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Discrete Mathematics

Discrete Mathematics. Chapter-8 Graphs. §8.1 Introduction to Graphs. Def 1. A (simple) graph G=(V,E) consists of a nonempty set V of vertices, and E, a set of unordered pairs of distinct elements of V called edges. eg. G=(V,E), where V={ v 1 ,v 2 ,…,v 7 }

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Discrete Mathematics

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  1. Discrete Mathematics Chapter-8 Graphs

  2. §8.1 Introduction to Graphs • Def 1. A (simple) graph G=(V,E) consists of a nonempty set V of vertices, and E, a set of unordered pairs of distinct elements of V called edges. • eg. G=(V,E), where V={ v1,v2,…,v7 } E={ {v1,v2}, {v1,v3}, {v2,v3} {v3,v4}, {v4,v5}, {v4,v6} {v4,v7}, {v5,v6}, {v6,v7} } V1 V5 V3 V4 V6 V7 V2

  3. multiedge eg. • Def 2.Multigraph simple graph + “兩點間允許多條邊” V1 V5 V3 V4 V6 V2 V7

  4. ( loop即 ) simple graph + multiedge + loop • Def 3. pseudo graph : eg.

  5. Table 1. Graph Terminology

  6. Note: directed multigraph 中 U U V V 是 multiedge 邊為(u,v),(u,v) 不是 multiedge 邊為(u,v),(v,u) Exercise : 3,5,6,7,9

  7. §8.2 Graph Terminology (undirected) • Def 1. Two vertices u and v in a graph G are called adjacent in G if {u,v} is an edge of G. • Def 2. The degree of a vertex v, denoted by deg(v), in an (undirected) graph is the number of edges incident with it. (Note : 點跟點相連 : adjacent 點跟邊相連 : incident ) (Note : loop要算2次)

  8. Example 1. What are the degree of the vertices in the graph H ? • Sol : deg(a)=4 deg(b)=6 b a c deg(c)=1 deg(d)=5 deg(e)=6 deg(f)=0 e d f H

  9. eg. “ f ” in Example 1. • Def. A vertex of degree 0 is called isolated. Thm 1. (The Handshaking Theorem) Let G=(V,E) be an undirected graph with n edges (i.e., |E|=n). Then Pf :每條edge {u,v}會貢獻一個degree給u跟v

  10. eg. Example 1. there are 11 edges, and Example 2. How many edges are there in a graph with 10 vertices each of degree 6 ? Sol : 10  6 = 2n => n=30

  11. Thm 2. An undirected graph has an even number of vertices of odd degree. • Def 3. G: directed graph , G=(V,E) (u,v)E : u is adjacent to v v is adjacent from u u : initial vertex , v : terminal vertex u v

  12. Def 4. G=(V,E) : directed graph vV deg(v) : # of edges with v as a terminal. (in-degree) deg+(v) : # of edges with v as a initial vertex

  13. c a b e d f a is adjacent to b , b is adjacent from a a : initial vertex of (a,b) b : terminal vertex of (a,b) end deg-(a)=2, deg+(a)=4 deg-(b)=2, deg+(b)=1 : : deg-(f)=0, deg+(f)=0 • Example 3.

  14. b a • Thm 3. Let G=(V,E) be a digraph. Then pf : 每個 edge貢獻一個 out–degree 給 a 一個 in–degree 給 b

  15. n 2 Note : | E | = Def : The complete graph on n vertices, denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices. Example 4. K3 K1 K2 K4

  16. C5 C6 • Example 5. The cycle Cn, n≧3, consists of n vertices v1,v2,…,vn and edges {v1,v2}, {v2,v3},…,{vn-1,vn},{vn,v1}. | E | = n

  17. W6 W5 • Example 6. Wn : (wheel), Cn中加一點連至其餘n點(n≧3) | V | = n + 1 | E | = 2n

  18. Def 5. A simple graph G=(V,E) is called bipartite if V can be partitioned into V1 and V2, V1∩V2=, such that every edge in the graph connect a vertex in V1 and a vertex in V2. Example 8. v1 v2 v3 ∴ C6 is bipartite. v4 v5 v6

  19. a c a b g e c b f d f d g e Example 10. Is the graph G bipartite ? Yes !

  20. Example 11. Complete Bipartite graphs (km,n) K3,3 K2,3 Note. | E | = mn

  21. a a b e e b c c d Def 6. A subgraph of a graph G=(V,E) is a graph H=(W,F) where W  V and F  E. (注意 F 要連接 W 裡的點) • Example 14. A subgraph of K5 subgraph of K5 K5

  22. b c a b c a b c a d e d d e f f • Example 15. • Def 7. The union of two simple graph G1=(V1,E1) and G2=(V2,E2) is the simple graph G1∪G2=(V1∪V2,E1∪E2) G1 G2 G1∪G2

  23. ex35上面A simple graph G=(V,E) is called regular if every vertex of this graph has the same degree. A regular graph is called n-regular if deg(v)=n , vV. eg. K4 : is 3-regular. Exercise : 5,7,21,23,25,35,37

  24. b c a d e §8.3 Representing Graphs and Graph Isomorphism ※Adjacency list Example 1. Use adjacency list to describe the simple graph given below. Sol :

  25. Example 2. (digraph) b a c e d

  26. a b c d ※Adjacency Matrices Def.G=(V,E) : simple graph,V={v1,v2,…,vn}. (順序沒關系) A matrix A is called the adjacency matrix of G if A=[aij]nxn , where a b c d a Example 3. b c undirected graph 的連 通矩陣必 “對稱” d b d c a b d c a

