LAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS

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1 Kragujevac Journal of Mathematics Volume 39() (015), Pages LAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS HARISHCHANDRA S. RAMANE 1, GOURAMMA A. GUDODAGI 1, AND IVAN GUTMAN Abstract. The Laplacian energy of a graph G with n vertices and m edges is defined as LE(G) = n µ i m/n, where µ 1, µ,..., µ n are the Laplacian eigenvalues of G. If two graphs G 1 and G have equal average vertex degrees, then LE(G 1 G ) = LE(G 1 ) LE(G ). Otherwise, this identity is violated. We determine a term Ξ, such that LE(G 1 ) LE(G ) Ξ LE(G 1 G ) LE(G 1 )LE(G )Ξ holds for all graphs. Further, by calculating LE of the Cartesian product of some graphs, we construct new classes of Laplacian non-cospectral, Laplacian equienergetic graphs. 1. Introduction Let G be a finite, simple, undirected graph with n vertices v 1, v,..., v n and m edges. In what follows, we say that G is an (n, m)-graph. Let A(G) be the adjacency matrix of G and let λ 1, λ,..., λ n be its eigenvalues. Let D(G) be the diagonal matrix whose (i, i)-th entry is the degree of a vertex v i. The matrix C(G) = D(G) A(G) is called the Laplacian matrix of G. The Laplacian polynomial of G is defined as ψ(g, µ) = det[µi C(G)], where I is an identity matrix. The eigenvalues of C(G), denoted by µ i = µ i (G), i = 1,,..., n, are called the Laplacian eigenvalues of G [16]. Two graphs are said to be Laplacian cospectral if they have same Laplacian eignevalues. The adjacency eigenvalues and Laplacian eigenvalues satisfies the following conditions: λ i = 0 and µ i = m. Key words and phrases. Laplacian spectrum, Laplacian energy, Cartesian product, Laplacian equienergetic graphs. 010 Mathematics Subject Classification. Primary: 05C50, 05C75 Received: September, 015. Accepted: November 11,

2 194 H. RAMANE, G. GUDODAGI, AND I. GUTMAN The energy of a graph G is defined as E = E(G) = λ i. It was introduced by one of the present authors in the 1970s, and since then has been much studied in both chemical and mathematical literature. For details see the book [15] and the references cited therein. The Laplacian energy of a graph was introduced a few years ago [13] and is defined as LE(G) = µ i(g) m n. This definition is chosen so as to preserve the main features of the ordinary graph energy E, see [18]. Basic properties and other results on Laplacian energy can be found in the survey [1], the recent papers [6 8, 11, 1, 17, 19], and the references cited therein.. Laplacian Energy of Union of Graphs Let G 1 and G be two graphs with disjoint vertex sets. Let for i = 1,, the vertex and edges sets of G i be, respectively, V i and E i. The union of G 1 and G is a graph G 1 G with vertex set V 1 V and the edge set E 1 E. If G 1 is an (, m 1 )-graph and G is an (n, m )-graph then G 1 G has n vertices and m 1 m edges. It is easy to see that the Laplacian spectrum of G 1 G is the union of the Laplacian spectra of G 1 and G. In [13] it was proven that if G 1 and G have equal average vertex degrees, then LE(G 1 G ) = LE(G 1 ) LE(G ). If the average vertex degrees are not equal, that is m 1 m n, then it may be either LE(G 1 G ) > LE(G 1 ) LE(G ) or LE(G 1 G ) < LE(G 1 )LE(G ) or, exceptionally, LE(G 1 G ) = LE(G 1 )LE(G ) [13]. In this section we study the Laplacian establish some additional relations between LE(G 1 G ) and LE(G 1 ) LE(G ). Theorem.1. Let G 1 be an (, m 1 )-graph and G be an (n, m )-graph, such that m 1. Then > m n LE(G 1 ) LE(G ) 4(n m 1 m ) n LE(G 1 G ) (.1) LE(G 1 ) LE(G ) 4(n m 1 m ) n.

