International Journal of Pure and Applied Mathematics Volume 117 No. 21 2017, 689-697 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu

+ The Cn Graph and Incidence Matrix 1K. Thiagarajan,2P. Mansoorand 3K. Umanath 1Department of Mathematics, P.S.N.A. College of Engineering and Technology, Dindigul, Tamilnadu, India. [email protected] 2Bharathiar University, Coimbatore, Tamilnadu, India. [email protected] 3Department of Mechanical Engineering, AMET University, Chennai, India. [email protected]

Abstract The definition of incidence matrix of a graph is observed. The matrix representation of a graph leads very new approach in computer science engineering and network problems. In this paper we define the graph Cn+ and some results on the incidence matrix of the graph Cn+ on the basis of some applications of computer science engineering in future. Key Words:Incidence matrix, binary matrix, Cycle, Cn+ graph.

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1. Introduction and Preliminaries

Graph theory is the study of graphs that concerns with the relationship among edges and vertices. It is a popular subject having its applications in various engineering and science subjects. This paper deals with the mathematical structures used to model pair wise relations between two different objects.

Graph: A graph G=(V,E) consist of a set of objects V={v1,v2, …}called vertices and another set E={e1,e2, …} such that each edge ek is identified with an unordered pair (vi,vj) of vertices.

Simple graph: A graph has neither self-loops nor parallel edges is called a simple graph.

Incidence:If a vertex viis an end vertex of some edge ej, then vi and ejare said to be incident with each other.

Degree of a vertex: The number of edges incident on a vertex vi with self-loop counted twice, is called the degree, d(vi) of vertex vi.

Sub graphs: A graph H is said to be subgraph of the graph G if all the vertices and all the edges of H are in G and each edge of H has the same end vertices in H as in G.

Walk: A walk is defined as a finite alternating sequence of vertices and edges, beginning and ending with vertices, such that each edge is incident with the vertices preceding and following it. No edge appears more than once in a walk. A vertex, however may appear more than once.

Cycle: A closed walk in which no vertex (except the initial and the final vertex) appears more than once is called a cycle. That is a cycle is a closed, non intersecting walk.

Determinant of a matrix:To every square matrix A = [ai,j] of order n, we can associate a number (real or complex) called determinant of the square matrix A.

Rank of Matrix: The rank of a matrix is defined as the maximum number of linearly independentcolumn vectors in the matrix or the maximum number of linearly independent row vectors in the matrix.

The following are some useful results on matrices:

AB  If E  , where A, B, C, D are block matrices which are pair wise CD commutes, then det(E ) det( AD BC ) .

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AB  If E  , where A, B, C, D are block matrices having same order, CD then ()()()E  A   D  CA1 B .

When A is not invertible, then ()()()EAD   .

 If det(A ) 0 , then ()An .

Incidence matrix:LetG be a graph with n vertices having no self loops, define an n×e matrix A = [ai,i], whose n rows correspond to the n vertices and the e columns corresponds to the e edges, as follows:

The matrix element

1, if evji is incident with aij,   . 0, otherwise

Such a matrix A is called incidence matrix. Matrix A for a given graph G is sometimes written as A(G).

Example: Consider the given graph G:

G

The incidence matrix of G is given by a b c d e f 1 1 0 1 0 0 0 2 1 1 0 0 1 1 . AG() 3 0 1 1 1 0 0  4 0 0 0 1 0 0 5 0 0 0 0 1 1

The incidence matrix contains only two elements 0 and 1. Such a matrix is called a binary matrix. Given any geometric representation of a graph without self loops, we can readily write its incidence matrix. On the other hand, if we are given an incidence matrix A(G), we can construct its geometric graph G without ambiguity. The incidence matrix and the geometric graph contain the same information they are simply two alternative ways of representing the same graph.

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The following observations about the incidence matrix A can readily made:

 Since every edge is incident on exactly two vertices, each column of A has exactly two 1’s.

 The number of 1’s in each row equals the degree of the corresponding vertex.

 A row with all zeros represents an isolated vertex.

 Parallel edges in a graph produce identical columns in its incidence matrix.

 If a graph G is disconnected and consists of two components, G1 and G2, the incidence matrix A(G) of graph G can be written in a block-diagonal form as

AGO()1 AG() , OAG()2

whereA(G1), A(G2) are the incidence matrices of components G1 and G2 respectively and O is the zero matrix. This observation results from the fact that no edge in G1 is incident on vertices of G2, and vice-versa.

