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**GRAPH Learning Outcomes Students should be able to:**

Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using adjacency matrices

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**Contents Introduction Paths and circuits**

Matrix representations of graphs

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**Introduction to Graphs**

DEF: A simple graph G = (V,E ) consists of a non-empty set V of vertices (or nodes) and a set E (possibly empty) of edges where each edge is associated with a set consisting of either one or two vertices called its endpoints. Q: For a set V with n elements, how many possible edges there?

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**Terminology Loop, parallel edges, isolated, adjacent**

Loop - an edge connects a vertex to itself Two edges connect the same pair of vertices are said to be parallel. Isolated vertex – unconnected vertex. Two vertices that are connected by an edge are called adjacent. An edge is said to be incident on each of its end points.

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**Example of a graph Vertex set = {u1, u2, u3}**

Edge set = {e1, e2, e3, e4} e1, e2, e3 are incident on u1 u2 and u3 are adjacent to u1 e4 is a loop e2 and e3 are parallel

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**Types of Graphs Directed – order counts when discussing edges**

Undirected (bidirectional) Weighted – each edge has a value associated with it Unweighted 21 April 2017 Graphs and Trees 6

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**Examples 21 April 2017 Graphs and Trees 7**

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**Special Graphs Simple – does not have any loops or parallel edges**

Complete graphs – there is an edge “between” every possible tuple of vertices Bipartite graph – V can be partitioned into V1 and V2, such that: (x,y)E (xV1 yV2) (xV2 yV1) Sub graphs G1 is a subset of G2 iff Every vertex in G1 is in G2 Every edge in G1 is in G2 Connected graph – can get from any vertex to another via edges in the graph 21 April 2017 Graphs and Trees 8

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Complete Graphs there is an edge “between” every possible tuple of vertices. |e| = C(n,2) = n. (n-1)/2

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Bipartite graph A graph is bipartite if its vertices can be partitioned into two disjoint subsets U and V such that each edge connects a vertex from U to one from V.

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Complete bipartite A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has m elements and V has n, then we denote the resulting complete bipartite graph by Km,n. The illustration shows K3,2

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Degree of Vertex Defined as the number of edges attached (incident) to the vertex. A loop is counted twice. 21 April 2017 Graphs and Trees 12

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Handshake Theorem If G is any graph, then the sum of the degrees of all the vertices of G equals twice the number of edges of G. Specifically, if the vertices of G are v1, v2, …, vn, where n is a nonnegative integer, then: The total degree of G = d(v1)+d(v2)+…+d(vn) = 2 (the number of edges of G) 21 April 2017 Graphs and Trees 13

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**Total degree of a graph is even**

Prove that the total of the degrees of all vertices in a graph is even. Since the total degree equals 2 times of edges, which is an integer, the sum of all degree is even. 21 April 2017 Graphs and Trees 14

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**Whether certain graphs exist**

Draw a graph with the specified properties or show that no such graph exists. (a) A graph with four vertices of degrees 1,1,2, and 3 (b) A graph with four vertices of degrees 1,1,3 and 3 (c) A simple graph with four vertices of degrees 1,1,3 and 3

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**Even no. of vertices with odd degree**

In any graph, there are an even number of vertices with odd degree Is there a graph with ten vertices of degrees 1,1,2,2,2,3,4,4,4, and 6? 21 April 2017 Graphs and Trees 16

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**Learning Outcomes Students should be able to:**

Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using adjacency matrices

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**Seven Bridges of Königsberg**

Is it possible for a person to take a walk around town, starting and ending at the same location and crossing each of the seven bridges exactly once? No 21 April 2017 Graphs and Trees 18

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Definitions Terminology - Walk, path, simple path, circuit, simple circuit. Walk from two vertices is a finite alternating sequence of adjacent vertices and edges Trivial walk from v to v consists of single vertex v0e1v1e2…envn 21 April 2017 Graphs and Trees 19

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Path Path – a walk that does not contain a repeated edge (may have a repeated vertex) v0e1v1e2…envn where all the ei are distinct Simple path – a path that does not contain a repeated vertex v0e1v1e2…envn where all the ei and vj are distinct

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**Circuit Closed walk – starts and ends at same vertex**

Circuit – a closed walk without repeated edge Simple circuit – a circuit with no repeated vertex except first and last

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Connectedness Connectedness – if there is a walk from one to the other

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Euler Circuits A circuit that contains every vertex and every edge of G. A sequence of adjacent vertices and edges that starts and ends at the same vertex, uses every vertex of G at least once, and uses every edge of G exactly once. 21 April 2017 Graphs and Trees 23

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**If a graph has an Euler circuit, every vertex has even degree.**

Contrapositive: if some vertex has odd degree, then the graph does not have an Euler circuit. 21 April 2017 Graphs and Trees 24

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If every vertex of nonempty graph has even degree and if graph is connected, then the graph has an Euler circuit. 21 April 2017 Graphs and Trees 25

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Euler Circuit A graph G has an Euler circuit if, and only if, G is connected and every vertex of G has even degree. 21 April 2017 Graphs and Trees 26

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Hamiltonian Path A path in an undirected graph which visits each vertex exactly once. 21 April 2017 Graphs and Trees 27

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**Hamiltonian Circuit A simple circuit that includes every vertex of G.**

A sequence of adjacent vertices and distinct edges in which every vertex of G appears exactly once, except for the first and last, which are the same. 21 April 2017 Graphs and Trees 28

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Hamiltonian Circuit Proved simple criterion for determining whether a graph has an Euler circuit No analogous criterion for determining whether a graph has a Hamiltonian circuit Nor is there an efficient algorithm for finding such an algorithm 21 April 2017 Graphs and Trees 29

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**Traveling Salesman Problem**

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**TSP One way to solve the general problem is to:**

Write down all Hamiltonian circuits Compute total distance for each Pick one for which total is minimal What if graph has 30 vertices: 29! =8.84 x 1030 different Hamiltonian circuits If each circuit could be found and total distance computed in a nanosecond, then would take: 2.8 x 1014 years!!! No known algorithm that is more efficient!!! Some that find “pretty good” solutions 21 April 2017 Graphs and Trees 31

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**Learning Outcomes Students should be able to:**

Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using adjacency matrices

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**Matrix Representations of Graphs**

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**Matrices and Connected Components**

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**Counting Walks of Length n**

Matrix multiplication 21 April 2017 Graphs and Trees 35

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**How do these graphs relate?**

= 21 April 2017 Graphs and Trees 36

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Summary

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