On the oriented incidence energy and decomposable graphs

Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 23:3 (2009), 243–249 DOI:10.2298/FIL0903243S ON THE ORIENTED INCIDENCE ENERGY AND DECOMPOSABLE GRAPHS∗ Dragan Stevanović, Nair M.M. de Abreu, Maria A.A. de Freitas Cybele Vinagre and Renata Del-Vecchio Abstract Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n ∈ N , there exists a set of n graphs with O(n) vertices having equal oriented incidence energy. 1 Introduction Let G = (V, E) be a finite, simple, undirected graph with vertices V = {1, 2, . . . , n} and m = |E| edges. Let G have adjacency matrix A with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn . The energy of G was defined by Gutman in [1] as E = E(G) = n X i=1 |λi |, (1) and it has a long known chemical applications; for details see the surveys [2, 3, 4]. Recently, Nikiforov [5] generalized a concept of graph energy to arbitrary matrix M by defining the energy E(M ) to be the sum of singular values of M . The singular values of a real (not necessarily square) matrix M are the square roots of the eigenvalues of the (square) matrix M M T , where M T denotes the transpose of M . Let edges of G be given an arbitrary orientation producing an oriented graph − → − → G , and let Q be the vertex-edge incidence matrix of G , whose (v, e) entry is equal to +1 if the vertex v is the head of the oriented edge e, −1 if v is the tail of e, and 0 otherwise. Then QQT = L = D − A is the Laplacian matrix of G, where ∗ This work was supported by the research grant 144015G of Serbian Ministry of Science and in part by Grant 300563/94-9 of National Research Council of Brazil.. 2000 Mathematics Subject Classifications. 05C50. Key words and Phrases. Laplacian-like energy, Incidence energy, Decomposable graphs. Received: October 20, 2009 Communicated by Dragan Stevanović 244 D. Stevanović, N.M.M. de Abreu et al D is the diagonal matrix of vertex degrees [6, 7]. Suppose that L has eigenvalues µ1 ≥ µ2 ≥ · · · ≥ µn = 0. The oriented incidence energy of G is then OIE(G) = E(Q) = n X √ µi , i=1 as observed in [8]. This invariant was introduced recently by Liu and Liu [9] under the name the Laplacian energy-like invariant and notation LEL(G). Due to its definition, it comes as no surprise that OIE(G) has a number of properties analogous to E(G) [9, 10]. OIE(G) was suggested as a new molecular descriptor in [11]: a correlating study of OIE and topological indices provided by TOPOCLUJ software package [12], on thirteen properties of octanes, revealed that OIE describes well the properties which are well accounted by the Wiener-based molecular descriptors: octane number MON, entropy S, volume MV, or refraction MR, particularly the AF parameter, but also more difficult properties like boiling point BP, melting point MP and logP. In a second set of polycyclic aromatic hydrocarbons, OIE was proved to be as good as the Randić index and better than the Wiener index in correlations to BP, MP and logP. A graph is decomposable if it can be contructed from isolated vertices by the operations of union and complement. The Laplacian spectrum of G1 ∪ · · · ∪ Gk is the union of Laplacian spectra of G1 ,. . . , Gk , while the Laplacian spectrum of the complement of n-vertex graph G consists of values n − µ, for each Laplacian eigenvalue µ of G, except for a single instance of eigenvalue 0 of G. Since the Laplacian spectrum of an isolated vertex