 Permutation of any two rows or a column in an incidence matrix simply corresponds to relabeling the vertices and edges of the same graph. 2. The Graph Cn+

Consider Cn(n≥ 3)withvertices v1,v2,…, vn. Corresponding to each vi (i = 1, 2, 3,…,n), place a new vertex ui and join by the edge (vi, ui). Then the resulting graph is known as Cn+. Such a graph has 2n vertices and 2n edges.

Example: The C4+ graph is as shown below:

C4+ 3. Incidence Matrix of Some Cn+Graphs

Now we consider some Cn+ graphs to find some results on the determinant and the rank of the incidence matrix for Cn+ graph.

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1. C3+ graph:

The incidence matrix is a b c d e f 1 1 0 1 0 1 0  2 1 1 0 0 0 1 3 0 1 1 1 0 0 AC()3   4 0 0 0 1 0 0 5 0 0 0 0 1 0  6 0 0 0 0 0 1

ACI()3 ie, AC()3 , where A(C3) is the3×3 incidence matrix of C3, I is the 0 I 3×3 identity matrix and 0 is the 3×3 zero matrix.

Now det ACAC (33 )  det ( )  2 .

2. The C4+ graph:

The incidence matrix is a b c d e f g h 1 1 0 0 1 1 0 0 0  2 1 1 0 0 0 1 0 0 3 0 1 1 0 0 0 1 0  4 0 0 1 1 0 0 0 1 AC()  4 5 0 0 0 0 1 0 0 0  6 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 1 0  8 0 0 0 0 0 0 0 1

ACI()4 ie, AC()4 , where A(C4) is the4×4 incidence matrix of C4, I is the 0 I 4×4 identity matrix and 0 is the 4×4 zero matrix.

Now det ACAC (44 )  det ( )  0.

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3. The C5+ graph:

The incidence matrix is a b c d e f g h i j 11000110000  21100001000 30110000100  40011000010 50001100001 AC()5   60000010000 70000001000  80000000100  90000000010 100000000001

ACI()5 ie, AC()5 , where A(C5) is the5×5 incidence matrix of C5, I is the 0 I 5×5 identity matrix and 0 is the 5×5 zero matrix.Here

det ACAC (55 )  det ( )  2 .

4. The C6+ graph:

abcdefghijkl 1100001100000   2110000010000 3011000001000  4001100000100 The incidence matrix is 5000110000010  6000011000001 AC() 6 7000000100000  8000000010000  9000000001000 10000000000100  11000000000010 12000000000001

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ACI()6 ie, AC()6 , where A(C6) is the6×6 incidence matrix of C6, I is the 0 I 6×6 identity matrix and 0 is the 6×6 zero matrix.

Now det ACAC (66 )  det ( )  0 .

On the basis of above observations, for n  3 , we may generalize as follows:

ACI()n AC()n , where A(Cn) is the n×nincidence matrix of Cn, I is the 0 I n×nidentity matrix and 0 is the n×n zero matrixand 2, if n is odd det ACAC (nn )  det ( )   . 0, if n is even 4. Results on the Incidence Matrix of the Cn+Graph

Theorem:IfAis the incidence matrix of the Cn+ graph on 2n ( ) vertices, then the determinant of A is 2, if n is odd det A   . 0, if n is even

Corollary 1: If A is the incidence matrix of the Cn+ graph, then the rank of A is 2nn , if is odd   A   . 2nn 1, if is even

Corollary 2: The complement graph of Cn+ graph need not be Planar. References [1] Harrary F., Graph theory, Addison Wesley, Reading Massachusetts, USA (1969). [2] Narsingh D., Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India, New Delhi (1990). [3] JuerissenR.H., The incidence matrix and labeling of a graph, Journal of Combinatorial Theory, Series B, 30(3) (1981). [4] ThiagarajanK., MansoorP., Complete Network through folding and Domination Technique, International Journal of Engineering Research 10(39) (2015). [5] Rajesh Kumar C., Uma MaheswariS., Matrix Representation of Hanoi Graphs, International Journal of Science and Research (2013).

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[6] Manju, K., Sabeenian, R.S., Surendar, A.”A review on optic disc and cup segmentation”,(2017) Biomedical and Pharmacology Journal, 10 (1), pp. 373-379. [7] Surendar, A., Rani, N.U.”High speed data searching algorithms for DNA searching”,(2016) International Journal of Pharma and Bio Sciences, 2016 (Special Issue), pp. 73-77. [8] B. Saichandana, G. Rachanasri, A. Surendar and B. Suniltej “controlling of wall lamp using arduino”, International Journal of Pure and Applied Mathematics, Volume 116 No. 24 2017, 349- 354, ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on- line version)

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