Graph 1 3 2 4 Types of graphs

Graph • 1 3 2 4

Types of graphs • 1 1 3 2 1 4 3 1 1 4 3 4 SYMMETRIC 1 1 1 2 3 1 1 2 1 1 1 3 1 1 1 4 0 1 3 2 SYMMETRIC & IRREFLEXIVE 1 2 2 4 4 3 1 1 2 4 1 1 1 4 1 2 1 3 (IRREFLEXIVE) RELATION 1 1 3 1 2 4 ? 2 1 3 3 4 2 3 1 3 4 2 2 4 4

More graph terminology • 1 2 1 is adjacent to 2 and 3 3 4 1 1 2 3 3 2 1 1 2 3 1 3 2 2 1 3 3 2 1 1 3 3 2 2

Isomorphism 3 3 3 1 2 3 2 4 4 3 6 2 1 4 3 5 6 3 5 3 3 (3, 3, 3, 3) 5 6 1 (3, 3, 3, 3) (3, 3, 2, 2, 2, 2)

Isomorphism • 1 1 3 3 2 2 3 2 4 2 1 5 self-complementary graph 3 3 2 3 3 2 1 1 4 2

Isomorphism • Yes, 2 3 3 2 4 2 1 5 1 4 5 1 Yes, there are 10 4 2 5 6 8 7 3

Handshake lemma 1 • 3 2 Is it possible for a self-complementary graph with 100 vertices to have exactly one vertex of degree 50 ? Answer. No 4 2 4 6 1 3 5 � definition of isomorphism definition of complement

Subgraphs, Paths and Cycles • 1 1 3 2 4 3 4 2 subgraph 1 1, {1, 2}, 2, {2, 3}, 3, {3, 1}, 1 is a cycle 3 2 4 1, {1, 2}, 2, {2, 1}, 1, {1, 3}, 3 is a walk (not path)

Special graphs • 1 2 5 4 3 3 2 4 1 5 111 110 100 010 101 001 011

Connectivity • 1 1, {1, 3}, 3, {3, 4}, 4 path from 1 to 4 3 2 connected components 1 4 3 2 connected 4 no path from 1 to 4 not connected = disconnected

Connectivity • Show that the complement of a disconnected graph is connected ! • What is the maximum number of edges in a disconnected graph ? • What is the minimum number of edges in a connected graph ? 1 2 3 4 5

Distance, Diameter and Radius • 1 3 2 1 3 2 4 4

Independent set and Clique • 1 3 2 4 1 2 3 4 5

Bipartite graph • 1 2 1 3 3 bipartite 1 2 4 3 4 2 not bipartite 4

Bipartite graph • 3 2 4 2 1 3 3 1 2 1 4 5 2 1 3 6 4 5 3 2 4 5 1 . . . ? �