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Graph terminologies & special type graphs
The document discusses various graph terminologies and special types of graphs. It defines undirected and directed graphs, and describes degrees, handshaking theorem, and other properties. Special graph types covered include complete graphs, cycles, wheels, n-cubes, and bipartite graphs. It provides examples of constructing new graphs from existing ones and an application using a bipartite graph model for employee skills and job assignments.
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Graph terminologies & special type graphs
- 1.
- 2. GRAPH TERMINOLOGIES & SPECIAL TYPE GRAPHS
- 3. TYPES OF GRAPHS Undirected Graphs Directed Graphs Special Simple Graphs Special Simple Graphs
- 4. UNDIRECTED GRAPHS Thegraph in which u and v(vertices) are endpoints of an edge of graph G is called an undirected graph G. U V LOOP
- 5. DEGREE The numberof edges for which vertex is an endpoint. The degree of a vertex v is denoted by deg(v).
- 6. DEGREE If deg(v)= 0, v is called isolated. If deg(v) = 1, v is called pendant.
- 7. THE HANDSHAKING THEOREM Let G = (V, E) be an undirected graph with E edges. Then 2|E| = vV deg(v) Note that This applies if even multiple edges and loops are present.
- 8. EXAMPLE How manyedges are there in a graph with 10 vertices each of degree 5? o vV deg(v) = 6·10 = 60 o 2E= vV deg(v) =60 o E=30
- 9. EXAMPLE How manyedges are there in a graph with 9 vertices each of degree 5? o vV deg(v) =5 · 9 = 45 o 2E= vV deg(v) =45 o 2E=45 o E=22.5 o Which is not possible.
- 10. DIRCTED GRAPHS When(u, v) is an edge of the graph G with directed edges. The vertex u is called the initial vertex of (u, v), and v is called the terminal or end vertex of (u, v). The initial vertex and terminal vertex of a loop are the same.
- 11. DEGREE The indegree of a vertex v, denoted deg−(v) is the number of edges which terminate at v. Similarly, the out degree of v, denoted deg+(v), is the number of edges which initiate at v. vV deg－ (v) = vV deg＋ (v)= |E|
- 12. EXAMPLE • Findthe in-degree and out-degree of each vertex in the graph G with directed edges. The Directed Graph G. 12
- 13. EXAMPLE The in-degreesin G are deg−(a) = 2 deg−(b) = 2 deg−(c) = 3 deg−(d) = 2 deg−(e) = 3 deg−(f ) = 0 The out-degrees in G are deg+(a) = 4 deg+(b) = 1 deg+(c) = 2 deg+(d) = 2 deg+(e) = 3 deg+(f ) = 0
- 14. SOME SPECIAL SIMPLEGRAPHS There are several classes of simple graphs. 14 Complete graphs Cycles Wheels n-Cubes Bipartite Graphs New Graphs from Old
- 15. COMPLETE GRAPHS Thecomplete graph on n vertices, denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices. The Graphs Kn for 1≦ n ≦6.
- 16. COMPLETE GRAPHS K5& K6 is important because it is the simplest non-planar graph. It cannot be drawn in a plane with nonintersecting edges.
- 17. CYCLE The cycleCn, n 3, consists of n vertices v1, v2, . . ., vn and edges {v1, v2}, {v2, v3 } ,. . . , {vn-1, vn} , and {vn , v1}. The cycles C3, C4, C5, and C6 are displayed below. The Cycles C3, C4, C5, and C6.
- 18. WHEEL We obtainthe wheel Wn when we add an additional vertex to the cycle Cn, for n 3, and connect this new vertex to each of the n vertices in Cn, by new edges. The Wheels W3, W4, W5, and W6. 18
- 19. N-CUBES Qn isthe graph with 2n vertices representing bit strings of length n. An edge exists between two vertices that differ by one bit position. 19
- 20. EXAMPLE A commonway to connect processors in parallel machines. Intel Hypercube.
- 21. EXAMPLE The n-cubeQn for n = 1, 2, and 3. 21
- 22. BIPARTITE GRAPH Asimple graph G is called bipartite if its vertex set V and be partitioned into two disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex in V2 . When this condition holds, we call the pair (V1 , V2 ) a bipartition of the vertex set V.
- 23. BIPARTITE GRAPH A Star network is a K(1,n) bipartite graph. V1(n=ODD) V2(n=EVEN)
- 24. BIPARTITE GRAPH V1={v1,v3,v5}; V1={v2,v4,v6} This is the graph of Hexagonal.
- 25.
- 26. NEW GRAPHS FROMOLD A sub-graph of a graph G= (V, E) is a graph H =(W, F), where W V and F E. A sub-graph H of G is a proper sub-graph of G if H G . 26
- 27. NEW GRAPHS FROMOLD The graph G shown below is a sub-graph of K5. A Sub-graph of K5. 27
- 28. NEW GRAPHS FROMOLD The union of two simple graphs G1= (V1, E1) & G2= (V2, E2) is the simple graph with vertex set V1 V2 and edge set E1 E2 . The union of G1 and G2 is denoted by G1 G2 . 28
- 29. NEW GRAPHS FROMOLD The Simple Graphs G1 and G2 Their Union G1∪G2. 29
- 30. APPLICATION OF SPYCIALTYPES OF GRAPHS Suppose that there are m employees in a group and j different jobs that need to be done where m j. Each employee is trained to do one or more of these j jobs. We can use a graph to model employee capabilities. We represent each employee by a vertex and each job by a vertex. For each employee, we include an edge from the vertex representing that employee to the vertices representing all jobs that the employee has been trained to do. 30
- 31. APPLICATION OF SPYCIALTYPES OF GRAPHS Note that the vertex set of this graph can be partitioned into two disjoint sets, the set of vertices representing employees and the set of vertices representing jobs, and each edge connects a vertex representing an employee to a vertex representing a job. Consequently, this graph is bipartite.
- 32. APPLICATION OF SPYCIALTYPES OF GRAPHS Modeling the Jobs for Which Employees Have Been Trained. To complete the project, we must assign jobs to the employees so that every job has an employee assigned to it and no employee is assigned more than one job. 32