Tree, function and graph
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The document outlines the fundamental concepts of functions in discrete mathematics, including definitions of domains, co-domains, one-to-one functions, onto functions, and bijective functions. It further discusses trees as a specific type of graph, including concepts such as spanning trees and minimum spanning trees, alongside various graph terminologies like diameter, isolated vertices, and bipartite graphs. Additionally, algorithms for constructing minimum spanning trees, namely Prim's and Kruskal's algorithms, are mentioned.
Tree, function and graph
- 2. Function Requirements There are rules for functions to be well defined, or correct. • No element of the domain must be left unmapped. • No element of the domain may map to more than one element of the co-domain.
- 4. Definition 1 • Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A.
- 5. Functions are specified as assignments as in Figure 1.
- 6. DEFINITION 2 If f is a function from A to B, we say that A is the domain of f and B is the co-domain of f.
- 7. Figure 2 represents a function f from A to B.
- 8. Is it a function? If not state the reasons.
- 9. One-to-One one-to-one or an injunction function never assign the same value to two different domain elements.
- 10. Explain why?
- 11. When every member of the codomain is the image of some element of the domain then functions are called onto functions or a surjection. Onto Functions.
- 12. Explain Why?
- 13. Check these!
- 14. Bijective(both one-to-one and onto) • If the function is both one-to- one and onto that function is bijective.
- 15. L23 15 Trees A very important type of graph in CS is called a tree: Real Tree transformation
- 16. L23 16 Trees A very important type of graph in CS is called a tree: Real Abstract Tree Tree transformation
- 17. Tree!A tree is a connected undirected graph with no simple circuits. Find which of them are not trees and why!
- 18. A tree with n vertices has n − 1 edges.
- 20. Spanning Trees • Let G be a simple graph. A spanning tree of G is a subgraph of G that is a tree containing every vertex of G. • A simple graph with a spanning tree must be connected, because there is a path in the spanning tree between any two vertices.
- 22. Other 4 Spanning trees of G
- 23. Minimum Spanning Trees A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible sum of weights of its edges.
- 24. Find Minimum Spanning Tree
- 25. We will discuss two algorithms for constructing minimum spanning trees.
- 26. Prim’s Algorithm
- 27. Solution
- 28. Try this
- 32. The diameter of a graph is the length of the shortest path between the most distanced nodes. The highest value of the topological distance of this matrix is the diameter of the graph (d=4). Diameter of a Graph
- 33. Find out the diameter of the following graph If P, A1, A2, A5, A3, A6, Q are the nodes then D=?
- 34. Isolated vertex • A vertex of degree zero is called isolated. It follows that an isolated vertex is not adjacent to any vertex.
- 35. Adjacent edge • When two vertices of a graph are connected by an edge, these vertices are adjacent vertices. • When two edges meet at the same vertex, those two edges are said to be adjacent.
- 36. Write the adjacent edges of vertex a.
- 37. An Unconnected, undirected graph • Clearly, the graph is not connected.However, the figure actually represents the undirected graph G8=(v , e) , given by
- 38. Write the degree of the entire graph.
- 39. L23 39 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 40. L23 40 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 41. L23 41 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 42. L23 42 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 43. L23 43 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 44. L23 44 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 45. L23 45 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 46. L23 46 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 47. L23 47 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 48. L23 48 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it:
- 49. L23 49 Bipartite Graphs EG: C4 is a bichromatic: And so is bipartite, if we redraw it: Q: For which n is Cn bipartite?
- 50. Check whether bipartite or not!
- 51. Best wishes!