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Module 5 – Networks and Decision Mathematics. Chapter 23 – Undirected Graphs. 23.1 Introduction and Definition. A network is a set of connections between people or destinations or activities.
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Module 5 – Networks and Decision Mathematics Chapter 23 – Undirected Graphs
23.1 Introduction and Definition • A network is a set of connections between people or destinations or activities. • The graphs are made up of vertices (or nodes) and edges (or arcs). The vertices are points and the edges are lines joining these points. Two edges can cross each other and not have a vertex at the point where they meet.
The degree of a vertex is the number of edges that come off it. • In the graph below, deg(A)= deg(C)= deg(D) =3 and deg(B)=5. • Each edge connects two vertices.
Multiple edges exist when two vertices are connected by morethan one edge. In the graph below, verticesEandFare connected bymultiple edges. • Number of edges = sum of the degrees of the vertices 2 • A loop connects a vertex to itself. For example, at vertex H there is a loop, so vertex H is of degree 3.
A simple graph is a graph with no multiple edges or loops. • An isolated vertex is not joined by any edges. • In the graph below, vertexJis an isolated vertex
A degenerate graph has all verticesisolated. • A graph is connected if there is a path between each pair of vertices (each vertex can be reached from any other vertex)
A subgraph of a graph is made up of some edges and vertices of the original graph. • The vertices must include all of the end-points of the selected edges. Some examples are given below
A bipartite graph is a graph whose set of vertices can be split into two disjoint sets (houses A, B, C) conneted to utility outlets (gas, water. Electricity) and each edge has a vertex in each set.
A path can be thought of as a sequence of edges. • E.g. A path in the graph below from A to F could be AB, BG, GE, EC, CD, DE, EG, GF
A circuit is a sequence of edges linking successive vertices that starts and finishes at the same vertex. • E.g. One circuit is A B C D F A. Another • circuit is E F D E.
A completegraph is a simple graph that has every vertex connected to every other vertex. • Note: For a complete graph on n vertices, each vertex is of degree (n - 1), and there are: n ( n – 1 ) 2 edges.
Example : For the graph below: (a) find the number of edges (b) find the number of vertices (c) find the degree of each vertex How many edges does a complete graph with 15 vertices have?
Matrix Representation • A graph can be represented by a table or an adjacency matrix in which the numbers denote how many edges connect each pair of vertices. Compare the graph below to it’s associatedadjacency matrix.
Noticein the matrix where vertex A meetsvertex A that a ‘2’ is used. When a ‘2’ appears in the matrix in a place representing how many edges are joining a vertex to itself, the ‘2’ represents one loop. A ‘4’ would represent two loops at the same vertex. If a ‘2’ appears in any other spot in the matrix it indicates that there are two edges joining those two vertices.