  27. 故矩陣 未必對稱 a b c d a a b b Example 5. (Pseudograph) (矩陣未必是0,1矩陣.) c d d c Def. If A=[aij] is the adjacency matrix for the directed graph, then 1 , if vi vj aij = 0 , otherwise

  28. G is isomorphic to H u2 u1 u4 u3 v1 v2 v4 v3 ※Isomorphism of Graphs • Def 1. The simple graphs G1=(V1,E1) and G2=(V2,E2) are isomorphic if there is an one-to-one and onto function f from V1 toV2 with the property that a~b in G1 iff f(a)~f(b) in G2,a,bV1 f is called an isomorphism. G H

  29. v1 v2 u2 u1 • Example 8. v4 v3 u3 u4 H G ※Isomorphism Graphs 必有 : (1) 相同的點數。 (2) 相同的邊數。 (3) 相同的degree分佈。 f(u1) = v1 f(u3) = v3 f(u2) = v4 f(u4) = v2

  30. b b a c a c e e d d ∵ H 有 degree = 1 的點，G 沒有 ∴ G  H ※給定二圖，判斷它們是否isomorphic的問題一般來說不易解，而且答案常是否定的。 Example 9. Show that G and H are not isomorphic. Sol :

  31. b s t e f w x g h d c z y v u a Example 10. Determine whether G and H are isomorphic. G H Sol : ∵ G 中 degree 為 3 的點有d, h, f, b 它們不能接成 4-cycle 但 H 中 degree 為3的點有s, w, z, v 它們可接成 4-cycle ∴ 不是 isomorphic. 另法 : G 中 degree 為 3 的點，旁邊都只連了另一個 deg = 3 的點 但 H 中 deg = 3 的點旁邊都連了 2 個 deg = 3 的點。

  32. H G v1 Example 11. Show that G H u1 u2 v3 u5 v2 u6 v6 v5 u4 u3 v4 Exercise : 3,7,14,17,19,23,37,39

  33. §8.4: Connectivity Def. 1,2 : • In an undirected graph, a path of length n from u to v is a sequence of adjacent vertices going from vertex u to vertex v. (e.g., P: u=x0, x1, x2, …, xn=v) • Note. A path of length n has n+1 vertices，n edges • A path is a circuit if u=v. • A pathtraversesthe vertices along it. • A path or circuit issimpleif it contains no vertex more than once. (simple circuit通常稱為cycle)

  34. Paths in Directed Graphs • Same as in undirected graphs, but the path must go in the direction of the arrows. Figure 5.

  35. Connectedness • Def. 3: • An undirected graph is connected (連通) iff there is a path between every pair of distinct vertices in the graph. • Def: • Connected component: maximal connected subgraph. (一個不連通的圖會有好幾個component) • Example 6

  36. Connectedness Def:A cut vertex separates one connected component into several components if it is removed. Def:A cut edge separates one connected component into two components if it is removed. Example 8.

  37. Connectedness in Digraphs Def. 4:A directed graph is strongly connected iff there is a directed path from a to b for any two vertices a and b. Example 9.

  38. Connectedness in Digraphs Def. 5: It is weakly connected iff the underlying undirected graph (i.e., with edge directions removed) is connected . • Note strongly implies weakly but not vice-versa. Example 9.

  39. Paths & Isomorphism • Note that connectedness, and the existence of a circuit or simple circuit of length k are graph invariants with respect to isomorphism. Example 12. Example 13.

  40. Counting Paths between Vertices Let A be the adjacency matrix of graph G. Theorem 2: The number of paths of length rfrom vi to vj is equal to (Ar)i,j. (The notation (M)i,j denotes mi,j where [mi,j] = M.) Example 14. Exercise: 15, 23, 25, 26

  41. §8.5: Euler & Hamilton Paths Def. 1: • An Euler circuit in a graph G is a simple circuit containing every edge of G. • An Euler path in G is a simple path containing every edge of G. Example 1.

  42. Useful Theorems Thm. 1:A connected multigraph has an Euler circuit iff each vertex has even degree. Thm. 2:A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree. Example 4.

  43. Hamilton Paths Def. 2: • A Hamilton circuit is a circuit that traverses each vertex in G exactly once. • A Hamilton path is a path that traverses each vertex in G exactly once. Example 5.

  44. Useful Theorems Thm. 3 (Dirac’s Thm.):If (but not only if) G is connected, simple, has n3 vertices, and deg(v)n/2 v, then G has a Hamilton circuit. Exercise: 3, 5, 7, 21, 26, 28, 42, 43.

  45. §8.6: Shortest Path Problems Def: Graphs that have a number assigned to each edge are called weighted graphs. Shortest path Problem: Determining the path of least sum of the weights between two vertices in a weighted graph.

  46. Dijkstra’s Algorithm Figure 4. Exercise: 3

  47. §8.7: Planar Graphs Def. 1: A graph is calledplanar if it can be drawn in the plane without any edge crossing. Example 1:K4 is planar. Example 3: K3,3 is nonplanar. Exercise: 2,3,4

  48. §8.8: Graph Coloring Def. 1: A coloring of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent vertices are assigned the same color. Def. 2: The chromatic number of a graph is the least number of colors needed for a coloring of this graph. (denoted by c(G))

  49. §8.8: Graph Coloring Example 1 Example 2~4:c(Kn)=n, c(Km,n)=2, c(Cn)=2,3. Example 3’: If G is a bipartite graph, c(G)=2. Theorem 1. (The Four Color Theorem)The chromatic number of a planar graph isno greater than four. Exercise: 6, 7

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