3 LAPLACIAN ENERGY 195 Proof. For the sake of simplicity, denote G 1 G by G. Then G is an ( n, m 1 m )- graphs. By the definition of Laplacian energy, (.) LE(G 1 G ) = 1 n µ i(g) (m 1 m ) n = µ i(g) (m n1n 1 m ) n µ i(g) (m 1 m ) n i= 1 = µ i(g 1 ) (m n 1 m ) n µ i(g ) (m 1 m ) n = µ i(g 1 ) m 1 m 1 (m 1 m ) n n µ i(g ) m m (m 1 m ) n n n µ i(g 1 ) m 1 n m 1 1 (m 1 m ) n n µ i(g ) m n n m (m 1 m ) n n. Since n m 1 > m, Eq. (.) becomes ( m1 LE(G 1 G ) LE(G 1 ) (m ) 1 m ) n ( LE(G ) n m (m ) 1 m ) n n = LE(G 1 ) LE(G ) 4(n m 1 m ) n which is an upper bound. To obtain the lower bound, we just have to note that in full analogy to the above arguments, LE(G 1 G ) µ i(g 1 ) m 1 n m 1 1 (m 1 m ) n

4 196 H. RAMANE, G. GUDODAGI, AND I. GUTMAN (.3) n µ i(g ) m n n m (m 1 m ) n n. Since n m 1 > m, the Eq. (.3) becomes ( m1 LE(G 1 G ) LE(G 1 ) (m ) 1 m ) n ( LE(G ) n m (m ) 1 m ) n n which is a lower bound. = LE(G 1 ) LE(G ) 4(n m 1 m ) n Corollary.1. [13] Let G 1 be an (, m 1 )-graph and G be (n, m )-graph such that m 1. Then = m n LE(G 1 G ) = LE(G 1 ) LE(G ). Corollary.. Let G 1 be an r 1 -regular graph on vertices and G be an r -regular graph on n vertices, such that r 1 > r. Then LE(G 1 ) LE(G ) n (r 1 r ) n LE(G 1 G ) LE(G 1 ) LE(G ) n (r 1 r ) n. Proof. Result follows by setting m 1 = r 1 / and m = n r / into Theorem.1. Theorem.. Let G be an (n, m)-graph and G be its complement, and let m > n(n 1)/4. Then LE(G)LE(G) [ 4m n(n 1) ] LE(G G) LE(G)LE(G) [ 4m n(n 1) ]. Proof. G is a graph with n vertices and n(n 1)/ m edges. Substituting this into Eq. (.1), the result follows. Theorem.3. Let G be an (n, m)-graph and G be the graph obtained from G by removing k edges, 0 k m. Then LE(G) LE(G ) k LE(G G ) LE(G) LE(G ) k. Proof. The number of vertices of G is n and the number of edges is m k. Substituting this in Eq. (.1), the result follows.

