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Verified questions
ADVANCED MATH
Write each of the following sets by listing their elements between braces. $\{x \in \mathbb{N}:-2<x \leq 7\}$
ADVANCED MATH
Given a positive integer n, use the sieve of Eratosthenes to find all primes not exceeding it.
ADVANCED MATH
The hypotenuse of a right triangle is 10 m long. How long is the side adjacent to the $21^\circ angle,$ to the nearest tenth of a metre?
ADVANCED MATH
Find the number n of leap years such that 1600 < n < Y, when (a) Y = 1825. (b) Y = 1950. (c) Y = 2075.