MTH 210 Introducing graphs II Dr Anthony Bonato

MTH 210 Introducing graphs II Dr. Anthony Bonato Ryerson University

Degrees • the degree of a node x, written deg(x) is the number of edges incident with x

Degrees Theorem - First Theorem of Graph Theory: • also called Handshake Theorem

Corollary 1. 2: In every graph, there an even number of odd degree nodes. • for example, there is no order 19 graph where each vertex has order 9 (i. e. 9 regular)

Subgraphs • a subgraph is a subset of the vertices and edges of a graph

S

a spanning subgraph

Special graphs • cliques (complete graphs): Kn – n nodes – all distinct nodes are adjacent • cocliques (independent sets): Kn – n nodes – no edges

• cycles Cn -n nodes on a circle • paths Pn -n nodes on a line -length is n-1

Connected graphs • a graph is connected if every pair of distinct vertices is joined by at least one path • otherwise, a graph is disconnected • connected components: maximal connected induced subgraphs

Examples of connected components

Exercises

MTH 210 Introducing graphs III Dr. Anthony Bonato Ryerson University

Special graphs, continued • bipartite graphs: union of two independent sets or colours

• bipartite cliques (bicliques, complete bipartite graphs) Ki, j: a set X of vertices of cardinality i, and one Y of cardinality j, such that all edges are present between X and Y, and these are the only edges

• hypercubes Qn -vertices are n-bit binary strings; two strings adjacent if they differ in exactly one bit Q 3

Petersen graph

Trees • a graph is a tree if it is connected and contains no cycles (that is, is acyclic or circuit-free)

• a graph is a forest if each component is a tree

Leaves • in a tree, a vertex of degree one is a leaf (or terminal vertex or endvertex) • all other vertices are internal

Key fact: If T is a tree, then T has at least two leaves (i. e. vertices with degree 1).

Key fact: If T is a tree, then there is exactly one path connecting any two vertices in T.

Key fact*: If T is a tree of order n, then T has n-1 edges.

Exercises