Normalized Laplacian spectrum of two new types of join graphs
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1 Journal of Linear and Topological Algebra Vol. 6, No. 1, 217, 1-9 Normalized Laplacian spectrum of two new types of join graphs M. Hakimi-Nezhaad a, M. Ghorbani a a Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, , ran. Received 1 November 216; Revised 7 February 217; Accepted 2 April 217. Abstract. Let G be a graph without an isolated vertex, the normalized Laplacian matrix L(G) is defined as L(G) = D 1 2 L(G)D 1 2, where D is a diagonal matrix whose entries are degree of vertices of G. The eigenvalues of L(G) are called as the normalized Laplacian eigenvalues of G. n this paper, we obtain the normalized Laplacian spectrum of two new types of join graphs. n continuing, we determine the integrality of normalized Laplacian eigenvalues of graphs. Finally, the normalized Laplacian energy and degree Kirchhoff index of these new graph products are derived. c 217 AUCTB. All rights reserved. Keywords: Join of graphs, normalized Laplacian eigenvalue, integral eigenvalue. 21 AMS Subject Classification: 5C5, 5C9, 15A18, 15A ntroduction Let G = (V, E) be a simple graph (namely a graph without loop and multiple edges) on n vertices and m edges and the degree of vertex v V (G) is denoted by deg(v). The eigenvalues of the adjacency matrix A = A(G) of G are called the eigenvalues of G denoted by λ 1 (G) λ 2 (G) λ n (G). Let λ 1 (G), λ 2 (G),..., λ s (G) be the distinct eigenvalues of G with multiplicity t 1, t 2,..., t s, respectively. The multiset {[λ 1 (G)] t 1, [λ 2(G)] t 2,..., [λ s(g)] t s } of eigenvalues of A is called the spectrum of G. Corresponding author. address: mghorbani@srttu.edu (M. Ghorbani). Print SSN: c 217 AUCTB. All rights reserved. Online SSN:
2 2 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) 1-9. The energy of graph G is a graph invariant introduced by van Gutman as E(G) = n λ i (G), i=1 see for more details [8, 1, 17]. Let L(G) = D(G) A(G) be the Laplacian matrix of graph G, where D(G) = [d ij ] is the diagonal matrix whose entries are degree of vertices, namely, d ii = deg(v i ) and d ij = for i j. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. The signless Laplacian eigenvalues of G are denoted by q 1 (G), q 2 (G),..., q n (G). f G is regular of degree r, then the eigenvalues of Q(G) are 2r, r + λ 2,..., r + λ n, see [1, p.14]. n [15], Jooyandeh et al. introduced the concept of incidence energy of a graph as RE(G) = n qi (G). (1) i=1 Let G be a graph without an isolated vertex, the normalized Laplacian matrix L(G) is defined as L(G) = D 1 2 L(G)D 1 2 which implies that its (i, j) entry is 1 if i = j, and it is 1/ deg(v i )deg(v j ), if i j and the vertices v i, v j are adjacent, and is zero otherwise. The eigenvalues of L(G) are the roots of g(x) = det(δ L(G)) and we call them as normalized Laplacian eigenvalues of G, denoted by δ 1 (G), δ 2 (G),..., δ n (G), where δ n (G) = for all graphs, see for more details [2, 3, 6]. The complement of graph G is denoted by G and a complete graph on n vertices is denoted by K n. Also a cycle graph with n vertices is denoted by C n. Clearly, K n is empty graph. The subdivision graph S(G) of a graph G is obtained by inserting an additional vertex in the middle of each edge of G. Equivalently, each edge of G is replaced by a path of length 2, see Figure 1. t is a well-known fact that P S(G) (x) = x m n Q G (x 2 ), where P G (x) and Q G (x) denote the characteristic polynomial and the signless Laplacian characteristic polynomial of G, respectively [7, p. 63]. From the definition, one can see that ± q 1 (G), ± q 2 (G),..., ± q n (G) and [] m n are all eigenvalues of S(G). G S(G) Figure 1. Two graphs G and subdivision S(G). ndulal in [13] defined two classes of join graphs as follows: Definition 1.1 The S vertex join of two graphs G 1 and G 2 denoted by G 1 G 2 is obtained from S(G 1 ) and G 2 by joining all vertices of G 1 with all vertices of G 2, see Figure 2.