5 LAPLACIAN ENERGY Laplacian Energy of Cartesian Product Let G be a graph with vertex set V 1 and H be a graph with vertex set V. The Cartesian product of G and H, denoted by G H is a graph with vertex set V 1 V, such that two vertices (u 1, v 1 ) and (u, v ) are adjacent in G H if and only if either u 1 = u and v 1 is adjacent to v in H or v 1 = v and u 1 is adjacent to u in G [14]. [ ] A0 A Lemma 3.1. [9] Let A = 1 be a symmetric block matrix. Then the A 1 A 0 spectrum of A is the union of the spectra of A 0 A 1 and A 0 A 1. Theorem 3.1. If µ 1, µ,..., µ n are the Laplacian eigenvalues of a graph G, then the Laplacian eigenvalues of G K are µ 1, µ,..., µ n and µ 1, µ,..., µ n. Proof. The Laplacian matrix of G K is [ ] C(G) I I C(G K ) = = I C(G) I [ ] C0 C 1 C 1 C 0 where C(G) is the Laplacian matrix of G and I is an identity matrix of order n. By Lemma 3.1, the Laplacian spectrum of G K is the union of the spectra of C 0 C 1 and C 0 C 1. Here C 0 C 1 = C(G). Therefore, the eigenvalues of C 0 C 1 are the Laplacian eigenvalues of G. Because C 0 C 1 = C(G) I, the characteristic polynomial of C 0 C 1 is ψ(c 0 C 1, µ) = det [ µi (C 0 C 1 ) ] = det [ µi (C(G) I) ] = det [ (µ )I C(G)) ] = ψ(g, µ ). Therefore the eigenvalues of C 0 C 1 are µ i, i = 1,,..., n. The Laplacian eigenvalues of the complete graph K n are n (n 1 times) and 0. The Laplacian eigenvalues of the complete bipartite regular graph K k,k are k, k (k times) and 0. The Laplacian eigenvalues of the cocktail party graph CP (k) (the regular graph on n = k vertices and of degree k ) are k (k 1 times), k (k times) and 0 [16]. Applying Theorem 3.1, we directly arrive at the following example. Example 3.1. LE(K n K ) = LE(K k,k K ) = { 4n 4, if n >, n, if n, { 8k 4, if k > 1, 6k, if k = 1, LE(CP (k) K ) = 10k 8, if k. Theorem 3.. Let G be an (n, m)-graph. Then [LE(G) n] LE(G K ) [LE(G) n].

6 = m m n m m n =. 198 H. RAMANE, G. GUDODAGI, AND I. GUTMAN Proof. Let µ 1, µ,..., µ n be the Laplacian eigenvalues of G. Then by Theorem 3.1, the Laplacian eigenvalues of G K are µ i, i = 1,,..., n and µ i, i = 1,,..., n. The graph G K has n vertices and m n edges. Therefore, (3.1) LE(G K ) = = µ i (m n) n µ i m n 1 µ i µ i m n 1. (m n) n Equation (3.1) can be rewritten as LE(G K ) µ i m n n µ i m n n = LE(G) n which is an upper bound. For lower bound, Eq. (3.1) can be rewritten as LE(G K ) µ i m n n µ i m n n = LE(G) n. Theorem 3.3. For a graph G with n vertices, LE(G K ) n. Proof. From Eq. (3.1) ( LE(G K ) µ i m ) n 1 ( µ i m ) n 1 4. Laplacian Equienergetic Graphs Two graphs G 1 and G are said to be equienergetic if E(G 1 ) = E(G ) []. For details see the book [15]. In analogy to this, two graphs G 1 and G are said to be Laplacian equienergetic if LE(G 1 ) = LE(G ). Obviously Laplacian cospectral graphs are Laplacian equienergetic. Therefore we are interested in Laplacian non-cospectral graphs with equal number of vertices, having equal Laplacian energies. Stevanović [4] has constructed Laplacian equienergetic threshold graphs. Fritscher et al. [10] discovered a family of Laplacian equienergetic unicyclic graphs. We now report some additional classes of such graphs. The line graph of the graph G, denoted by L(G), is a graph whose vertices corresponds to the edges of G and two vertices in L(G) are adjacent if and only if the corresponding edges are adjacent in G [14]. The k-th iterated line graph of G is defined

