MAT 2720 Discrete Mathematics Section 8.1 Introduction

HW… This HW due on noon Monday 5/14/12. Put it in my mailbox or hand it to me in my office. Solutions will be available after 3pm that day. Remind me if you do not get it shortly after 3pm.

HW… Optional early submission: If you turn in your HW by 5pm Friday 5/11/12, the grader will try to grade it and return it to you by 3pm Monday 5/14/12 (No guarantee). You will also get 2 bonus points for this HW assignment (guaranteed).

Goals Introduction to Graph Theory Definitions and Examples More Definitions

Graph Theory Real World Problems Graph Theory Problems

Tons of Applications Computer Technology Related Networks Biological System Networks

Example 1 Graph Structure of the Web

Example 2 Phylogenetic Tree

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

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

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

Example 5 Is G=(V,E) bipartite?

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