Milovanović Bounds for Seidel Energy of a Graph

Transcription

1 Advances in Theoretical and Applied Mathematics. ISSN Volume 10, Number 1 (2016), pp Research India Publications Milovanović Bounds for Seidel Energy of a Graph M. R. Rajesh Kanna Post Graduate Department of Mathematics, Maharani s Science College for Women, J. L. B. Road, Mysore , India. mr.rajeshkanna@gmail.com R. Pradeep Kumar Department of Mathematics, The National Institute of Engineering, Mysuru , India. pradeep.mysore@gmail.com Mohammad Reza Farahani Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, Tehran, 16844, Iran. mrfarahani88@gmail.com Abstract In this paper Seidel Energy of Cocktail Party graph and Crown graph are computed. Recently Milovanović et al. gave a sharper lower bounds for energy of a graph. Similar bounds for Siedel energy of a graph are established. AMS Subject Classification: Primary 05C50, 05C69. Keywords: Seidel set, Seidel matrix, Seidel eigenvalues, Seidel energy. 1. Introduction The concept of energy of a graph was introduced by I. Gutman [5] in the year Let G be a graph with n vertices and m edges and let A = (a ij ) be the adjacency matrix of the graph. The eigenvalues λ 1,λ 2,...,λ n of A, assumed in non increasing order, are the eigenvalues of the graph G. As A is real symmetric, the eigenvalues of G are real with

2 38 M.R. Rajesh kanna, R. Pradeep kumar, Mohammad Reza Farahani sum equal to zero. The energy E(G) of G is defined to be the sum of the absolute values of the eigenvalues of G. i.e., E(G) = λ i. For details on the mathematical aspects of the theory of graph energy see the reviews [6], papers [2, 3, 7] and the references cited there in. The basic properties including various upper and lower bounds for energy of a graph have been established in [9, 11] and it has found remarkable chemical applications in the molecular orbital theory of conjugated molecules [4, 8]. Further studies on covering energy and dominating energy can be found in [1, 13] Seidel Energy Let G be a simple graph of order n with vertex set V ={v 1,v 2,...,v n } and edge set E. The Seidel matrix of G is the n n matrix defined by S(G) := (s ij ), where 1 if v i v j E s ij = 1 if v i v j / E 0 if v i = v j The characteristic polynomial of S(G) is denoted by f n (G, λ) = det(λi S(G)). The Seidel eigenvalues of the graph G are the eigenvalues of S(G). Since S(G) is real and symmetric, its eigenvalues are real numbers. The Seidel energy [14] of G is defined as SE(G) := λ i. 2. Siedel Energy of Some Standard Graphs Definition 2.1. The Cocktail party graph is denoted by K n 2,is a graph having the n vertex set V = {u i,v i } and the edge set E ={u i u j,v i v j : i = j} {u i v j,v i u j : 1 i<j n}. Theorem 2.2. For n 2, the Siedel energy of Cocktail party graph K n 2 is 6n 6.

3 Milovanović Bounds for Seidel Energy of a Graph 39 Proof. Let K n 2 be the Cocktail party graph with vertex set V = S(K n 2 ) = Characteristic equation is Siedel eigen values are Siedel energy, n {u i,v i }. Then (λ + 1) n (λ 3) n 1 [λ + (2n 3)] =0. λ = 1[ntimes],λ= 3[(n 1)times],λ= (2n 3) SE(K n 2 ) = 1 n + 3 (n 1) + (2n 3) =6n 6. Definition 2.3. The Crown graph Sn 0 for an integer n 2 is the graph with vertex set {u 1,u 2,...,u n,v 1,v 2,...,v n } and edge set {u i v j : 1 i, j n, i = j}. Sn 0 coincides with the Complete bipartite graph K n,n with horizontal edges removed. Theorem 2.4. For n 2, the siedel energy of the Crown graph Sn 0 is equal to 6n 6. Proof. For the Crown graph Sn 0 with vertex set V ={u 1,u 2,...,u n,v 1,v 2,...,v n }. Then S(Sn 0 ) = (2n 2n)

4 40 M.R. Rajesh kanna, R. Pradeep kumar, Mohammad Reza Farahani Characteristic equation is Siedel eigen values are Siedel energy, (λ 1) n (λ + 3) n 1 [λ (2n 3)] =0. λ = 1[ntimes],λ= 3[(n 1) times],λ= 2n 3 SE(Sn 0 ) = 1 n + 3 (n 1) + (2n 3) =6n Properties of Seidel Eigenvalues Lemma 3.1. Let G be a simple graph with vertex set V ={v 1,v 2,...,v n },edge set E. Ifλ 1,λ 2,...,λ n are the eigenvalues of Seidel matrix S(G) then (i) λ i = 0. (ii) λ 2 i = n(n 1). Proof. i) We know that the sum of the eigenvalues of S(G) is the trace of S(G) λ i = a ii = 0. (ii) Similarly the sum of squares of the eigenvalues of S(G) is trace of [S(G)] 2 λ 2 i = j=1 a ij a ji = ii ) (a 2 + a ij a ji i =j = ii ) (a (a ij ) 2 i<j [ ( n = m( 1) 2 2 n + 2 = n 2 n. m )(1) 2]

