1. Miniconference on the Mathematics of Computation
MTH 210
Introducing graphs II
Dr. Anthony Bonato
Ryerson University

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

3. Degrees Theorem - First Theorem of Graph Theory:
also called Handshake Theorem

4. 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)

5. Subgraphs
a subgraph is a subset of the vertices and edges of a graph

6. S

7. a spanning subgraph

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

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

10. 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

11. Examples of connected components

12. Exercises

13. Introducing graphs III
Miniconference on the Mathematics of Computation
MTH 210
Introducing graphs III
Dr. Anthony Bonato
Ryerson University

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

15. 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

16. hypercubes Qn
-vertices are n-bit binary strings; two strings adjacent if they differ in exactly one bit Q3

17. Petersen graph

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

19. a graph is a forest if each component is a tree

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

21. 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).

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

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

24. Exercises