Graph Theory: Trees
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The document discusses properties and theorems related to trees in graph theory. Some key points include: - A tree is a connected acyclic graph with n vertices that has n-1 edges. - There is a one-to-one correspondence between labeled trees with n vertices and sequences of n-2 labels, as proven by Cayley's theorem. - Every connected graph has at least one spanning tree, which is a subgraph that contains all vertices. Fundamental circuits are formed when a chord is added to a spanning tree. - Cyclic interchange can be used to generate all possible spanning trees by adding and removing edges.
Graph Theory: Trees
- 1. Trees A.B.M. Ashikur Rahman Wee-3: Trees 1
- 2. Definition • A connected acyclic graph Wee-3: Trees 2
- 3. Properties of Tree • Theorem 3.1 – There is one and only one path between every pair of vertices in a tree, T. • Theorem 3.2 – If in a graph G there is one and only one path between every pair of vertices, G is a tree. • Theorem 3.3 – A tree with n vertices has n-1 edges. Wee-3: Trees 3
- 4. Properties of Tree • Theorem 3.4 – Any connected graph with n vertices and n-1 edges is a tree. • Theorem 3.5 – A graph is a tree if and only if it is minimally connected. • Theorem 3.6 – A graph G with n vertices, n-1 edges and no circuits is connected. Wee-3: Trees 4
- 5. Summary of the Properties • A graph G with n vertices is called a tree if – • G is connected & is circuitless, or • G is connected & has n-1 edges, or • G is circuitless and has n-1 edges, or • There is exactly one path between every pair of vertices in G, or • G is a minimally connected graph. Wee-3: Trees 5
- 6. Properties of tree Question: Cycle in tree. Possible? Answer: Wee-3: Trees 6 In graph theory: Nope. In jungle: May be.
- 7. Pendant Vertices in a Tree • Theorem 3.7 – In any tree (with two or more vertices), there are at least two pendant vertices. • Finding monotonically increasing subsequence of 4, 1, 13, 7, 0, 2, 8, 11, 3 Wee-3: Trees 7
- 8. Distance & Centers in a Tree • Distance • distance d(vi, vj) between two of its vertices vi and vj is the length of the shortest path • Metric – is a function f that satisfies the conditions below • Nonnegativity: f(x,y) ≥ 0, and f(x,y) = 0 if and only if x = y. • Symmetry: f(x,y) = f(y,x). • Triangle inequality: f(x,y) ≤ f(x,z) + f(z,y) for any z. Wee-3: Trees 8
- 9. Distance & Centers in a Tree • Theorem 3.8 – The distance between vertices of a connected graph is a metric. • Eccentricity(also referred to as assosiated number or separation) E(v) of a vertex v in a graph G is distance from v to the vertex farthest from v in G; that is • Center : A vertex with minimum eccentricity in graph G is called a center of G. Wee-3: Trees 9
- 10. Distance & Centers in a Tree • Theorem 3.9 – Every tree has either one or two centers. • Radius & Diameter Wee-3: Trees 10
- 11. Rooted & Binary Trees • Rooted trees with 4 vertices • Binary Trees • Internal Vertex Wee-3: Trees 11
- 12. Properties Binary Trees • The number of vertices n in a binary tree is always odd. This because there is exactly one vertex of even degree and the remaining n-1 vertices are of odd degrees. Since the number of vertices of odd degree is even, n-1 is even. Hence n is odd. • Let p be the number of pendant vertices in a binary tree T. Then n-p-1 is the number of vertices of degree three. Therefore the number of edges in T equals Hence Wee-3: Trees 12
- 13. Properties Binary Trees • Height : The maximum level, lmax, of any vertex in a binary tree is called the height of the tree. Wee-3: Trees 13
- 14. Weighted Path Length • Minimize: ∑wjlj • A Coke machine is to identify, by a sequence of tests, the coin that is put into the machine. Only pennies, nickels, dimes, and quarters can go through the slot. Let us assume that the probabilities of a coin being a penny, a nickel, a dime, and a quarter are .05, .15, .5, and .30, respectively. Each test has the effect of partitioning the four types of coins into two complementary sets and asserting the unknown coin to be in one of the two sets. Thus for four coins we have 23 - 1 such tests. If the time taken for each test is the same, what sequence of tests will minimize the expected time taken by the Coke machine to identify the coin? Wee-3: Trees 14
- 15. Weighted Path Length • Minimize: ∑wjlj Wee-3: Trees 15
- 16. Counting Trees • On 1857, Arthur Cayley discovered tree while studying structural isomers of saturated hydrocarbons CkH2k+2 molecule. • The total number of vertices in such a graph is n = 3k + 2 • And total number of edge • e = .5 x ( sum of degree ) • e = .5 x ( 4k + 2k +2) • e = 3k + 1 This structure is tree. Wee-3: Trees 16
- 17. Counting Trees • When • n = 4 • 16 labeled trees • 2 unlabeled trees Wee-3: Trees 17
