Chapter I PRELIMINARIES

Transcription

1 Chapter I PRELIMINARIES

2 6, i CHAPTER I PRELIMINARIES Definition 1.1 Let V = { VI,V2,V3,... } be a set and E = {ei, e2, e3,... } be another set such that each ek is an unordered pair of elements of V so that ek = (Vi, Vj) = (Vj, Vi) for some i andj. Then the ordered pair G = (V, E) is called a graph. The elements of V are called the vertices or points and the elements of E are called the edges or lines. The vertices Vi, Vj associated with ek are called the end vertices of ek or simply ends of Definition 1.2 Two vertices Vi and Vj of a graph G are said to be adjacent if there is an edge joining Vi and Vj. Two edges ei and ej are said to be adjacent ifthey have a common end vertex. Definition 1.3 An edge with identical ends is called a loop or self- loop. Definition 1.4 Edges having the same end vertices are called multiple edges or parallel edges. Edges with distinct ends are called links. Definition 1.5 A graph that has neither loops nor multiple edges is called a simple graph.

3 7 Definition 1.6 A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is an infinite graph. Definition 1.7 If Vi and Vj are the ends of the edge ek, then we say that ek is incident on Vi and Vj. Consider the graph shown in figure 1.1 v '--'------"-t~ Vs Figure 1.1 Figure 1.1 is a graph with 6 vertices and 10 edges. e2 is a loop. e7 and eg are parallel edges. e6 and eg are adjacent. e6 is ~ link. Definition 1.8 The number of edges incident on a vertex Vi (with self-loop counted twice), is called the degree of the vertex Vi and is denoted by d(vi).

4 8 In figure 1.1, d(vl) = 4; d(v2) = 5 and so on. Definition 1.9 A graph in which all the vertices are of equal degree is called a regular graph. Definition 1.10 A vertex ofdegree zero is called an isolated vertex Definition 1.11 A vertex of degree one is called a pendant vertex and the corresponding edge is called a leaf Figure 1.2. In figure 1.2, vertex V2 is a pendent vertex. Vertex Vs is an isolated vertex. Definition 1.12 A graph G with p vertices and q edges is called {p,q)-graph where p is called the order of the graph and q is called the size of the graph G.

5 9 Definition 1.13 A (p,q)-graph with p 1= 0, q = is called a vertex graph and is denoted by <p. Definition 1.14 A (P, q)-graph with p = q = is called a null graph or an empty graph and is denoted by <1>. Vs Figure 1.3 A vertex graph with 5 vertices is shown in figure 1.3. Definition 1.15 Two graphs 0 and 0' are said to be isomorphic to each other if there is a one to one correspondence between their vertices and between their edges such that the incidence relationship is preserved.

6 10 VI V5' e6' V2 e3 e4 V4 e V4' V5 V3' e2 e5 e4' e2' VI'~_ " el' V2' G H Figure 1.4 The graphs G and H given in figure 1.4 are isomorphic. Definition 1.16 A graph H. is said to be a subgraph of a graph G if all the vertices and all the edges ofh are in G. Figure 1.5 (a)

7 11 Vs Figure 1.5(b) Figure 1.5(c) Graphs given in Figure 1.5(b) and 1.5(c) are subgraphs of the graph in Figure 1.5(a). Definition 1.17 Two subgraphs of a graph are said to be edge disjoint if they have no edges in common. Similarly two subgraphs of a graph are said to be vertex disjoint ifthey have no vertex in common. The two subgraphs given in figure 1.5(b) and 1.5(c) are edge disjoint subgraphs ofthe graph given in figure 1.5(a). Definition 1.18 Let G = (V, E) be any graph. Let VI be a non- empty subset of the vertex set V. The subgraph of G with vertex set V I and edge set as the set ofthose edges of G have both ends in V I is called the subgraph ofg induced by V I and it is denoted by G[VI] or < V I >. < V I > is also called an induced subgraph ofg.

8 12 VI e_ '------iiik Vs e "'---- V3 G Figure 1.6(a) Figure 1.6(b) Graph shown in figure 1.6(b) is an induced subgraph ofthe graph given in 1.6(a). Definition 1.19 A spanning subgraph of a graph G IS containing all the vertices ofg. a subgraph of G Figure 1.7(a) Figure 1.7(b) Graph given in figure 1.7(b) is a spanning subgraph ofg', displayed in figure 1.7(a).

