MAT 2720 Discrete Mathematics Section 8 1 Introduction

MAT 2720 Discrete Mathematics Section 8. 1 Introduction to Graphs http: //myhome. spu. edu/lauw

HW l l This HW has discovery-based problems. You are highly encouraged to work in a group of two for this HW assignment. Discussions are very helpful and is a integral part of the learning process. You need to start your HW ASAP. You are not helping yourself (time-wise) if you wait for Thursday morning.

Goals l Introduction to Graph Theory • Definitions and Examples • More Definitions

Graph Theory Real World Problems Graph Theory Problems

Tons of Applications l l l Computer Technology Related Networks Biological System Networks (You have read the paragraphs from the last HW)

Example 1 Graph Structure of the Web

Example 2 Phylogenetic Tree

Definitions l l A graph G consists of a set of elements called vertices V and a set of elements called edges E. Each edge joins two vertices.

Edges l l An edge may be labeled by a pair of vertices, for instance e = (u, v). Note that in this context, (u, v) is not an ordered pair, i. e. (u, v)=(v, u)

Definitions l l l e is said to be incident on u and v. Two vertices are adjacent vertices if they are joined by an edge. Isolated vertex is a vertex without incident edges.

Example 3

Definitions l l l In a graph, two or more edges joining the same pair of vertices are parallel edges. An edge joining a vertex to itself is a loop. A graph with no parallel edges or loops is a simple graph.

Definitions l A weighted graph is a graph where each edge is assigned a numerical label or “weight”.

Definitions l A graph is a directed graph or digraph if each edge has been associated with an ordered pair of vertices, i. e. each edge has a direction.

Example 4

Definitions l The complete graph on n vertices, denoted by Kn, is the simple graph with n vertices in which there is an edge between every pair of distinct vertices.

Definitions l A graph G=(V, E) is a bipartite graph if there exists subsets V 1 and V 2 of V such that 1. 2. each edge in E is incident on one vertex in V 1 and one vertex in V 2

Definitions l 1. 2. The complete bipartite graph on m and n vertices, denoted by Km, n, is the simple graph whose vertex set is partitioned into in sets V 1 with m vertices and V 2 with n vertices the edge set consists of all edges of the form with

Group Explorations Is G=(V, E) bipartite?

Templates Terminology: Without Loss of Generality (WLOG) is a term used in proofs to indicate that an assumption is being made that does not introduce new restrictions to the problem