7 LAPLACIAN ENERGY 199 as L k (G) = L(L k 1 (G)) where L 0 (G) G and L 1 (G) L(G). If G is a regular graph of order n 0 and of degree r 0, then L(G) is a regular graph of order = n 0 r 0 / and of degree r 1 = r 0. Consequently, the order and degree of L k (G) are [3, 4]: n k = 1 n k 1r k 1 and r k = r k 1 where n i and r i stand for the order and degree of L i (G), i = 0, 1,,.... Therefore [3,4], (4.1) n k = n k 1 0 r k i = n 0 k and i=0 k 1 ( i r 0 i1 ) i=0 (4.) r k = k r 0 k1. Theorem 4.1. [3] If λ 1, λ,..., λ n are the adjacency eigenvalues of a regular graph G of order n and of degree r, then the adjacency eigenvalues of L(G) are λ i r i = 1,,..., n, and n(r )/ times. Theorem 4.. [] If λ 1, λ,..., λ n are the adjacency eigenvalues of a regular graph G of order n and of degree r, then the adjacency eigenvalues of G, the complement of G, are n r 1 and λ i 1, i =, 3,..., n. Theorem 4.3. [16] If λ 1, λ,..., λ n are the adjacency eigenvalues of a regular graph G of order n and of degree r, then its Laplacian eigenvalues are r λ i, i = 1,,..., n. For G being a regular graph of degree r 3, and for k, expressions for E ( L k (G) ) and E ( L k (G) ) were reported in [0, 1]. Theorem 4.4. If G is a regular graph of order n and of degree r 4, then LE(L (G) K ) = 4nr(r ). Proof. Let λ 1, λ,..., λ n be the adjacency eigenvalues of G. Then by Theorem 4.1, the adjacency eigenvalues of L(G) are } λ i r i = 1,,..., n, and (4.3) n(r )/ times. Since L(G) is a regular graph of order nr/ and of degree r, by Eq. (4.3) the adjacency eigenvalues of L (G) are λ i 3r 6 i = 1,,..., n, and (4.4) r 6 n(r )/ times, and nr(r )/ times.

8 00 H. RAMANE, G. GUDODAGI, AND I. GUTMAN Since L (G) is a regular graph of order nr(r 1)/ and of degree 4r, by Theorem 4.3 and Eq. (4.4), the Laplacian eigenvalues of L (G) are (4.5) r λ i i = 1,,..., n, and 4r 4 r n(r )/ times, and nr(r )/ times. Using Theorem 3.1 and Eq. (4.5), the Laplacian eigenvalues of L (G) K are (4.6) r λ i i = 1,,..., n, and r n(r )/ times, and 4r 4 nr(r )/ times, and r λ i i = 1,,..., n, and r n(r )/ times, and 4r nr(r )/ times. The graph L (G) K is a regular graph of order and of degree 4r 5. By Eq. (4.6), the Laplacian energy of L (G) K is computed as LE(L (G) K ) = r λi (4r 5) r (4r 5) n(r ) 4r 4 (4r 5) nr(r ) r λi (4r 5) (4.7) r (4r 5) n(r ) 4r (4r 5) nr(r ) = λi 3r 5 r 5 n(r ) 1 nr(r ) λ i 3r 7 r 7 n(r ) 3 nr(r ). If d max is the greatest vertex degree of a graph, then all its adjacency eigenvalues belongs to the interval [ d max, d max ] [5]. In particular, the adjacency eigenvalues of a regular graph of degree r satisfy the condition r λ i r, i = 1,,..., n.

9 LAPLACIAN ENERGY 01 If r 4 then λ i 3r 5 > 0, λ i 3r 7 > 0, r 5 > 0, and r 7 > 0. Therefore by Eq. (4.7), and bearing in mind that n λ i = 0, LE(L (G) K ) = n(r ) nr(r ) λ i n(3r 5) (r 5) = 4nr(r ). n(r ) 3nr(r ) λ i n(3r 7) (r 7) Corollary 4.1. Let G be a regular graph of order n 0 and of degree r 0 4. Let n k and r k be the order and degree, respectively of the k-th iterated line graph L k (G) of G, k. Then (4.8) LE(L k (G) K ) = 4n k r k (r k ) = 4n k 1 (r k 1 ), k LE(L k (G) K ) = 4n 0 (r 0 ) ( i r 0 i1 ), i=0 ( ) LE(L k rk (G) K ) = 8(n k n k 1 ) = 8n k. r k From Eq. (4.8) we see that the energy of L k (G) K, k is fully determined by the order n and degree r 4 of G. Theorem 4.5. If G is a regular graph of order n and of degree r 3, then ) LE (L (G) K = (nr 4)(4r 6) 4. Proof. Let λ 1, λ,..., λ n be the adjacency eigenvalues of a regular graph G of order n and of degree r 3. Then the adjacency eigenvalues of L (G) are as given by Eq. (4.4). Since L (G) is a regular graph of order nr(r 1)/ and of degree 4r, by Theorem 4. and Eq. (4.4), the adjacency eigenvalues of L (G) are λ i 3r 5 i =, 3,..., n, and r 5 n(r )/ times, and (4.9) 1 nr(r )/ times, and (/) 4r 5.

