Graph Theory Emily Belyea Paton Vinal Paul Friedman Emily Belyea Paton Vinal Paul Friedman

Definitions  A graph consists of two types of elements, vertices and edges. Every edge has two endpoints in the set of vertices, and connects or joins the two endpoints.  A cycle is edges connected that make a continuous circuit.  A graph consists of two types of elements, vertices and edges. Every edge has two endpoints in the set of vertices, and connects or joins the two endpoints.  A cycle is edges connected that make a continuous circuit. Vertex Edge A vertex is simply drawn as point or dot. The vertex set of G is usually represented by V(G), or V when there is no danger of confusion. The order of a graph is the number of its vertices, for example, |V(G)|. An edge is drawn as a line connecting two vertices, called end vertices. An edge with end vertices x and y is represented by xy. The edge set of G is usually written as E(G), or E when there is no possible confusion. cycle

 A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph, meaning there can only be one line connecting two vertices.  A tree is a connected graph that contains no cycles.  A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph, meaning there can only be one line connecting two vertices.  A tree is a connected graph that contains no cycles. Bridge tree

 A weighted graph associates a label, a weight, with every edge found in the graph. Weights are usually listed as real numbers. The numbers are usually restricted to rational numbers or integers. Certain algorithms require further restrictions on weights.  Acyclic graph is a graph that contains no cycles.  A weighted graph associates a label, a weight, with every edge found in the graph. Weights are usually listed as real numbers. The numbers are usually restricted to rational numbers or integers. Certain algorithms require further restrictions on weights.  Acyclic graph is a graph that contains no cycles Weighted acyclic graph

The Problem  Our farm is installing a new irrigation system for a field. The well is located off the field and is the water source for the system. There is a need to find the amount of piping that will be used for the system. Each vertex represents a sprinkler head that provides water to the field. What is the minimum amount of piping that can be used for the field?

The Field WELL

Total weight of the Tree  In order to find the total weight of the tree, add up the total of the weighted edges.  =518  In order to find the total weight of the tree, add up the total of the weighted edges.  =518

Minimal Spanning Tree Algorithm 1.Each edge of the graph needs to be labeled, starting with the smallest amount weighted to the largest. (e1,e2,e3…en) weighted as: the weight of e1<e2<en 2.Start the graph sequence with e1. 3.Continue with the next smallest weighted edge, and continue until there is none left without making a circuit. 1.Each edge of the graph needs to be labeled, starting with the smallest amount weighted to the largest. (e1,e2,e3…en) weighted as: the weight of e1<e2<en 2.Start the graph sequence with e1. 3.Continue with the next smallest weighted edge, and continue until there is none left without making a circuit.

The Field (labeled weighted) WELL e5 e2 e8 e4 e11 e10 e12 e14 e17 e1 e13e18 e6 e19 e20 e24 e15 e22 e23 e21 e3 e9 e7 e16

The Field (removing unnecessary pipes) WELL e5 e2 e8 e4 e11 e10 e12 e14 e17 e1 e13e18 e6 e15 e22 e21 e3 e9 e7 e16 The dotted lines are the lines that are unnecessary for the irrigation system, they are removed.

The Field as a Minimal Spanning Tree WELL e5 e2 e8 e4 e11 e10 e12 e14 e17 e1 e13e18 e6 e15 e22 e21 e3 e9 e7 e16

Totaling the Minimal Spanning Tree  =406  The new total saves the farm 112 feet of piping, saving 21.62%  =406  The new total saves the farm 112 feet of piping, saving 21.62%

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