- 18. Cayley’s Theorem • Theorem 3.10 – The number of labeled trees with n(≥2) vertices is nn-2. • Proof: • Let the n vertices of the tree T be labeled 1, 2, 3, .., n. • Remove the pendant vertex(and the edge incident on it) having the smallest label. • Repeat until n-2 steps or only 2 vertices are left. • The tree T defines the sequence : (b1, b2, …, bn-2) uniquely. Wee-3: Trees 18
- 19. Cayley’s Theorem • Remove a1 = 2, get b1 = 1 • Remove a2 = 4, get b2 = 1 • Remove a3 = 1, get b3 = 3 • Remove a4 = 3, get b4 = 5 • Remove a5 = 6, get b5 = 5 • Remove a6 = 7, get b6 = 5 • Remove a7 = 5, get b7 = 9 • Only vertex 8 and 9 are left • So our sequence of b = (1,1,3,5,5,5,9); is only unique to this labeled graph Wee-3: Trees 19 5 24 3 7 6 8 9 1
- 20. Cayley’s Theorem • Conversely given a sequence b of n-2 labels, an n-vertex tree can be constructed uniquely • First construct the sequence of a = (1, 2, 3, …, n). • Pick the first number in sequence a which is not in b and pair them. • Remove both the vertices from the list. • Continue pairing until sequence of b has no element left. • Finally join the last two remaining vertices of sequence a. Wee-3: Trees 20
- 21. Cayley’s Theorem Reconstruction: b = (1,1,3,5,5,5,9) a = (1,2,3,4,5,6,7,8,9) *Follow the class lecture for reconstruction process Wee-3: Trees 21
- 22. Cayley’s Theorem & Proof • For example, given a sequence (4, 4, 3, 1, 1) • Given sequence is the sequence of b • So, we construct a = (1, 2, 3, 4, 5, 6, 7) • Pair (2, 4) as a1 = 2, b1 = 4 • Pair (5, 4) as a2 = 5, b2 = 4 • Pair (4, 3) as a3 = 4, b3 = 3 • Pair (3, 1) as a4 = 3, b4 = 1 • Pair (6, 1) as a5 = 6, b5 = 1 • Finally pair (1, 7) Wee-3: Trees 22
- 23. Cayley’s Theorem & Proof • For each of the n-2 element in sequence b we can choose any one of the n numbers, thus froming nn-2; (n-2) tuples, each defining a distinct labeled tree of n vertices. • Since each tree defines one of the sequence uniquely, there is a 1-to-1 correspondence between the trees and the nn-2 sequences. • Thus the theorem is proved. Wee-3: Trees 23
- 24. Spanning Trees Wee-3: Trees 24 • A tree T is said to be a spanning tree of connected graph G if T is a subgraph of G and T contains all vertices of G. • Maxiaml Tree/Subgraph, skeleton • Sppaning forest of disconnected graph • Finding Spanning Tree • Find Circuits • delete an edge from the circuit • continue
- 25. Spanning Trees • Theroem 3.11 – Every connected graph has at least one spanning tree. Wee-3: Trees 25 • Branch (an edge which is part of spanning tree T and graph G), • Chord (an edge which is not part of spanning tree T but of graph G)
- 26. Spanning Trees • Theroem 3.12 – With respect to any of its spanning trees, a connected graph of n vertices and e edges has n-1 tree branches and e-n+1 chords. • Electrical network with e edges and n nodes, how to be sure that we don’t have a circuit? • How many walls have to be broken to drain the lands? Wee-3: Trees 26
- 27. Spanning Trees: Rank & Nullity • n, e, k are independent of each other except for two constrains • n-k ≥ 0 • e-n+k ≥ 0 • Rank, r = n-k; number of branches • Nullity, μ = e-n+k; number of chords • Rank + Nullity = total number of edges Wee-3: Trees 27
- 28. Fundamental Circuits • Theroem 3.13 – A connected graph G is a tree if and only if adding an edge between any two vertices in G creates exactly one circuit. • Consider a spanning tree T in a connected graph G. Adding any chord to T will create exactly one circuit. Such a circuit, formed by adding a chord to a spanning tree, is called a fundamental circuit. • How many fundamental circuits does a graph have? Wee-3: Trees 28
- 29. Fundamental Circuits • Has 8 fundamental circuits • A circuit may be fundamental with respect to a spanning tree but not other spanning tree of the same graph • Has 75 total circuits • A set of edges comprising a circuit will not be fundamental if it has more than one chord Wee-3: Trees 29
- 30. Finding All Spanning Tree • Cyclic interchange: • start with a given spanning tree • add a chord forming a fundamental circuit • delete an appropriate branch Wee-3: Trees 30
- 31. Cyclic interchange Wee-3: Trees 31 • Can we start from any spanning tree and get a desired spanning tree by a number of cyclic exchanges? • Can we get all spanning trees of a given graph in this fashion? • How long will we have to continue exchanging edges? • Is there a preferred spanning tree for starting?
- 32. Cyclic interchange Wee-3: Trees 32 • The distance between two spanning trees of a graph G is defined as the number of edges of G present in one tree but not in the other • Written as d(Ti ,Tj) d(Ti ,Tj) = 1 2 N(Ti ⊕ Tj ) *N(g) denote the number of edges in a graph g
- 33. Metric Wee-3: Trees 33 The distance between the spanning trees of a graph is a metric. That is, it satisfies- 1. d(Ti, Tj) ≥ 0 and d(Ti, Tj) = 0 if and only if Ti = Tj, 2. d(Ti, Tj) = d(Tj, Ti), 3. d(Ti, Tj) ≤ d(Ti, Tk) + d(Tk, Tj).
- 34. Spanning Trees In A Weighted Graph • Edges have weights • Minimal spanning tree • Kruskal’s or Prim’s algorithm Wee-3: Trees 34