Lecture 16 graph introduction

Lecture 16 graph introduction

• 1.
  Lecture 16 Introduction toGraph Abirami Sivaprasad
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  Vertices and Edges Definition:A graph is a collection (nonempty set) of vertices and edges Vertices: can have names and properties(set of nodes ) Edges: connect two vertices, can be labeled, can be directed Adjacent vertices: there is an edge between them
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  ExampleGraph1 Vertices: A,B,C,D Edges: AB,AC, BC, CD A B C D A C B D Two ways to draw the same graph
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  Undirected graphs When theedges in a graph have no direction, the graph is called undirected
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  Directed Graph When theedges in a graph have a direction, the graph is called directed (or digraph) Note: if the graph is directed, the order of the vertices in each edge is important !!
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  More definitions :Path A list of vertices in which successive vertices are connected by edges A B C B A C D A B C A B C A B C D B A B A C A B C D
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  More definitions : SimplePath No vertex is repeated. A B C D D C A D C B A B A B C A B C D
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  More definitions :Cycle Simple path with distinct edges, except that the first vertex is equal to the last A B C A B A C B C B A C A B C D A graph without cycles is called acyclic graph.
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  More definitions :Loop An edge that connects the vertex with itself A B C D
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  Connected and Disconnectedgraphs Connected graph: There is a path between each two vertices Disconnected graph : There are at least two vertices not connected by a path. Examples of disconnected graphs: A B C D A B C D
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  Graphs and Trees Tree:an undirected graph with no cycles, and a node chosen to be the root A B C E D Source graph:
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  Graphs and Trees A C EB D C A E B D Tree1: root A Tree2: root C
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  A spanning treeof an undirected graph A sub-graph that contains all the vertices, and no cycles. If we add any edge to the spanning tree, it forms a cycle, and the tree becomes a graph A B C D A B C D graph spanning tree
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  Examples A B C D AB C D A B C D All spanning trees of the graph on the previous slide
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  Complete graphs Graphs withall edges present – each vertex is connected to all other vertices Dense graphs: relatively few of the possible edges are missing Sparse graphs: relatively few of the possible edges are present A B C D E A complete graph
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  Weighted graphs andNetworks Weighted graphs – weights are assigned to each edge (e.g. road map) Networks: directed weighted graphs (some theories allow networks to be undirected) A B D C 3 2 4 1 2
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  Graph Representation Adjacency matrix Adjacencylists
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  Adjacency matrix –undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD A B C D A 0 1 1 1 B 1 0 0 1 C 1 0 0 1 D 1 1 0 0 The matrix is symmetrical A B C D
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  Adjacency matrix –directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA A B C D A 0 1 1 0 B 0 0 0 1 C 0 0 0 0 D 1 0 0 0 A B C D
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  Adjacency lists –undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD Heads lists A B C D B A D C A D A B A B C D
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  Adjacency lists –directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA Heads lists A B C B D C = D A A B C D
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  Thank u