Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Definitions and examples of key graph terminology, including neighbors, degree, complete graphs, and bipartite graphs. It covers concepts such as adjacency, degree theorem, complete graphs, and bipartite graphs. Students of Discrete Mathematics can use this document as a reference for understanding the basics of graph theory.
Typology: Lecture notes
1 / 13
This page cannot be seen from the preview
Don't miss anything!
Two vertices u and v in an undirected graph G = (V , E ) are called adjacent (or neighbors) in G if {u , v } โ E. In other words, u and v are neighbors if there is an edge e which has u and v as its endpoints. Such an edge e is called incident with the vertices u and v and e is said to connect u and v.
v
u
The neighbourhood of a vertex v in an undirected graph G = (V , E ) is the set of all neighbours of v , which we will denote by N(v ). If A โ V , then we define N(A) =
โ v โ A N(v^ )^ to be the set of all neighbours of some vertex in A.
v
w q
u
z r
t
Let G = (V , E ) be an undirected graph with m edges. Then:
โ
v โ V
deg(v ) = 2 m
By induction on the number of edges. Alternatively, there is also a simple combinatorial proof โ every edge is incident to exactly two vertices and each edge will be counted twice in the sum.
A graph has 8 vertices with degree 4 and 2 vertices with degree 2. How many edges does this graph have?
Let G = (V , E ) be an undirected graph with m edges. Then:
โ
v โ V
deg(v ) = 2 m
By induction on the number of edges. Alternatively, there is also a simple combinatorial proof โ every edge is incident to exactly two vertices and each edge will be counted twice in the sum.
A graph has 8 vertices with degree 4 and 2 vertices with degree 2. How many edges does this graph have? Using the theorem, we see (^) โ
v โ V
deg(v ) = 8 ร 4 + 2 ร 2 = 36_._
Thus, the graph must have 36/2=18 edges. Remark: there are many different graphs which satisfy these conditions.
For n โ N, the complete graph on n vertices, denoted K n , is the undirected graph which contains n vertices and exactly one edge between any distinct pair of vertices.
(empty graph)
For n โฅ 3, the cycle graph on n vertices, denoted C n , is the undirected graph which contains n vertices V = {v 1 , v 2 ,... , v n } and edges E = {{v 1 , v 2 } , {v 2 , v 3 } ,... , {v n โ 1 , v n } , {v n, v 1 }}
Remark: Our definition of undirected graph does not allow us to define C 2 , but we may do it using pseudographs. Why?
Question: How can we determine if the following graph is bipartite?
Try to rearrange the vertices in two clusters such that there are no edges between any two vertices in the same cluster.
Question: How can we determine if the following graph is bipartite?
Try to rearrange the vertices in two clusters such that there are no edges between any two vertices in the same cluster. After rearranging, the above graph becomes:
Given n , m โ N+, we denote with K m,n the complete bipartite graph on n and m vertices which is defined to be the bipartite graph whose vertex set is partitioned into two subsets - one on n vertices and the other on m vertices, where each of the n vertices from the first partition is adjacent to all m vertices of the second partition.
Remark 1: โn , m โ N+ : K n,m = K m,n Remark 2: K 0 ,m does not exist, because we require partitions of the vertex set to be nonempty. Question: In general K m,n is not complete. For what values of n and m is K m,n complete? Answer: K 1 , 1 = K 2. If n 6 = 1 and m 6 = 1, then K m,n is not complete.
