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ICS 241

ICS 241. Discrete Mathematics II William Albritton, Information and Computer Sciences Department at University of Hawai’i at Manoa For use with Kenneth H. Rosen’s Discrete Mathematics & Its Applications (5 th Edition) Based on slides originally created by

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ICS 241

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  1. ICS 241 • Discrete Mathematics II • William Albritton, Information and Computer Sciences Department at University of Hawai’i at Manoa • For use with Kenneth H. Rosen’s Discrete Mathematics & Its Applications (5th Edition) • Based on slides originally created by • Dr. Michael P. Frank, Department of Computer & Information Science & Engineering at University of Florida

  2. Section 8.2: Graph Terminology • Adjacency • Let G be an undirected graph with edge set E. Let eE be (or map to) the pair {u,v}. Then we say: • u, v are adjacent / neighbors / connected. • Edge e is incident with vertices u and v. • Edge econnectsu and v. • Vertices u and v are endpoints of edge e.

  3. Degree of a Vertex • Let G be an undirected graph, vV a vertex. • The degree of v, deg(v), is its number of incident edges. (Except that any self-loops are counted twice.) • A vertex with degree 0 is called isolated. • A vertex of degree 1 is called pendant.

  4. Handshaking Theorem • Let G be an undirected (simple, multi-, or pseudo-) graph with vertex set V and edge set E. Then • Corollary: Any undirected graph has an even number of vertices of odd degree • Note: this means that vertices of odd degrees always occur in one or more pairs

  5. Directed Adjacency • Let G be a directed (possibly multi-) graph, and let e be an edge of G that is (or maps to) (u,v). Then we say: • u is adjacent tov, v is adjacent fromu • ecomes from u, e goes to v. • e connects u to v, e goes from u to v • the initial vertex of e is u • the terminal vertex of e is v

  6. Directed Degree • Let G be a directed graph, v a vertex of G. • The in-degree of v, deg(v), is the number of edges going to v. • The out-degree of v, deg(v), is the number of edges coming from v. • The degree of v, deg(v):deg(v)+deg(v), is the sum of v’s in-degree and out-degree.

  7. Directed Handshaking Theorem • Let G be a directed (possibly multi-) graph with vertex set V and edge set E. Then: • Note that the degree of a node is unchanged by whether we consider its edges to be directed or undirected.

  8. Class Exercise • Exercise 7 (p. 555) • Each pair of students should use only one sheet of paper while solving the class exercises

  9. Special Graph Structures • Special cases of undirected graph structures: • Complete graphsKn • CyclesCn • WheelsWn • n-CubesQn • Bipartite graphs • Complete bipartite graphsKm,n

  10. Complete Graphs • For anynN, a complete graph on n vertices,Kn, is a simple graph with n nodes in which every node is adjacent to every other node:u,vV: uv{u,v}E. K1 K4 K3 K2 K5 K6

  11. Cycles • For anyn3, a cycle on n vertices, Cn, is a simple graph where V={v1,v2,… ,vn}and E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}. C3 C4 C5 C6 C8 C7

  12. Wheels • For anyn3, a wheelWn, is a simple graph obtained by taking the cycle Cnand adding one extra vertex vhub and n extra edges {{vhub,v1}, {vhub,v2},…,{vhub,vn}} W3 W4 W5 W6 W8 W7

  13. n-cubes (hypercubes) • For any nN, the hypercube Qn is a simple graph consisting of two copies of Qn-1 connected together at corresponding nodes. Q0 has 1 node. • Note: this represents bit strings, in which adjacent vertices differ in exactly 1 bit position Q0 Q1 Q4 Q2 Q3

  14. Bipartite Graphs • Def’n.: A graph G=(V,E)is bipartite (two-part) if V = V1∩V2where V1V2=and eE: v1V1,v2V2: e={v1,v2}. • In English: The graph canbe divided into two partsin such a way that all edges go between the two parts. V2 V1

  15. Class Exercise • Exercises 19, 21, 23 (p. 555) • Each pair of students should use only one sheet of paper while solving the class exercises • If can traverse an ODD number of distinct edges to return to a vertex, then NOT bipartite (p. 550)

  16. Complete Bipartite Graphs • For m,nN, the complete bipartite graphKm,nis a bipartite graph where |V1| = m, |V2| = n, and E = {{v1,v2}|v1V1  v2V2}. • That is, there are m nodes in the left part, n nodes in the right part, and every node in the left part is connected to every node in the right part. K4,3

  17. Class Exercise • Exercise 25 (p. 555) • Each pair of students should use only one sheet of paper while solving the class exercises

  18. Subgraphs • A subgraph of a graph G=(V,E) is a graph H=(W,F)whereWVand FE. G H

  19. a b c a b c e d d f Graph Unions • The unionG1G2of two simple graphs G1=(V1, E1)and G2=(V2,E2)is the simplegraph(V1V2, E1E2). 

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