consists of single eigenvalue 0, it is easy to conclude that the Laplacian spectrum of every decomposable graph consists of integers only [13, 14]. Much work on graph energy has appeared in literature, especially in the last decade, and a good deal of it studies graphs with equal energy [15]-[24]. Two graphs G1 and G2 of the same order, noncospectral with respect to L, are said to be OIE-equienergetic if OIE(G1 ) = OIE(G2 ). Three pairs of connected OIEequienergetic graphs were presented in [25] and, based on the computer search among small graphs, it was suggested that OIE-equienergetic graphs occur relatively rarely. However, note that the graphs G802 , G804 and G1202 from [25] are all decomposable graphs. Our goal here is to show that, for any given n ∈ N , there exists a set of n mutually OIE-equienergetic decomposable graphs with O(n) vertices. Let A = {a1 , . . . , ak } be a multiset of positive integers such that ai ≥ 3, i = ∗ 1, . . . , k. The graph SA , formed from the union of stars S³a1 −1 , Sa2´−1 , . . . , Sak −1 by Pk adding a vertex adjacent to all other vertices, has n = i=1 ai − k + 1 vertices and m = 2n − k − 2 edges. It is decomposable since it can be represented as ∗ SA = K1 ∪ k [ i=1 K1 ∪ ai−2 K1 On the oriented incidence energy and decomposable graphs 245 and its Laplacian spectrum is given by [n, a1 , . . . , ak , 2n−2k−1 , 1k−1 , 0], where exponents denote multiplicities. Thus, ∗ OIE(SA )= √ n+ k X √ i=1 √ ai + (n − 2k − 1) 2 + k − 1. (2) Let S be the set of of finite multisets of positive integers each of which is at least three. Let ρ be an equivalence relation on S defined by AρB ⇔ |A| = |B|, k X ai = i=1 k X bi and i=1 k X √ ai = i=1 k p X bi . i=1 From (2) we see that AρB ⇒ ∗ ∗ OIE(SA ) = OIE(SB ). ∗ ∗ and SB Moreover, if A and B are distinct equivalent multisets, then the graphs SA are noncospectral, while they have the same order and size. Therefore, in order to construct sets of OIE-equienergetic decomposable graphs, we need to find nontrivial equivalence classes of ρ in S. Construction of equivalence classes containing pairs of triplets is given in Section 2, while operations for constructing large equivalence classes in S/ρ are discussed in Section 3. A few nontrivial equivalence classes found by initial computer search are given in Table 1. P i ai = 37 40 24 42 43 P i bi {a1 , . . . , ak } {25,6,6} {27,9,4} {12,4,4,4} {20,9,9,4} {27,4,4,4,4} {b1 , . . . , bk } {24,9,4} {25,12,3} {9,9,3,3} {16,16,5,5} {25,9,3,3,3} P √ ai =√ i bi 5 + 2 √6 5 + 3 √3 6 + 2 √3 8 + 2 √5 8+3 3 P √ i Table 1: A few equivalence classes in S. 2 Equivalence classes containing triplets Proposition 1. Let a, b, c, d, e, f be positive integers such that abc = def . Then {a2 c, b2 c, (d + e)2 f } ρ {(a + b)2 c, d2 f, e2 f }. 246 D. Stevanović, N.M.M. de Abreu et al Proof. Both multisets have and the sum of square roots of their √ √ three elements elements is equal to (a + b) c + (d + e) f . From abc = def it follows that the sum of their elements are also equal, (a2 + b2 )c + (d2 + e2 )f + 2def = (a2 + b2 )c + 2abc + (d2 + e2 )f, so that these two triplets belong to the same equivalence class of ρ. For example, the first pair of triplets in Table 1 is obtained by setting (a, b, c, d, e, f ) = (1, 1, 6, 2, 3, 1), while the second pair of triplets is obtained for (a, b, c, d, e, f ) = (2, 3, 1, 2, 1, 3). We can construct infinitely many new pairs of triplets from Proposition 1 by