3 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) n 2 Figure 2. Graphs K 2 K n2 and C 4 K 2. Definition 1.2 The S edge join of two graphs G 1 and G 2 denoted by G 1 G 2 is obtained from S(G 1 ) and G 2 by joining all vertices of S(G 1 ) corresponding to the edges of G 1 with all vertices of G 2, see Figure 3. n 2 Figure 3. Graphs K 2 K n2 and C 4 K 2. The following results are crucial throughout this paper. Lemma 1.3 [7] Let G be a r regular (n, m) graph with adjacency matrix A(G) and incidence matrix R. Let L(G) be its line graph. Then RR T = A(G) + r and R T R = A(L(G)) + 2. Moreover, if G has an eigenvalue equals to r with an eigenvector V, then G is bipartite and R T V =. Lemma 1.4 [7] Let G be an r regular (n, m) graph with eigenvalues r, λ 2,..., λ n. Then {[2r 2] 1, [λ i + r 2] 1, [ 2] m n }, (2 i n) consist the spectrum of line grah. Also V is an eigenvector belonging to the eigenvalue 2 if and only if RV =. 2. The normalized Laplacian spectra of two new join graphs n this section, we determine the normalized Laplacian spectrum of G 1 G 2 and G 1 G 2, where G i is r i regular (i = 1, 2). Consider two graphs G 1, G 2 with V (G 1 ) = {u 1, u 2,..., u n1 }, E(G 1 ) = {e 1, e 2,..., e m1 } and V (G 2 ) = {v 1, v 2,..., v n2 }. Then 1 k and k are two vectors of order k with all elements equal to 1 and, respectively. Moreover i j denotes an i j matrix whose entries are zero, J i j is one whose entries are 1 and n is the identity matrix of order n. Theorem 2.1 For i = 1, 2, let G i be r i regular graph with n i vertices with incidence matrix R and eigenvalues r i = λ 1 (G i ) λ 2 (G i ) λ n (G i ). Then the normalized
4 4 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) 1-9. Laplacian spectrum of G 1 G 2 is {[] 1, [ 1 ± λj(g 1)+r 1 ] 1, [ 2(n 2 +r 1 ) 1 λ k(g 2) ] 1, n 1+r 2 [1] m 1 n 1 }, (2 j n 1 ), (2 k n 2 ) together with the roots of x 2 3n 1 + 2r 2 x + 2n 1n 2 + 2r 1 n 1 + n 2 r 2 n 1 + r 2 (n 1 + r 2 )(n 2 + r 1 ) Proof. Let G i be an r i regular graph with n i vertices and an adjacency matrix A(G i ), i = 1, 2. Let R be the incidence matrix of G 1. Then by a proper labeling of vertices, the normalized Laplacian matrix of G 1 G 2 can be written as =. L(G 1 G 2 ) = D 1 2 L(G1 G 2 )D 1 2 = n1 m 1 n1 2(n2+r 1) T m 1 n 1 J n2 n 1 J n1 n 2 (n1+r 2)(n 2+r 1) m1 m1 n 2 n2 m 1 n2 A(G2) n 1+r 2, where D = (n 2 + r 1 ) n1 n1 m 1 n1 n 2 m1 n 1 2 m1 m1 n 2 n2 n 1 n2 m 1 (n 1 + r 2 ) n2, and (n 2 + r 1 ) n1 n1 m 1 J n1 n 2 L(G 1 G 2 ) = R T m 1 n 1 2 m1 m1 n 2 J n2 n 1 n2 m 1 (n 1 + r 2 ) n2 A(G) 1 n1 is an eigenvector corresponding to eigenvalue r 1 of G and the other eigenvectors are orthogonal to 1 n1. Let X be an eigenvector of G 1 corresponding to an eigenvalue λ j (G 1 ) r 1, 2 j n 1. Then Φ = [ αx R T X ] T is eigenvector of L(G 1 G 2 ) corresponding to the normalized eigenvalue δ of graph G 1 G 2. Then L.Φ = θφ. Thus n1 m 1 n1 T m 1 n 1 J n2 n 1 J n1 n 2 m1 m1 n 2 n2 m 1 n2 A(G 2) αx R T X αx = θ R T X This implies that (α RRT )X 2(n2+r 1) θαx α (1 )RT X = θr T X The last equality follows from Lemma 1.3. By solving these equations, we get α =
