Journal of Science and Arts Year 17, No. 1(38), pp , 2017

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1 Journal of Science and Arts Year 17, No. 1(38), pp , 2017 BOUNDS FOR THE ENERGY OF ( ) ORIGINAL PAPER ( ) USING 2-ADJACENCY SERIFE BUYUKKOSE 1, SEMIHA BASDAS NURKAHLI 1, NURSAH MUTLU 1 Manuscript received: ; Accepted paper: ; Published online: Abstract. Let be a simple connected graph. We redefined adjacency for any two vertices if there is connected two edges between any two vertices and. In this case, this vertices are called 2-adjacency and denoted by. In this study, we redefined the energy of Adjacency matrix ( ) and Laplacian matrix ( ) by using 2-adjacency definition. Also, we found some bounds for the energy of ( ) and ( ) of a graph, respectively by using 2- adjacency. Keywords: 2-adjacency, energy, bound. 1. INTRODUCTION Let ( ) be a simple graph with vertices and edges having vertex set ( ) and edge set ( ). For, the degree of, the set of neighbors of and the average of the degrees of the vertices adjacent to are denoted by, ( ) and, respectively. Let ( ) be the diagonal matrix of vertex degrees of a graph. Also, let ( ) be the adjacency matrix of and ( ) ( ) be defined as the matrix ( ), where ( )={ It follows immediately that if is a simple graph, then ( ) is a symmetric ( ) matrix where all diagonal elements are zero. We will denote the characteristic polynomial of by ( ) ( ( )) 1 Gazi University, Faculty of Sciences, Department of Mathematics, Ankara, Turkey. sbuyukkose@gazi.edu.tr; semiha.basdas@gazi.edu.tr ; nursah.mutlu@gazi.edu.tr ISSN:

2 50 Bounds for the energy of Since ordered as ( ) is a real symmetric matrix, its eigenvalues must be real and may be ( ( )) ( ( )) ( ( )). Denoted ( ( )) simply by ( ). The sequence of eigenvalues is called the spectrum of. The largest eigenvalue ( ) is often called the spectral radius of. We now give some known upper bounds for the spectral radius ( ). The energy of the graph is defined as ( ) The Laplacian matrix of a graph is defined as ( ) ( ), where { The Laplacian matrix of is ( ) ( ) ( ). Clearly, ( ) is a real symmetric matrix. From this fact and Gersgorin s theorem, it follows that its eigenvalues are nonnegative real numbers. Moreover, since its rows sum to, is the smallest eigenvalue of ( ). We assume that the Laplacian eigenvalues are. Especially the largest eigenvalue of ( ) is called Laplacian spectral radius of, denoted by ( ). The Laplacian energy of a graph as put forward by Gutman and Zhou is defined as ( ) Firstly, we now give some known upper bounds for ( ). Let be a simple graph with vertices and edges.

3 Bounds for the energy of 51 1) In 1971, McClelland [10] discovered the first upper bound for ( ) as follows: ( ) (1) 2) If, then [6] ( ) ( ) [ ( ) ] ( ) Moreover, equality holds if and only if is either, or a non-complete connected strongly regular graph with two non-trivial eigenvalues both with absolute value ( ). If, then ( ) Equality holds if and only if is disjoint union of edges and isolated vertices. 3) If is a graph with vertices, edges and degree sequence then [13] ( ) ( ) [ ] ( ) with equality holds if and only if is either a complete bipartite graph, a non-complete connected strongly regular graph with two non-trivial eigenvalues both with absolute value ( ) or. following: Secondly, many researchers have investigated upper bounds for L ( ) is the ISSN:

4 52 Bounds for the energy of Let be a simple graph with vertices and edges. 1) B.J.McClelland s bound [10]: ( ) ( ) where ( ). 2) J.H.Koolen, V.Moulton, I.Gutman s bound [7]: ( ) ( ) [ ( ) ] ( ) 2. BOUNDS FOR THE ENERGY OF GRAPHS USING 2-ADJACENCY Let be a simple connected graph. We redefined adjacency for any two vertices and. And then, we called this vertices as 2-adjacency. If this vertices are 2-adjacency, we denoted it by. In this study, we defined 2-adjacency matrix and denoted by ( ). We redefined the 2-adjacency energy of the graph by using 2-adjacency definition. We found an upper bound for the 2-adjacency energy of the graph. Then we expanded to k-adjacency of two vertices and redefined the k-adjacency matrix and obtained a new bound for the k-adjacency energy of a graph. Definition 2.1 If there are two connected edges between vertices of the this vertices are called 2-adjacent and denoted by. and, then Definition 2.2 The 2-degree of any vertex in graph is the number of 2-adjacency vertices regarding the vertex and the 2-degree of a vertex is denoted by. We can define the k-adjacency by generalizing 2-adjacency concept further.

5 Bounds for the energy of 53 Definition 2.3 If there are k connected edges between vertices of the and, then this vertices are called k-adjacency and denoted by. Definition 2.4 The k- degree of any vertex in graph is the number of k-adjacency regarding the vertex and the k-degree of a vertex is denoted by. Given a graph ( ) with the 2-adjaceny degree matrix ( ) for is a diagonal matrix defined as: ( ) { where the 2-degree of a vertex is the number of 2-adjacency regarding the vertex. Given a simple, connected graph with vertices, its 2-adjacency matrix ( ) ( ) defined as: ( ) { we denoted by Because the 2-adjacency matrix is real, symmetric matrix, its eigenvalues are real and. denote the largest eigenvalue of ( ). Definition 2.5 The 2-adjacency energy of the graph is defined as ( ) Theorem 2.6 If is a graph with vertices, edges and degree sequence then ISSN:

6 54 Bounds for the energy of ( ) ( ) ( ) ( ( ) ) Proof: Among the known lower bounds for is the following [14]: If you apply for 2-adjacency to this inequality, it is obtained By the Cauchy-Schwartz inequality, ( ) ( ) ( )( ( ) ) Hence ( ) ( )( ( ) ) Note that the function ( ) ( )( ) decreases for and ( ( ) ), we see that ( ) ( ( ( ) ) ). Since, it is obtained

7 Bounds for the energy of 55 ( ) ( ( ) ) ( ) ( ( ) ) Given a simple, connected graph with vertices, its 2-adjacency self-adjacency matrix ( ) ( ) defined as: ( ) { Because the 2-adjacency self-adjacency matrix is real, symmetric matrix, its eigenvalues are real and we denoted by. denote the largest eigenvalue of ( ). Definition 2.7. The 2-adjacency self energy of the graph is defined as ( ) Theorem 2.8 If is a graph with vertices, edges and degree sequence then ( ) ( ( ) ) ( ) ( ( ) ) Conjecture 2.9. Let be a simple, connected graph. Then ( ) ( ) ( ). ISSN:

8 56 Bounds for the energy of When we generalize for k-adjacency to find bound, we obtain following result. Theorem If is a graph with vertices, edges and degree sequence then ( ) ( ) ( ) ( ( ) ) Specially, if we get k=1, Theorem 2.10 will turn (3). 3. BOUNDS FOR THE LAPLACIAN ENERGY OF GRAPHS USING 2-ADJACENCY Let be a simple connected graph. We redefined adjacency for any two vertices and. And then we called this vertices as 2-adjacency. If this vertices are 2-adjacency then denoted by. In this study, we defined 2-adjacency Laplacian matrix and denoted by ( ). We found an upper bound for the energy of ( ) Then we expanded to k-adjacency of two vertices and redefined the k-adjacency Laplacian matrix and obtained a new bound for the 2- adjacency Laplacian energy of a graph. Given a simple, connected graph with vertices, its 2-adjacency Laplacian matrix ( ) ( ) defined as: ( ) { where is 2-degree of the vertex. Because the 2-adjacency Laplacian matrix is real, symmetric matrix, its eigenvalues are real and we denoted by.