10 0 H. RAMANE, G. GUDODAGI, AND I. GUTMAN Since L (G) is a regular graph of order nr(r 1)/ and of degree (nr(r 1)/) 4r5, by Theorem 4.3 and Eq. (4.9), the Laplacian eigenvalues of L (G) are (/) r λ i i =, 3,..., n, and (/) r n(r )/ times, and (4.10) (/) 4r 4 nr(r )/ times, and 0. Using Theorem 3.1 and Eq. (4.10), the Laplacian eigenvalues of L (G) K are (/) r λ i i =, 3,..., n, and (4.11) (/) r n(r )/ times, and (/) 4r 4 nr(r )/ times, and 0 1 time, and (/) r λ i i =, 3,..., n, and (/) r n(r )/ times, and (/) 4r 6 nr(r )/ times, and. The graph L (G) K is a regular graph of order and of degree (/) 4r 6. By Eq. (4.11), ) ( LE (L (G) K = r λ i 4r 6) i= ( r ( 4r 4 ( 0 4r 6) i= ) n(r ) 4r 6 ) nr(r ) 4r 6 ( r λ i 4r 6) ( r 4r 6 ) n(r )

11 (4.1) = LAPLACIAN ENERGY 03 ( 4r 6 ( 4r 6) λi 3r 6 r 6 n(r ) i= nr(r ) r 4 n(r ) 0 nr(r ) ) nr(r ) 4r 6 4r 6 λi 3r 4 i= 4r 4. All adjacency eigenvalues of a regular graph of degree r satisfy the condition r λ i r, i = 1,,..., n [5]. Therefore if r 3, then λ i 3r 6 0, λ i 3r 4 0, r 6 0, r 4 0, ( /) 4r 6 < 0, and ( /) 4r 4 < 0. Then from Eq. (4.1), and bearing in mind that n λ i = r, we get ) LE (L (G) K = i= λ i (n 1)(3r 6) (r 3)n(r ) nr(r ) i= 4r 6 λ i (n 1)(3r 4) i= (r )n(r ) = (nr 4)(r 3) 4. 4r 4 Corollary 4.. Let G be a regular graph of order n 0 and of degree r 0 3. Let n k and r k be the order and degree, respectively of the k-th iterated line graph L k (G) of G, k. Then ) LE (L k (G) K = (n k r k 4)(4r k 6) 4 (4.13) = (n k 1 4)(r k 1 ) 4, [ ] ) k n LE (L k 0 (G) K = ( i r k 0 i1 ) 4 ( k r 0 k1 ) 4, i=0