5 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) ± λ j (G 1 ) + r 1, 2 j n 1. Therefore, Φ k = [ ± λ j (G 1 ) + r 1 X R T X ] T is an eigenvector of L(G1 G 2 ) corresponding to the normalized eigenvalues θ k of graph G 1 G 2, where θ k = 1 ± 2(n 2+r 1) (k = 1, 2). Now consider the m 1 n 1 linearly independent eigenvectors Z l (1 l m 1 n 1 ) of L(G 1 ) corresponding to the eigenvalue 2. Then by Lemma 1.4, RZ l =. Consequently Ω l = [ Z l ] T is an eigenvector of L(G 1 G 2 ) with an eigenvalue 1, since L.Ω = n1 m 1 n1 T m 1 n 2 2(n2+r 1) J n2 n 1 (n1+r 2)(n 2+r 1) J n1 n 2 m1 m1 n 2 n2 m 1 n2 A(G 2) Z l = n1 m 1 On the other hand, 1 n2 is an eigenvector of G 2 corresponding to the eigenvalue r 2, in which all other eigenvectors are orthogonal to 1 n2. Let Y be the eigenvector of G 2 corresponding to the eigenvalue λ j (G 2 ) r 2 (2 j n 2 ). By a similar arguments we can deduce that Ψ = [ Y ] T is eigenvector of L(G1 G 2 ) corresponding to the eigenvalue 1 λ j(g 2 ) (2 j n 2 ). Let Λ = [ αj n1 1 βj n1 1 γj n1 1 ] T for some vector (α, β, γ) (,, ). The equation L(G 1 G 2 ).Λ = xλ yields that Λ is an eigenvector of A corresponding to eigenvalue x if and only if Λ = [ α β γ ] T is an eigenvector of the matrix M, where M = 1 2 2(n2+r 1) n 1 (n1+r 2)(n 2+r 1) r 1 n A(G 2) Z l The characteristic polynomial of M is x 3 3n1+2r2 x 2 + 2n1n2+2r1n1+n2r2 (n 1+r 2)(n 2+r 1) x = and this completes the proof. Corollary 2.2 f G 2 = Kn2 then the normalized Laplacian spectrum of G 1 G 2 is {[] 1, [1] m1 n1+n2, [ 1 ± ] 1, 2(n 2+r 1) [2] 1 }, (2 j n 1 ). Theorem 2.3 For i = 1, 2, let G i be an r i regular graph on n i vertices with incidence matrix R and the eigenvalues r i = λ 1 (G i ) λ 2 (G i ) λ n (G i ). Then the normalized Laplacian spectrum of G 1 G 2 is {[] 1, 1± r, 1 λ k(g 2 ) 1(n 2+2) m 1 +r 2, [1] m 1 n 1 }, (2 j n 1 ), (2 k n 2 ), together with the roots of x 2 3m 1+2r 2 m 1+r 2 x + 2m 1n 2 +4m 1 +n 2 r 2 (m 1 +r 2 )(2+n 2 ) =. Proof. t is not difficult to see that the normalized Laplacian matrix of G 1 G 2 can be written as follows: L(G 1 G 2 ) = D 1 2 L(G1 G 2 )D 1 2 n1 m n1 1 r1(n 2+2) = T m 1 n 1 r1 (n 2 +2) J n2 m 1 n2 n 1 (m1 +r 2 )(n 2 +2) n1 n 2 J m1 n 2 m1 (m1 +r 2 )(n 2 +2) n2 A(G 2) m 1 +r 2,
6 6 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) 1-9. where D = (n 2 + 2) n1 n1 m 1 n1 n 2 m1 n 1 r 1 m1 m1 n 2 n2 n 1 n2 m 1 (m 1 + r 2 ) n2, and (n 2 + 2) n1 n1 m 1 n1 n 2 L(G 1 G 2 ) = R T m 1 n 1 r 1 m1 J m1 n 2 n2 n 1 J n2 m 1 (m 1 + r 2 ) n2 A(G) Continuing of the proof is similar to that of Theorem 2.1. Corollary 2.4 f G 2 = Kn2, then the normalized Laplacian spectrum of G 1 G 2 is {[] 1, [1] m 1 n 1 +n 2, [ 1 ± ] 1, r 1(n 2+2) [2] 1 }, (2 j n 1 ). 3. ntegrality of two new join graphs The graph G is L integral if all its L eigenvalues are integral. Since all L eigenvalues of G are in interval [, 2], we can deduced that G is L integral if and only if its eigenvalues are, 1, or 2. The complete bipartite graphs are such graphs; in fact, no other connected graphs are L integral, see [14]. Proposition 3.1 [14] Let G be a connected graph. Then the following statements are equivalent. 