1 Number Theory and Graph Theory

Chapter 6

Basic concepts and definitions of graph theory

By

A. Satyanarayana Reddy

Department of Mathematics Shiv Nadar University Uttar Pradesh, India

E-mail: [email protected] 2

Module-2: Some known graph families and their properties

Objectives

• Complete graph, null graph, complete bipartite graph

• Path graph, cycle graph and wheel graph

• Regular graph, Cayley graph and intersection graphs.

1. The simple graph X that has an edge for each pair of vertices is called a complete graph,

denoted Kn, where n is the number of vertices of X. 1

3 3 4 2 3

4 5 1 2 1 2 1 2 K2 K3 K4 K5

• There is an edge between any two vertices (any two vertices are adjacent).

• It is also called universal graph or clique.

n(n−1) • Total number of edges are 2 . • The degree of each vertex is n − 1.

• Every simple graph with n vertices is obtained from Kn by deleting some edges.

2. A graph with no edge is called a null graph. We denote null graph with n vertices by Nn 3 3 4

1 2 1 2 1 2 N2 N3 N4 3

3. The cycle graph on n vertices, say 1,2,...,n, denoted Cn, is a simple graph in which {i, j} is an edge if and only if i − j ≡ ±1 (mod n). 2 2 2 3

3 C4 1 C C3 1 5 1 4 3 4 5

• All the vertices form a cycle. Here cycle we mean, let V = {1,2,3,...,n} be the vertices

of cycle graph Cn, then there is an edge from 1 to 2, then 2 to 3 and so on n − 1 to n and n to 1.

• The degree of each vertex is 2.

• There are n edges.

4. A graph on n vertices, say 1,2,...,n, is called a path graph, denoted Pn, if for each i,1 ≤ i ≤ n − 1, the set {i,i + 1} is an edge. 1 2 1 2 3 1 2 3 4

P2 P3 P4

5. A graph X = (V,E) is called bipartite, with bipartition (V1,V2), if V = V1 ∪V2, V1 ∩V2 = /0

and each edge of X joins a vertex in V1 with a vertex in V2. For example, in the following

graph V = V1 ∪V2 where V1 = {a,b,c,d},V2 = {e, f ,g,h,i}. There is no edge between any

two vertices of V1. Similarly, there no edge between any two vertices of V2. a b c d

e f g h i 4

It is easy to check that

(a) every cycle graph of even length is bipartite, and

(b) every path graph is bipartite.

Whereas, the complete graph Kn, for n ≥ 3 and the cycle graph C2n+1, for n ≥ 1 are not bipartite graphs.

6. A complete bipartite graph is a bipartite graph X with bipartition (V1,V2), in which every

vertex in V1 is joined to every vertex in V2. A complete bipartite graph is denoted by Km,n,

where |V1| = m and |V2| = n.

1 1 3 1 2 3 K2,2 K1,1 2 K2,1 2 4

• There is an edge between every vertex in V1 to every vertex in V2.

• The degree of every vertex in V1 is n whereas in V2, it is m.

• The length of every cycle is even.

• The number of edges are mn and the number of vertices are m + n.

• The graphs K1,m or Km,1 are also called star graphs.

7. Wheel graph: Wheel graph is obtained from a cyclic graph by adding one vertex to it, called the hub, and edges from this vertex to all other vertices.

• It is denoted by WHn.

• It consists of n + 1 vertices and 2n edges.

• The degree of every vertex excluding the hub is 3. For the hub it is n. 5

3 4 3

4 5

1 2 1 2 WH3 WH4

8. A graph (digraph) X is said to be k- regular if d(v) = k (d+(v) = d−(v) = k) for all v ∈ V.

• The degree of every vertex is k.

• If the number of vertices are odd then using the hand-shaking lemma, its regularity cannot be odd.

• Complete graph is n − 1 regular.

• Complete bipartite graph is regular graph if m = n.

• Cyclic graph is 2-regular.

• Path is regular if it has 2 vertices.

• Wheel graph is regular only when n = 3.

• The Johnson graphs (see Example 2.1 in module 1 of chapter 6) and k-cube graphs (see Example 2.3 in module 1 of chapter 6) are regular.

• The Petersen graph (see Example 2.1 in module 1 of chapter 6) is 3-regular.

9. Let G be a group and let S be a non-empty subset of G that does not contain the identity element of G. Then the Cayley digraph/graph associated with the pair (G,S), denoted Cay(G,S), has the set G as its vertex set and for any two vertices x,y ∈ G, (x,y) is an edge if xy−1 ∈ S.

