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 fi; jg 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 = f1;2;3;:::;ng 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 fi;i + 1g 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 = fa;b;c;dg;V2 = fe; f ;g;h;ig. 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 jV1j = m and jV2j = 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 2 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 2 G; (x;y) is an edge if xy−1 2 S. Observe that Cay(G;S) is a graph if and only if S is closed with respect to inverse (S = S−1 = fs−1 : s 2 Sg). 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 = fe;(12);(13);(23);(123);(132)g be the symmetric group on 3 symbols and S = f(12);(13);(23)g. 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 = fS1;S2;:::;Sng. 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 jV1j = x, then jV2j = 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 2jEj = kjVj. Hence k divides 2jEj but gcd(2;k) = 1 hence k divides jEj. 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.