1) G is bipartite with three L eigenvalues, 2) G is L integral, 3) G is complete bipartite. The following propositions give a necessary and sufficient conditions in which S vertex and S edge joins of graphs are L integral. Proposition 3.2 f G i is r i regular connected graph (i = 1, 2), then G 1 G 2 and G 1 G 2 are L integral if and only if G 1 = K2 and G 2 = Kn2. Proof. Since K 2 K n2 = K 2,n2 and K 2 K n2 = K 1,n2 +2 are two graphs of order n with the normalized Laplacian eigenvalues {[] 1, [1] n 1, [2] 1 }, these graphs are L integral. Conversely, suppose that G 1 G 2 is L integral, G 1 = K2 and G 2 = Kn2. By Theorem 2.1, the normalized Laplacian spectrum of G 1 G 2 is {[] 1, [ 1± 2(n 2+r 1) [1] m 1 n 1 }, (2 j n 1 ), (2 k n 2 ) together with the roots of x 2 3n 1 + 2r 2 x + 2n 1n 2 + 2r 1 n 1 + n 2 r 2 n 1 + r 2 (n 1 + r 2 )(n 2 + r 1 ) =. ] 1, [ 1 λ k (G 2 ) ] 1, Since G 1 is connected graph, G 1 G 2 is connected. Hence, [] 1 is an eigenvalue of G 1 G 2. Thus, the other eigenvalues must be equal to [1] n 1, [2] 1. Since 1 λ j(g 2 ) < 2, 2 j n 2, we have λ j(g 2 ) =. So, λ j (G 2 ) =, 2 j n 2 and it implies that G 2 = Kn2,
7 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) a contradiction. Since 1 λj(g 1)+r 1 2(n 2 +r 1 ) < 2, 2 j n 1, we achieve λj(g 1)+r 1 2(n 2 +r 1 ) = and thus λ j (G 1 ) = r 1, 2 j n 1, a contradiction. On the other hand, 1 < 1+ λj(g 1)+r 1 2(n 2 +r 1 ) 2, 2 j n 1, yields that λj(g 1)+r 1 2(n 2 +r 1 ) = 1. So, λ j (G 1 ) = r 1 + 2n 2 >, 2 j n 1, a contradiction. As well as, for two eigenvalues that are the roots of x 2 3n 1 + 2r 2 x + 2n 1n 2 + 2r 1 n 1 + n 2 r 2 n 1 + r 2 (n 1 + r 2 )(n 2 + r 1 ) =. (2) Since, G is connected and the normalized Laplacian eigenvalues are integral, one can deduce that Eq.(2) is equal to (x 1) 2 = x 2 2x + 1 or (x 1)(x 2) = x 2 3x + 2. n the first case, one can conclude easily that 2n1n2+2r1n1+n2r2 (n 1+r 2)(n 2+r 1) = 1 which yields that n 1 n 2 = r 2 n 1, for r 1 1 which comes to a contradiction, because for example, there is no an 8-regular graph of order 5. Thus, the second case holds, namely x 2 3n 1 + 2r 2 x + 2n 1n 2 + 2r 1 n 1 + n 2 r 2 n 1 + r 2 (n 1 + r 2 )(n 2 + r 1 ) = x 2 3x + 2 and the proof is completed. Proposition 3.3 Let G 1 be an empty graph of order n 1 and G 2 be r regular graph of order n 2, then G 1 G 2 and G 1 G 2 are L integral if and only if G 2 = Kn2 or G 2 is complete bipartite graph. Proof. By using Proposition 3.1, Theorems 2.1 and 2.3, the proof is straight forward. The normalized Laplacian energy of G is defined as LE(G) = n i=1 δ i(g) 1, see [4, 12] for details. Proposition 3.4 Let G 1 is an r 1 regular of graph of order n 1 and G 2 = Kn2. Then 1) LE(G ( 1 G 2 ) = RE(G 1) ) 2r 1, 2) LE(G ( 1 G 2 ) = RE(G 1) ) 2r 1. r1(n 2+2) Proof. By Corollary 2.2, Corollary 2.4, respectively, we have LE(G 1 G 2 ) = n δ i (G 1 G 2 ) 1 i=1 n 1 = 1 + ± λ j (G 1 ) + r 1 2(n 2 + r 1 ) + 1 n 1 λ j (G 1 ) + r 1 = (n 2 + r 1 ) ( = RE(G 1) 2r 1 r1 (n 2 + 2) ),
