Graph theory algorithms

Graph theory and the Graph ADT and graph theory and complex networks and graph theory data science
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Dr.SethPatton,United Kingdom,Teacher
Published Date:22-07-2017
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ECE 250 Algorithms and Data Structures Graph theory and the Graph ADT Douglas Wilhelm Harder, M.Math. LEL Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada © 2006-2013 by Douglas Wilhelm Harder. Some rights reserved.Graph theory 2 Outline A graph is an abstract data type for storing adjacency relations – We start with definitions: • Vertices, edges, degree and sub-graphs – We will describe paths in graphs • Simple paths and cycles – Definition of connectedness – Weighted graphs – We will then reinterpret these in terms of directed graphs – Directed acyclic graphsGraph theory 3 Outline We will define an Undirected Graph ADT as a collection of vertices V = v , v , ..., v 1 2 n – The number of vertices is denoted by V = n – Associated with this is a collection E of unordered pairs v , v termed i j edges which connect the vertices There are a number of data structures that can be used to implement abstract undirected graphs – Adjacency matrices – Adjacency listsGraph theory 4 Graphs Consider this collection of vertices V = v , v , ..., v 1 2 9 where V = nGraph theory 5 Undirected graphs Associated with these vertices are E = 5 edges E = v , v , v , v , v , v , v , v , v , v 1 2 3 5 4 8 4 9 6 9 – The pair v , v indicates that both vertex v is adjacent to vertex v and j k j k vertex v is adjacent to vertex v k jGraph theory 6 Undirected graphs We will assume in this course that a vertex is never adjacent to itself – For example, v , v will not define an edge 1 1 The maximum number of edges in an undirected graph is VV1  V 2 EV O  2 2 Graph theory 7 An undirected graph Example: given the V = 7 vertices V = A, B, C, D, E, F, G and the E = 9 edges E = A, B, A, D, A, E, B, C, B, D, B, E, C, E, C, F, D, EGraph theory 8 Degree The degree of a vertex is defined as the number of adjacent vertices degree(A) = degree(D) = degree(C) = 3 degree(B) = degree(E) = 4 degree(F) = 1 degree(G) = 0 Those vertices adjacent to a given vertex are its neighborsGraph theory 9 Sub-graphs A sub-graph of a graph a subset of the vertices and a subset of the edges that connected the subset of vertices in the original graphGraph theory 10 Vertex-induced sub-graphs A vertex-induced sub-graph is a subset of a the vertices where the edges are all edges in the original graph that originallyGraph theory 11 Paths A path in an undirected graph is an ordered sequence of vertices (v , v , v , ..., v ) 0 1 2 k where v , v is an edge for j = 1, ..., k j – 1 j – Termed a path from v to v 0 k – The length of this path is k Graph theory 12 Paths A path of length 4: (A, B, E, C, F)Graph theory 13 Paths A path of length 5: (A, B, E, C, B, D)Graph theory 14 Paths A trivial path of length 0: (A)Graph theory 15 Simple paths A simple path has no repetitions other than perhaps the first and last vertices A simple cycle is a simple path of at least two vertices with the first and last vertices equal – Note: these definitions are not universal theory 16 Connectedness Two vertices v , v are said to be connected if there exists a path i j from v to v i j A graph is connected if there exists a path between any two vertices An unconnected graph A connected graphGraph theory 17 Weighted graphs A weight may be associated with each edge in a graph – This could represent distance, energy consumption, cost, etc. – Such a graph is called a weighted graph Pictorially, we will represent weights by numbers next to the edgesGraph theory 18 Weighted graphs The length of a path within a weighted graph is the sum of all of the edges which make up the path – The length of the path (A, D, G) in the following graph is 5.1 + 3.7 = 8.8Graph theory 19 Weighted graphs Different paths may have different weights – Another path is (A, C, F, G) with length 1.2 + 1.4 + 4.5 = 7.1Graph theory 20 Weighted graphs Problem: find the shortest path between two vertices – Here, the shortest path from A to H is (A, C, F, D, E, G) with length 5.7

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