9 Bounds for the energy of 57 denotes the largest eigenvalue of ( ). Also, every row sum and column sum of is zero. There are equality ( ) ( ) ( ) for 2-adjacency Laplacian matrix. Definition 3.1 The 2-adjacency Laplacian energy of the graph is defined as ( ) Theorem 3.2. If and is a graph on vertices with edges, then ( ) ( ) [ ( ) ] where ( ). via Proof: We first introduce the auxiliary eigenvalues,, defined The 2-adjacency Laplacian eigenvalues satisfy the conditions ( ) where ( ) ISSN:

10 58 Bounds for the energy of with denoting the 2-degree of the -th vertex of. Since ( ), we have ( ). Using this together with the Cauchy-Schwartz inequality, we obtain the inequality ( ) ( ) ( ) [ ( ) ] Hence ( ) ( ) [ ( ) ] Since, it is obtained ( ) ( ) [ ( ) ] Given a simple, connected graph with vertices, its 2-adjacency self-laplacian matrix ( ) ( ) defined as: ( ) { where is 2-degree of the vertex. Because the 2-adjacency self-laplacian matrix is real, symmetric matrix, its eigenvalues are real and we denoted by. denotes the largest eigenvalue of ( ). There are equality

11 Bounds for the energy of 59 ( ) ( ) ( ) for 2-adjacency self-laplacian matrix. Definition 3.3. The 2-adjacency self-laplacian energy of the graph is defined as ( ) Theorem 3.4. If and is a graph on vertices with edges, then ( ) ( ) [ ( ) ] where ( ) Conjecture 3.5. Let be a simple, connected graph. Then ( ) ( ) ( ). When we generalize for k-adjacency to find bound, we obtain following result. Corollary 3.6 If and is a graph on vertices with edges, then ( ) ( ) [ ( ) ] where ( ) Specially, if we get k=1, Corollary 3.6 will turn (5). ISSN:

12 60 Bounds for the energy of REFERENCES [1] Anderson, W.N., Morley, T.D., Linear Multilinear Algebra, 18(2), 141, [2] Başdaş Nurkahlı, S., Büyükköse, Ş., Bounds for the Laplacian eigenvalue of graphs using 2-adjacency, Gazi University Master Thesis, [3] Berman, A., Zhang, X.D., J.Combin.Theory Ser.B., 83, 233, [4] Collatz, L., Sinogowitz, U., Abh. Math. Sem. Univ. Hamburg, 21, 63, [5] Das, K.C., Kumar, P., Discrete Mathematics, 281, 149, [6] Koolen, J.H., Moulton, V., Adv. Appl. Math., 26, 47, [7] Koolen, J.H., Moulton, V., Gutman, I., Chem. Phys. Lett., 320, 213, [8] Li, J.S., Zhang, X.D., Linear Algebra and Application, 265, 93, [9] Liu, H., Lu, M., Tian, F., Journal of Mathematical Chemistry, 41(1), 45, [10] McClelland, B.J., J. Chem. Phys., 54, 640, [11] Merris, R., Linear Algebra, 285(1-3), 33, [12] Ocak, N.G., Büyükköse, Ş., Bounds for the eigenvalue of graphs using 2-adjacency, Gazi University Master Thesis, [13] Zhou, B., MATCH Commun. Math. Comput. Chem., 51, 111, [14] Zhou, B., Australasian J. Combin., 22, 301, [15] Zhou, B., Gutman, I., Linear Algebra and its Applications, 414, 29,

BOUNDS FOR LAPLACIAN SPECTRAL RADIUS OF THE COMPLETE BIPARTITE GRAPH

Volume 115 No. 9 017, 343-351 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu BOUNDS FOR LAPLACIAN SPECTRAL RADIUS OF THE COMPLETE BIPARTITE GRAPH