Miniconference on the Mathematics of Computation 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 are 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 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

Introducing graphs III Miniconference on the Mathematics of Computation 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 Q3

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 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