8 8 M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) 1-9. LE(G 1 G 2 ) = n δ i (G 1 G 2 ) 1 i=1 n 1 = 1 + ± λ j (G 1 ) + r 1 r 1 (n 2 + 2) + 1 n 1 λ j (G 1 ) + r 1 = r 1 (n 2 + 2) ( = RE(G 1) 2r 1 r1 (n 2 + 2) ), where the last equality follows from RE(G 1 ) n 1 2r 1 = λ j (G 1 ) + r 1. The degree Kirchhoff index of graph G is defined by Chen et al. in [5] as Kf (G) = i<j d i r ij d j, where r ij denotes the resistance-distance between vertices v i and v j in a graph G. They proved that Kf (G) = 2m n 1 i=2 δ i (G), where δ i(g) is normalized Laplacian eigenvalues of G of order n (2 i n), see [11, 18] for details. Proposition 3.5 Let G 1 is an r 1 regular graph of order n 1 with m 1 edges and G 2 = K n2. Then ( 1 Kf (G 1 G 2) = 2(2m 1 + n 1n 2) 2 + m1 n1 + n2 + (n 1 1) 2(n 2 + r ) 1) (n 1 1) 2(n 2 + r 1 ) ± (RE(G 1 ), 2r 1 ) 1 Kf (G 1 G 2 ) = 2m 1 (2 + n 2 )( 2 + m (n 1 1) r 1(n 2 + 2) ) 1 n 1 + n 2 + (n 1 1) r 1 (n 2 + 2) ± (RE(G 1 ). 2r 1 ) Proof. t is not difficult to see that the number of edges of G 1 G 2 and G 1 G 2 are E(G 1 G 2 ) = 2m 1 +n 1 n 2 and E(G 1 G 2 ) = m 1 (2+n 2 ), respectively. On the other hand, since G 1 is r 1 regular by using Eq.(1), and so n 1 RE(G 1 ) = λ j (G 1 ) + r 1 j=1 RE(G 1 ) n 1 2r 1 = λ j (G 1 ) + r 1. Now, Corollaries 2.2, 2.4 complete the proof.
9 References M. Hakimi-Nezhaad et al. / J. Linear. Topological. Algebra. 6(1) (217) [1] A. E. Brouwer, W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 212. [2] S. Butler, Eigenvalues and Structures of Graphs, Ph.D. dissertation, University of California, San Diego, 28. [3] M. Cavers, The normalized Laplacian matrix and general Randić index of graphs, Ph.D. University of Regina, 21. [4] M. Cavers, S. Fallat, S. Kirkland, On the normalized Laplacian energy and general Randic index R 1 of graphs, Linear Algebra Appl. 433 (21), [5] H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discr. Appl. Math. 155 (27), [6] F. R. K. Chung, Spectral Graph Theory, American Math. Soc. Providence, [7] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Academic Press, New York, 198. [8]. Gutman, The energy of a graph, Steiermrkisches Mathematisches Symposium (Stift Rein, Graz, 1978), Ber. Math. Statist. Sekt. Forsch. Graz 13 (1978), [9]. Gutman, B. Mohar, The Quasi-Wiener and the Kirchhoff indicescoincide, J. Chem. nf. Comput. Sci. 36 (1996), [1]. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohner, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin, 21, [11] M. Hakimi-Nezhaad, A. R. Ashrafi,. Gutman, Note on degree Kirchhoff index of graphs, Trans. Comb. 2 (3) (213), [12] M. Hakimi-Nezhaad, A. R. Ashrafi, A note on normalized Laplacian energy of graphs, J. Contemp. Math. Anal. 49 (5) (214), [13] G. ndulal, Spectrum of two new joins of graphs and infinite families of integral graphs, Kragujevac J. Math. 36 (1) (212), [14] E. R. Van Dam, G. R. Omidi, Graphs whose normalized Laplacian has three eigenvalues, Linear Algebra Appl. 435 (1) (211), [15] M. R. Jooyandeh, D. Kiani, M. Mirzakhah, ncidence energy of a graph, MATCH Commun. Math. Comput. Chem. 62 (29), [16] D. J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993), [17] X. Li, Y. Shi,. Gutman, Graph energy, Springer, New York, 212. [18] B. Zhou, N. Trinajstić, On resistance-distance and Kirchhoff index, J. Math. Chem. 46 (29),
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