9 13 Definition A walk of a graph G is an alternating sequence of vertices and edges say Vo elvl e2 V2...Vn-l en Vn beginning and ending with vertices in which each edge is incident with two vertices immediately preceding and following it. The above walk may also be called VO-Vn walk. The walk VO-Vn is said to be a closed walk if Vo = Vn and is open otherwise. Definition 1.21 An open walk in which no vertex appears more than once is called a path. The number ofedges in a path is called the length ofthe path. The terminal vertices of a path are of degree one, and the rest of the vertices (called intermediate vertices) are of degree two. Generally a path on n vertices is denoted by Pn. Remark: In the above vo- Vnwalk if all the vertices are distinct it will be termed as a Vo- Vnpath it may also be written as VOVIV2... Vn by omitting the edges. Definition 1.22 A closed path is called a circuit. A circuit is also called a cycle, elementary chain, circular path or polygon. A cycle on n vertices is denoted by en.

10 ',:- '14 Vs Figure 1.8 In figure 1.8, VlelV2e2v3e4V4 is a walle VI and V4 are the terminal vertices of the walk. VlelV2eS V6e7 Vs is a path. Length of this path is 3. o Figure 1.9

11 15 Three different circuits C 6, CJ, C 2 are shown in figure 1.9. Definition 1.23 A graph G is said to be connected if there is at least one path between every pair ofvertices in G. Otherwise G is disconnected. Graph in figure 1.2 is disconnected. A vertex graph with more than one vertex is disconnected. The graph in figure 1.1 is connected. Definition 1.24 A disconnected graph consists of two or more connected subgraphs ofthe graph. Each ofthese connected subgraphs is called a component. The graph in figure 1.2 consists oftwo components. Definition 1.25 The union of two graphs G 1 = (V}, E 1 ) and G 2 = (V 2, E 2 ) is another graph G = (V, E) whose vertex set V = V I U V2 and the edge set E = E1u E 2. Definition 1.26 The intersection denoted by G1nG 2 of two graph~ G1and G 2 is a graph G consisting of those vertices and edges that are in both G1 and G 2. For any graph G, G u G = G ; G ng = G. Ifvj is a vertex in a graph G, then G - Vi denotes a subgraph of G obtained from G.by

12 16 deleting Vi from G. [Deletion of a vertex implies the deletion of all edges incident on that vertex.] If ej is an edge in G, th.en G - ej is a subgraph of G obtained by deleting ej from G. [Deletion of an edge does not imply deletion ofits end vertices.] Definition 1.27 A graph in which there exists an edge between every pair of distinct vertices is called a complete graph. Generally K p denotes a complete graph in p vertices. Figure 1.10 A complete graph on 6 vertices is displyed in figure Definition 1.28 A tree is a connected graph without any circuit. Definition 1.29 A spanning tree of a connected graph G is a spanning subgraph ofg which is also a tree.

13 . 17 Definition 1.30 Let S be a set and F = {Sl, S2,... Sp} a non- empty family of distinct non- empty subsets of S whose union is S. The intersection graph of F is denoted by Q(F) and is defmed by V(Q(F)) = F, witb Si and Sj adjacent whenever i ::j:. j and I Si n Sj I::j:. O. Then a graph G is an intersection graph on S if there exists a family F of subsets of S for which G ~ Q(F). Theorem 1.31 Every graph is an intersection graph. Definition 1.32 The intersection number m(g) of a given graph G is the minimum number of elements in a set S such that G is an intersection graph on S. Definition 1.33 A graph G is said to be a bipartite graph or bigraph if its vertex set V can be partitioned into two subsets VI and V2 such that each edge ofg has one end in V 1 and the other in V2. Ifevery vertices in VI is joined with all the vertices ofv2, then the bipartite graph is called as a complete bipartite graph and is denoted by Kro,n where m is the number ofvertices in VI and n is the number ofvertices in V 2 Definition 1.34 Let G = (V, E) be a graph and S c V. S is said to be an independent set if no two vertices ofs are adjacent in G. \.1 ~ "

14 18 Next we give some important terminologies In the field of algebra. Definition 1.35 A non empty set with an associative binary operation is called a semigroup. We write a multiplicative semigroup as (Sg,.) or simply as Sg. Definition 1.36 Let (Sg,.) be a semigroup. A non-empty subset T of Sg is called a subsemigroup of Sg ifit is closed with respect to multiplication. ie. for all x,yet, xyet. Definition 1.37 A non empty set SG, together with a binary operation I called a group ifthe following axioms are satisfied: *.' is i) The binary operation * is associative on SG. ii) There is an element e in SG such that e * x = x * e = x for all x ESG. (The element e is called the identity element for * on SG') iii) For each a in SG there is an element a' in SG such that at * a = a * a' = e (The element a' is called the inverse ofa). This group is denoted by (Sa,...).