5 Milovanović Bounds for Seidel Energy of a Graph Bounds for Seidel Energy Similar to McClelland s [11] bounds for energy of a graph, bounds for SE(G) are given in the following theorem. Theorem 4.1. Let G be a simple graph with n vertices and m edges and P = dets(g) then (n 2 n) + n(n 1)P n 2 SE(G) n(n 2 n). Proof. Cauchy Schwarz inequality is If a i = 1,b i = λ i then ( ) 2 ( a i b i a 2 i )( ) ( 2 ( )( λ i ) 1 [SE(G)] 2 n(n 2 n) [Lemma 3.1] SE(G) n(n 2 n) λ 2 i b 2 i ) Since arithmetic mean is not smaller than geometric mean we have 1 1 λ i λ j n(n 1) λ i λ j n(n 1) i =j i =j [ n ] 1 = λ i 2(n 1) n(n 1) [ n = λ i 2 n n = λ i ] 2 n = dets(g) 2 n = P 2 n i =j λ i λ j n(n 1)P 2 n

6 42 M.R. Rajesh kanna, R. Pradeep kumar, Mohammad Reza Farahani ( Now consider, [SE(G)] 2 = ) 2 λ i = λ i 2 + λ i λ j i =j [SE(G)] 2 (n 2 n) + n(n 1)P n 2 [From (4.1)] i.e., SE(G) (n 2 n) + n(n 1)P n 2 Recently Milovanović [12] et al. gave a sharper lower bounds for energy of a graph. In this paper similar bounds for minimum dominating Seidel energy of a graph are established. Theorem 4.2. Let G be a graph with n vertices and m edges. Let λ 1 λ 2... λ n be a non-increasing order of Seidel eigenvalues of S(G) then SE(G) [ n ] ( n(n 2 n) α(n)( λ 1 λ n ) 2 where α(n) = n 1 1 [ n ] ) and [x] denotes 2 n 2 the integral part of a real number. Proof. Let a,a 1,a 2,...a n,aand b, b 1,b 2,...b n,bbe real numbers such that a a i A and b b i B i = 1, 2,...nthen the following inequality is valid. n a i b i a i b i α(n)(a a)(b b) where [ n ] ( α(n) = n 1 1 [ n 2 n 2] ) and equality holds if and only if a 1 = a 2 =... = a n and b 1 = b 2 =... = b n. If a i = λ i, b i = λ i, a = b = λ n and A = B = λ 1, then ( 2 n λ i 2 λ i ) α(n)( λ 1 λ n ) 2 But λ i 2 = n 2 n

7 Milovanović Bounds for Seidel Energy of a Graph 43 and SE(G) n(n 2 n) [13] then the above inequality becomes n(n 2 n) (SE(G)) 2 α(n)( λ 1 λ n ) 2 i,e., SE(G) n(n 2 n) α(n)( λ 1 λ n ) 2 Theorem 4.3. Let G be a graph with n vertices and m edges. Let λ 1 λ 2... λ n > 0 be a non-increasing order of eigenvalues of S(G) then SE(G) n2 n + n λ 1 λ n. ( λ 1 + λ n ) Proof. Let a i = 0, b i, r and R be real numbers satisfying ra i b i Ra i, then the following inequality holds. [Theorem 2, [12]] bi 2 + rr a i (r + R) a i b i Put b i = λ i, a i = 1,r = λ n and R = λ 1 then λ i 2 + λ 1 λ n 1 ( λ 1 + λ n ) λ i i.e., n 2 n + λ 1 λ n n ( λ 1 + λ n )SE(G) SE(G) n2 n + n λ 1 λ n. ( λ 1 + λ n ) References [1] C. Adiga, A. Bayad, I. Gutman, S.A. Srinivas, The minimum covering energy of a graph, Kragujevac J. Sci., 34 (2012) [2] D. Cvetković, I. Gutman (eds.), Applications of Graph Spectra (Mathematical Institution, Belgrade, 2009). [3] D. Cvetković, I. Gutman (eds.), Selected Topics on Applications of Graph Spectra (Mathematical Institute Belgrade, 2011). [4] A. Graovac, I. Gutman, N. Trinajstić, Topological Approach to the Chemistry of Conjugated Molecules (Springer, Berlin, 1977). [5] I. Gutman, The energy of a graph. Ber. Math-Statist. Sekt. Forschungsz. Graz, 103, 1 22 (1978).

8 44 M.R. Rajesh kanna, R. Pradeep kumar, Mohammad Reza Farahani [6] I. Gutman, X. Li, J. Zhang, in Graph Energy, ed. by M. Dehmer, F. Emmert Streib., Analysis of Complex Networks. From Biology to Linguistics, (Wiley - VCH, Weinheim, 2009), pp [7] I. Gutman, in The energy of a graph: Old and New Results, ed. by A. Betten, A. Kohnert, R. Laue, A. Wassermann, Algebraic Combinatorics and Applications (Springer, Berlin, 2001), pp [8] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986). [9] Huiqing Liu, Mei Lu and Feng Tian, Some upper bounds for the energy of graphs, Journal of Mathematical Chemistry, Vol. 41, No. 1, (2007). [10] J.H. Koolen, V. Moulton, Maximal energy graphs, Adv.Appl. Math., 26, (2001). [11] B.J. McClelland, Properties of the latent roots of a matrix: The estimation of π- electron energies, J. Chem. Phys., 54, (1971). [12] I. Ž. Milovanović, E. I. Milovanović, A. Zakić, A Short note on Graph Energy, MATH Commun. Math. Comput. Chem, 72 (2014), [13] M. R. Rajesh Kanna, B. N. Dharmendra, and G. Sridhara, Minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics, 85, No. 4 (2013), [ [14] Willem H. Haemers, Seidel Switching and Graph Energy, Math. Commun. Math. Comput. Chem, 68 (2012),