taking distinct factorizations of positive integers into three factors a, b, c and d, e, f . For example, 10 can be factorized in distinct ways as 10 = 2 · 5 · 1 = 1 · 1 · 10, which gives a new pair of equivalent triplets (4, 25, 40) and (49, 10, 10). Previous proposition can be easily generalized: Proposition 2. For a given k ∈ N , let ai , bi , ci , di , ei , fi be positive integers such that k k X X ai bi ci = di ei fi . i=1 i=1 Then the multisets A = {a2i ci , b2i ci , (di + ei )2 fi : i = 1, . . . , k} and B = {(ai + bi )2 ci , d2i fi , e2i fi : i = 1, . . . , k} belong to the same equivalence class of ρ. Proof. Both A and PkB have 3k elements and the sum of square roots of their elements is equal to i=1 (ai + bi )ci + (di + ei )fi . For the sum of elements of A and B, we have X x = k X (a2i + b2i )ci + (d2i + e2i )fi + 2di ei fi i=1 x∈A = k X X (a2i + b2i )ci + (d2i + e2i )fi + 2ai bi ci = y. i=1 y∈B This proposition has even more freedom than Proposition 1. For example, 10 can be written in distinct ways as 10 = 1 · 1 · 4 + 2 · 3 · 1 = 1 · 1 · 5 + 1 · 1 · 5, yielding (a1 , b1 , c1 , a2 , b2 , c2 ) = (1, 1, 4, 2, 3, 1) and (d1 , e1 , f1 , d2 , e2 , f2 ) = (1, 1, 5, 1, 1, 5). Proposition 2 now gives equivalent multisets {4, 4, 20, 4, 9, 20} and {16, 5, 5, 25, 5, 5}. On the oriented incidence energy and decomposable graphs 3 247 Operations in S/ρ We can introduce two operations to S which agree with ρ to construct equivalence classes with more than two multisets. First, declare scalar to be a positive integer. Then for scalar α and multiset A ∈ S, the product αA is defined as αA = {αa : a ∈ A}. The second operation is the union A ⊎ B of multisets A and B, which preserves multiplicities of their elements: if a appears m times in A and n times in B, then a appears m + n times in A ∪ B. Proposition 3. For any α ∈ N and A, B, C, D ∈ S, AρB A ρ B, C ρ D ⇒ ⇒ αA ρ αB, A ⊎ C ρ B ⊎ D. Proof. The sum of elements in αA is α times√the sum of elements in A. Similarly, the sum of square roots of elements in αA is α times the sum of square roots of elements in A. Thus, from A ρ B it follows that αA ρ αB. Next, we have X X X X X X x= x+ x= x+ x= x, x∈A⊎C and, similarly, X √ x∈A⊎C x= x∈A X√ x∈A x+ x∈C X√ x∈C x∈B x= X√ x∈D x+ x∈B X√ x∈B⊎D x= x∈D X √ x. x∈B⊎D Thus, A ⊎ C ρ B ⊎ D. These two operations now provide a simple way to create arbitrarily large equivalence classes. Namely, for any A ρ B, n ∈ N and α1 , . . . αn ∈ N , it follows from Proposition 3 that α1 A ⊎ α2 A ⊎ · · · ⊎ αn−1 A ⊎ αn A ρ α1 B ⊎ α2 A ⊎ · · · ⊎ αn−1 A ⊎ αn A ρ α1 B ⊎ α2 B ⊎ · · · ⊎ αn−1 A ⊎ αn A ρ ... ρ α1 B ⊎ α2 B ⊎ · · · ⊎ αn−1 B ⊎ αn A ρ α1 B ⊎ α2 B ⊎ · · · ⊎ αn−1 B ⊎ αn B. Thus, this equivalence class contains at least n+1 multisets, each of them containing n|A| elements. In particular, take A = {25, 6, 6}, B = {24, 9, 4} and α1 = · · · = αn = 1. Then for any n ∈ N , we have a set of n+1 OIE-equienergetic noncospectral decomposable graphs ∗ ∗ ∗ ∗ SA⊎A⊎···⊎A , SB⊎A⊎···⊎A , SB⊎B⊎···⊎A , . . . , SB⊎B⊎···⊎B , each of which has 34n + 1 vertices and 65n edges. 248 4 D. Stevanović, N.M.M. de Abreu et al Concluding remarks Our last example shows that for any n ∈ N , there exists a set of n OIE-equienergetic noncospectral graphs with O(n) vertices. Propositions 1, 2 and 3 provide means to construct an abundance of further examples of OIE-equienergetic noncospectral graphs. 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