13 May 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Introduction to Discrete Mathematics Graph Theory Part2

13 May 2009Instructor: Tasneem Darwish2 Outlines Definitions and Examples Paths and Cycles Isomorphism of Graphs Trees

13 May 2009Instructor: Tasneem Darwish3 A simple graph is a graph which has no loops and every pair of vertices is connected by a unique edge. Definitions and Examples

13 May 2009Instructor: Tasneem Darwish4 The complete graph Kn with n vertices can be described as follows:  It has vertex set V = {v1, v2,...,vn}  The graph Kn is clearly regular with degree n − 1. Definitions and Examples

13 May 2009Instructor: Tasneem Darwish5 The properties of a bipartite graph where the vertex set V has the partition {V1,V2}, are as follows:  The graph need not be simple.  each edge must join a vertex of V1 to a vertex of V2.  Given v1 ∈ V1 and v2 ∈ V2, there may be more than one edge joining them or no edge joining them.  There are no loops in the graph. Definitions and Examples

13 May 2009Instructor: Tasneem Darwish6 The complete bipartite graph on n and m vertices, denoted Kn,m, has |V1| = n and |V2| = m. and it must be simple. Definitions and Examples

13 May 2009Instructor: Tasneem Darwish7 The adjacency matrix is necessarily symmetric If there are no loops at v i then its degree is the sum of the entries in the i th column (or i th row) of the matrix Definitions and Examples

13 May 2009Instructor: Tasneem Darwish8 Examples The following is the adjacency matrix A of the graph represented in the figure. For this graph two properties can be concluded:  by considering the leading diagonal we note that there is only one loop—from v1 to itself.  the last row (or column) of zeros indicates that v4 is an isolated vertex connected to no vertices at all (including itself). Definitions and Examples

13 May 2009Instructor: Tasneem Darwish9 Examples The null graph with n vertices has the n × n zero matrix as its adjacency matrix, since there are no edges whatsoever. 3.A complete graph has adjacency matrix with zeros along the leading diagonal and ones everywhere else. Definitions and Examples

13 May 2009Instructor: Tasneem Darwish10 ∑ is a subgraph of Γ if we can obtain a diagram for ∑ by erasing some of the vertices and/or edges from a diagram of Γ Definitions and Examples

13 May 2009Instructor: Tasneem Darwish11 Example 10.4 Graphs ∑ and Γ have vertex sets V Γ = {v1, v2, v3, v4, v5} and V ∑ = {v1, v2, v4, v5} and respective adjacency matrices ∑ is a subgraph of Γ Definitions and Examples

13 May 2009Instructor: Tasneem Darwish12 Paths and Cycles

13 May 2009Instructor: Tasneem Darwish13 An edge sequence is any finite sequence of edges which can be traced on the diagram of the graph without removing pen from paper. An edge sequence may repeat edges and go round loops several times. In a path we are not allowed to ‘travel along’ the same edge more than once In a simple path we do not ‘visit’ the same vertex or the same edge more than once. The edge sequence or path is closed if we begin and end the ‘journey’ at the same place. Paths and Cycles

13 May 2009Instructor: Tasneem Darwish14 Examples Let Γ be the graph represented below; examples of edge sequences in Γ are: (i) e1, e3, e4, e5, e3; (ii) e3, e3; (iii) e2, e3, e4; (iv) e4, e3; (v) e4, e5, e2. (i) is a closed edge sequence (ii) is a closed sequence (iii) is a cycle (iv) is a simple path from v2 to v1. (v) is a path with initial and final vertices v2, v1 respectively Paths and Cycles

13 May 2009Instructor: Tasneem Darwish15 An arbitrary graph naturally splits up into a number of connected subgraphs, called its (connected) components Examples 10.6 Each of the following graphs has two component Paths and Cycles

13 May 2009Instructor: Tasneem Darwish16 Examples 10.7 In the complete bipartite graph K 4,4 the vertices have been partitioned into the sets {1, 2, 3, 4} and {a, b, c, d}. The graph is connected and every vertex has degree 4, so K 4,4 is Eulerian One Eulerian path beginning at the vertex 1 has the following vertex sequence: 1, a, 2, b, 3, c, 4, d, 1, c, 2, d, 3, a, 4, b, 1 Paths and Cycles

13 May 2009Instructor: Tasneem Darwish17 Example 10.8 The following figure illustrates Hamiltonian cycles in two graphs. Paths and Cycles