Observe that Cay(G,S) is a graph if and only if S is closed with respect to inverse (S = S−1 = {s−1 : s ∈ S}). Also, the graph is k-regular if S has k elements. The set S is called the 6

connection set of the graph. For example, G = S3 = {e,(12),(13),(23),(123),(132)} be the symmetric group on 3 symbols and S = {(12),(13),(23)}. Then, S is a generating set of G as every permutation can be expressed as product of transpositions. The corresponding Cayley graph Cay(G,S) is K3,3, as shown below. e (12)

(13) (123)

(132) (23)

Cayley graphs on cyclic groups are called circulant graphs. An important class of circulant graphs are Payley graphs. We define them in the following manner.

Let q be a prime power such that q ≡ 1 (mod 4), and Fq denote a finite field of order q, the

Payley graph P(q) of order q is a graph whose vertices are elements of the finite field Fq in which two vertices are adjacent if and only if their difference is a non-zero square in Fq. Since q−1 q−1 in Fq there are 2 non-zero squares (quadratic residues) hence P(q) is 2 -regular graph. As we aware −1 is a square in modulo q if and only if q ≡ 1 (mod 4). Consequently a − b is a square if and only if b − a is a square. To know more about Payley graphs refer Algebraic graph theory by Chris Godsil and Gordan Royal. On the other hand if q ≡ 3 (mod 4), then P(q) is a digraph. The following graph is P(13). 7

3 2 1 4 0 5

12 6 11 7 10 8 9

10. Let S1,S2,...,Sn be non-empty subsets of a set S. Then we define the intersection graph, overlap graph, containment graph and disjointedness graphs as follows.

The vertex set of all these graphs is V = {S1,S2,...,Sn}. There is an edge between Si and S j for i 6= j if

Intersection graph Si ∩ S j 6= /0.

Overlap graph Si ∩ S j 6= /0. and Si * S j and S j * Si

Containment graph Si ⊂ S j or S j ⊂ Si.

Disjointedness graph Si ∩ S j = /0.

Depending on the nature or the geometric configurations of the sets S1,S2,...,Sn, many im- portant subclasses of intersection graphs were generated and studied. Few of them are interval graphs, tolerance graphs, chordal graphs, unit disk graphs, line graphs, circular arc graphs etc. For example, a graph is an interval graph if it can be represented as intersection graph

of intervals. Here S1,S2,...,Sn are intervals. Following figure illustrates the construction of interval graph with five intervals. 8

5 5

4 4

3 3

1 2 1 2

Many problems which are difficult to solve for general graphs are easy to solve once we confine to interval graphs. Further interval graphs have a number of applications in different areas, for example, archeology, scheduling, biology etc.

Few Solved Problems

1. What is the smallest number n such that the complete graph Kn has at least 500 edges? n(n−1) n(n−1) Solution:Since number of edges in Kn is 2 , hence 2 ≥ 500. Or equivalently n(n − 1) ≥ 1000. Thus smallest value of n is 33.

n2 2. Prove that the number of edges in a bipartite graph with n vertices is at most 4 .

Solution:Let X = (V1 ∪V2,E) be a bipartite graph with n vertices. Let |V1| = x, then |V2| = n − x. By definition, X has at most f (x) = x(n − x) edges. Since f 0(x) = n − 2x and f 00 = −2, n n2 consequently f (x) has its maximum value at x = 2 and the value is 4 .

3. Let X be a k-regular graph, where k is an odd number. Then prove that number of edges in X is a multiple of k. Solution:We have 2|E| = k|V|. Hence k divides 2|E| but gcd(2,k) = 1 hence k divides |E|.

4. For what values of m and n, the complete bipartite graph Km,n has more number of vertices than edges.

Solution:We know that the number of vertices and edges in Km,n are m+n and mn, respectively. Further m,n ≥ 1 and hence m + n ≥ mn is possible only if either m = 1 or n = 1. 9

5. Show that the cycle graph C4 is not an interval graph. d

d c a c

a b b

Solution:Assume C4 is an interval graph (see the figure above). Since there is no edge between a and c, there is no overlap between their corresponding intervals. Further, b and d both are adjacent to a and c. Hence, there is an overlap among their corresponding intervals which is again shown above. Which is a contradiction as b and d are not adjacent.