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Lec13_Graph_Theory_Part_1

# Lec13_Graph_Theory_Part_1 - Discrete Structures TDS1191...

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[Lecture 13][Graph Theory Part 1] Discrete Structures 1 Discrete Structures TDS1191 - Lecture 13 - Graph Theory Part 1

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[Lecture 13][Graph Theory Part 1] Discrete Structures 2 Definition A graph G = (V, E) where V : set of vertices and E : set of pairs, called edges . Definition of Graph v z w u u w v The vertex set of G is often denoted by V(G) and the edge list of G by E(G). Hence the first graph is expressed fully formally as G 1 = ({u, v, w, z}, {(u,w), (w,u), (v,v), (v,w), (w,v), (w,z), (z,w)}) Vertices – points or dots in a graph Edges – line segments joining vertices u w z
[Lecture 13][Graph Theory Part 1] Discrete Structures 3 v z w u Connected non-simple graph Parallel edges loop l u w v z Disconnected simple graph Definition A simple graph is a graph that does not have any loops or parallel edges . Terminology Let G = (V, E) - An edge e that connects a point v back to itself is called loop . - Two distinct edges with same set of endpoints are said to be parallel . - A vertex with no edges connected to it is called isolated . - A graph is called connected if there is a path from any vertex to any other vertex, otherwise the graph is disconnected . Terminology Let G = (V, E) - An edge e that connects a point v back to itself is called loop . - Two distinct edges with same set of endpoints are said to be parallel . - A vertex with no edges connected to it is called isolated . - A graph is called connected if there is a path from any vertex to any other vertex, otherwise the graph is disconnected . Types of Graphs

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[Lecture 13][Graph Theory Part 1] Discrete Structures 4 Definition: A complete graph is a graph in which every 2 distinct vertices are joined by exactly one edge . The complete graph on n vertices is denoted by K n. Definition: A complete graph is a graph in which every 2 distinct vertices are joined by exactly one edge . The complete graph on n vertices is denoted by K n. K 1 K 2 K 3 K 4 K 5 Cycles : The cycle C n , n 3, consists of n vertices v 1 v 2 v n and edges { v 1 ,v 2 } { v 2 ,v 3 },…., { v n-1 ,v n } and { v n ,v 1 }. The cycles C 3 ,C 4 and C 5 are displayed below. Cycles : The cycle C n , n 3, consists of n vertices v 1 v 2 v n and edges { v 1 ,v 2 } { v 2 ,v 3 },…., { v n-1 ,v n } and { v n ,v 1 }. The cycles C 3 ,C 4 and C 5 are displayed below. C 3 C 4 C 5 Some Special Simple Graphs
[Lecture 13][Graph Theory Part 1] Discrete Structures 5 Wheels : A wheel W n is obtained when we add an additional vertex to the cycle C n, for n 3, and connect this new vertex to each of the n vertices in C n, with new edges. Wheels : A wheel W n is obtained when we add an additional vertex to the cycle C n, for n 3, and connect this new vertex to each of the n vertices in C n, with new edges. W 3 W 4 W 5 n- Cubes : The n- dimensional cube, denoted by Q n is the graph that has vertices representing the 2 n bit strings of length n. Two vertices are adjacent if and only if the bit strings that they represent differ in exactly one bit position.

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Lec13_Graph_Theory_Part_1 - Discrete Structures TDS1191...

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