15 19 Definition 1.38 If L is a subset of SG closed under the group operation of SG and L itself is a group under this induced operation, then L is a subgroup of SG and is denoted by L ~ SG. Definition Let SG be a group. Then all subgroups of SG other than SG are proper subgroups of SG. Also {e} is the trivial subgroup of SG. All other subgroups are non-trivial. Definition 1.40 Let SG and SG' be any two groups. A mapping \ji : SG ~SG' is said to be an isomorphism if i) \jj is bijective. ii) \jj(x y) = \ji(x). \ji(y) for all x, Y ESG. The groups SG and SG' are then isomorphic and is denoted by SG ~ SG' Definition 1.41 Let L be a subgroup of a group SG and let ae SG. The left coset al ofl is the set {al: IEL}. The right coset La is similarly defined. Definition 1.42 A subgroup H ofa group SG is said to be normal if g-l Lg = L for all g E SG.

16 20 Definition 1.43 IfN is a normal subgroup of a group So, the group of cosets of N under the induced operation is called the factorgroup of So modulo N and is denoted by SG / N. The cosets are residue classes ofso modulo N. Definition 1.44 A normal series ofa group So is a finite sequence Lo,L 1, ; L n ofnormal subgroups ofso such that L i < L i + I ; Lo= {e} and Consider the group Z under addition then {O} < 8Z < 4Z < Z {O} < 9Z < Z are two normal series ofz. Definition A normal series {K j } is a refinement of a normal series {L j } of a group So if {Lj } C {K j }; ie. ifeach L i is one ofthe K j. The series {O} < 72Z < 24Z < 8Z < 4Z < Z is a refmement of {O} < 72Z < 8Z < Z.

17 21 Definition 1.46 Two normal senes {Ld and {K j } of the same group G are isomorphic if there is one to one correspondence between the collections of factor groups {L i + 1 / L i } and {K j + 1 / K j } such that the corresponding factor groups are isomorphic. Two isomorphic normal series must have the same number of groups. Consider Z IS The two series {O} < < 5 > < ZI5 and {O} < < 3 > < ZI5 are isomorphic. Since ZI5 / < 5 > and < 3 > / {O} are isomorphic to Z5, isomorphic to < 5 > / {O} or to Z3' ZI51 < 3 > is Definition 1.47 A lattice is a partially ordered set S in which each pair of elements has greatest lower bound and least upper bound. If x, yes then greatest lower bound is denoted by x 1\ y and least upper bound is denoted by x v y. Definition 1.48 A lattice S is said to be complete if every non-empty subset ofs has a greatest lower bound and least upper bound....

18 22 Now we just present two basic defmitions one in algebra and the other related to topology. Definition 1.49 Let p be a collection of subsets of a set S. Then Cf'is called a field iff S E Cf' and p is closed under complementation and finite union. (a) SE P (b) IfAE pthen ACE Cf' (c) IfA], A 2,..., AnE p then UA j n i=1 E Cf' It follows that Cf'is closed under finite intersection. For, if AI, A 2,..., An E Cf', then If(c) is replaced by closure under countable union, ie, (d) IfAt, A 2,A 3..., UA j <X) i=1 Epthen Cf'is called a a-field. pis also closed under countable intersection. If p is a field, a countable union of sets in Cf' can be expressed as the limit of an increasing sequence of sets in P and conversely. ie. <X) n <X) if A = UA j J then UA j t A; conversely if Ant A, then A = UA j ~ ~ ~

19 23 This shows that a-field IS a field that IS closed under limits of increasing sequence. Definition 1.50 Let X be a non empty set and 't be a collection of subset ofx. 't is said to be a topology on X ifit satisfies i) X, <D E't ii) The union ofthe elements ofany subcollection of't is in 't. iii) The intersection of the elements of any finite subcollection of't is 10 'to Then pair (X, 't) is called a topological space.