Lec16_17_18_Graph

Lec16_17_18_Graph - Graph Theory and Representation Graph...

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Graph Theory and Representation
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Graph Algorithms Graphs and Theorems about Graphs Graph implementation Graph Algorithms Shortest paths minimum spanning tree
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What can graphs model? Cost of wiring electronic components together. Shortest route between two cities. Finding the shortest distance between all pairs of cities in a road atlas. Flow of material (liquid flowing through pipes, current through electrical networks, information through communication networks, parts through an assembly line, etc). State of a machine. Used in Operating systems to model resource handling (deadlock problems). Used in compilers for parsing and optimizing the code.
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What is a Graph? Informally a graph is a set of nodes joined by a set of lines or arrows. 1 1 2 3 4 4 5 5 6 6 2 3
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A directed graph , also called a digraph G is a pair ( V , E ), where the set V is a finite set and E is a binary relation on V . The set V is called the vertex set of G and the elements are called vertices. The set E is called the edge set of G and the elements are edges (also called arcs ). A edge from node a to node b is denoted by the ordered pair ( a , b ). 1 2 3 4 5 6 V = { 1, 2, 3, 4, 5, 6, 7 } | V | = 7 E = { (1,2), (2,2), (2,4), (4,5), (4,1), (5,4),(6,3) } | E | = 7 Self loop 7 Isolated node
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An undirected graph G = ( V , E ) , but unlike a digraph the edge set E consist of unordered pairs. Our text uses the notation ( a, b ) to refer to a directed edge, and { a , b } for an undirected edge. A D E F B C V = { A, B, C, D, E, F } | V | = 6 E = { {A, B}, {A,E}, {B,E}, {C,F} } | E | = 4 Some texts use (a, b) also for undirected edges. So ( a , b ) and ( b , a ) refers to the same edge.
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Degree of a Vertex in an undirected graph is the number of edges incident on it. In a directed graph , the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it. A D E F B C The degree of B is 2. 1 2 4 5 The in degree of 2 is 2 and the out degree of 2 is 3. Self-loop
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Cyclic and Acyclic A path from a vertex to itself is called a cycle (e.g., v1 v2 v4 v1) If a graph contains a cycle, it is cyclic Otherwise, it is acyclic A path is simple if it never passes through the same vertex twice. 1 2 4 5
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Simple Graphs Simple graphs are graphs without multiple edges or self-loops. We will consider only simple graphs. Proposition: If G is an undirected graph then Σ deg( v ) = 2 | E | Proposition: If G is a digraph then Σ indeg( v ) = Σ outdeg( v ) = |E | v G v G v G
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A weighted graph is a graph for which each edge has an associated weight , usually given by a weight function w: E R . 1 2 3 4 5 6 .5 1.2 .2 .5 1.5 .3 1 4 5 6 2 3 2 1 3 5
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A path is a sequence of vertices such that there is an edge from each vertex to its successor. A path from a vertex to itself is called a cycle . A graph is called cyclic if it contains a cycle; otherwise it is called acyclic A path is simple if each vertex is distinct.
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Lec16_17_18_Graph - Graph Theory and Representation Graph...

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