12 04 H. RAMANE, G. GUDODAGI, AND I. GUTMAN ) LE (L k (G) K = 8n kr k r k 4(r k 1). From Eq. (4.13) we see that the energy of L k (G) K, k is fully determined by the order n and degree r 3 of G. Theorem 4.6. Let G 1 and G be two Laplacian non-cospectral, regular graphs of the same order and of the same degree r 4. Then for any k, L k (G 1 ) K and L k (G ) K is a pair of Laplacian non-cospectral, Laplacian equienergetic graphs possessing same number of vertices and same number of edges. Proof. If G is any graph with n vertices and m edges, then G K has n vertices and m n edges. Hence by repeated applications of Eqs. (4.1) and (4.), L k (G 1 ) K and L k (G ) K have same number of vertices and same number of edges. By Eqs. (4.5) and (4.6), if G 1 and G are not Laplacian cospectral, then L k (G 1 ) K and L k (G ) K are not Laplacian cospectral for all k 1. Finally, Eq. (4.8) implies that L k (G 1 ) K and L k (G ) K are Laplacian equienergetic. Theorem 4.7. Let G 1 and G be two Laplacian non-cospectral, regular graphs of the same order and of the same degree r 3. Then for any k, L k (G 1 ) K and L k (G ) K is a pair of Laplacian non-cospectral, Laplacian equienergetic graphs possessing same number of vertices and same number of edges. Proof. The proof is similar to that of Theorem 4.6 by using Eqs. (4.10), (4.11), and (4.13). Acknowledgments. H. S. Ramane is thankful to the University Grants Commission (UGC), Govt. of India, for support through research grant under UPE FAR-II grant No. F 14-3/01 (NS/PE). G. A. Gudodagi is thankful to the Karnatak University, Dharwad, for support through UGC-UPE scholarship No. KU/SCH/UGC-UPE/014-15/901. References [1] E. O. D. Andriantiana, Laplacian energy, in: I. Gutman, X. Li (Eds.), Graph Energies Theory and Applications, Univ. Kragujevac, Kragujevac, 016, pp [] R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (004), [3] F. Buckley, Iterated line graphs, Congr. Numer. 33 (1981), [4] F. Buckley, The size of iterated line graphs, Graph Theory Notes N. Y., 5 (1993), [5] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory and Application, Academic Press, New York, [6] G. Dahl, The Laplacian energy of threshold graphs and majorization, Linear Algebra Appl. 469 (015), [7] K. C. Das, S. A. Mojallal, Relation between energy and (signless) Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 74 (015), [8] K. C. Das, S. A. Mojallal, I. Gutman, On Laplacian energy in terms of graph invariants, Appl. Math. Comput. 68 (015), [9] P. J. Davis, Circulant Matrices, Wiley, New York, 1979.

13 LAPLACIAN ENERGY 05 [10] E. Fritscher, C. Hoppen, V. Trevisan, Unicyclic graphs with equal Laplacian energy, Lin. Multilinear Algebra 6 (014), [11] H. A. Ganie, S. Pirzada, E. T. Baskoro, On energy, Laplacian energy and p-fold graphs, El. J. Graph Theory Appl. 3 (015), [1] I. Gutman, E. Milovanović, I. Milovanović, Bounds for Laplacian type graph energies, Miskolc Math. Notes 16 (015), [13] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (006) [14] F. Harary, Graph Theory, Addison Wesley, Reading, [15] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 01. [16] B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O. R. Ollermann, A. J. Schwenk (Eds.), Graph Theory, Combinatorics and Applications, Wiley, (1991), pp [17] S. Pirzada, H. A. Ganie, On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl. 486 (015), [18] S. Radenković, I. Gutman, Total π-electron energy and Laplacian energy: How far the analogy goes?, J. Serb. Chem. Soc. 7 (007), [19] H. S. Ramane, S. B. Gudimani, Laplacian polynomial and Laplacian energy of some cluster graphs, Int. J. Sci. Innov. Math. Res. (014) [0] H. S. Ramane, I. Gutman, H. B. Walikar, S. B. Halkarni, Equienergetic complement graphs, Kragujevac J. Sci. 7 (005), [1] H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett. 18 (005), [] H. Sachs, Über selbstkomplementare Graphen, Publ. Math. Debrecen 9 (196), [3] H. Sachs, Über Teiler, Faktoren und charakteristische Polynome von Graphen, Teil II, Wiss. Z. TH Ilmenau 13 (1967), [4] D. Stevanović, Large sets of noncospectral graphs with equal Laplacian energy, MATCH Commun. Math. Comput. Chem. 61 (009), Department of Mathematics, Karnatak University, Dharwad , India address: hsramane@yahoo.com address: gouri.gudodagi@gmail.com Faculty of Science, University of Kragujevac, Kragujevac, Serbia, and State University of Novi Pazar, Novi Pazar, Serbia address: gutman@kg.ac.rs