Laplacian Sum-Eccentricity Energy of a Graph. 1. Introduction

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1 Mathematics Iterdiscipliary Research 2 (2017) Laplacia Sum-Eccetricity Eergy of a Graph Biligiriragaiah Sharada, Mohammad Issa Sowaity ad Iva Gutma Abstract We itroduce the Laplacia sum-eccetricity matrix LS e of a graph G, ad its Laplacia sum-eccetricity eergy LS ee ηi, where ηi ζ i 2m ad where ζ1, ζ2,..., ζ are the eigevalues of LSe. Upper bouds for LS ee are obtaied. A graph is said to be twieergetic if ηi ζi. Coditios for the existece of such graphs are established. Keywords: Sum-eccetricity eigevalues, sum-eccetricity eergy, Laplacia sum-eccetricity matrix, Laplacia sum-eccetricity eergy Mathematics Subject Classificatio: 05C Itroductio Let G be a simple coected graph with vertex set V(G) ad edge set E(G), of order V(G) ad size E(G) m. Let A (a ij ) be the adjacecy matrix of G. The eigevalues λ 1, λ 2,..., λ of A are the eigevalues of the graph G [6]. Sice A is a symmetric matrix with zero trace, these eigevalues are real with sum equal to zero. The eergy of the graph G is defied as the sum of the absolute values of its eigevalues [10, 16]: E(G) λ i. After the itroductio of the graph eergy cocept i the 1970s [10], several other graph eergies have bee put forward ad their mathematical properties extesively studied; for details see the recet moograph [12] ad the survey [11]. I the last few years, a whole class of graph eergies was coceived, based o the eigevalues of matrices associated with a particular topological idex. Thus, let T I be a topological idex is of the form T I T I(G) F (v i, v j ) v iv j E(G) Correspodig author ( mohammad_d2007@hotmail.com) Academic Editor: Hassa Yousefi-Azari Received 19 November 2017, Accepted 12 December 2017 DOI: /mir c 2017 Uiversity of Kasha

2 210 B. Sharada, M. I. Sowaity ad I. Gutma where F is a pertietly chose fuctio with the property F (x, y) F (y, x). The a matrix TI ca be associated to T I, defied as F (v i, v j ) if v i v j E(G) (TI) ij 0 otherwise. If τ 1, τ 2,..., τ are the eigevalues of the matrix TI, the a eergy ca be defied as E T I E T I (G) τ i. (1) The most extesively studied such graph eergy is the Radić eergy [2,3,7,12], based o the eigevalues of the Radić matrix R, where (R) ij 1 di d j if v i v j E(G) 0 otherwise ad where d i is the degree of the i-th vertex of G. I a aalogous maer the harmoic eergy [14], ABC eergy [9], geometric arithmetic eergy [23], Zagreb eergy [15], ad sum-eccetricity eergy [22, 26] were put forward. To ay eergy E T I of the form (1), a Laplacia eergy LE T I ca be associated, defied as LE T I LE T I (G) θ i 2m (2) where θ 1, θ 2,..., θ are the eigevalues of the matrix LTI D TI, ad where D D(G) is the diagoal matrix of vertex degrees. The first such Laplacia eergy, based o the adjacecy matrix A, was itroduced i 2006 [13] ad its theory is owadays elaborated i full detail, see [12]. It is worth otig that this Laplacia eergy foud iterestig egieerig applicatios i image processig [18, 25, 27]. Bearig this i mid, it is purposeful to study other Laplacia graph eergies. Some recet studies alog these lies are [1, 5, 8, 20, 21]. I this paper we study the Laplacia versio of the sum-eccetricity eergy. I order to defie it, we eed some preparatios. The distace d(u, v) betwee two vertices u ad v i a (coected) graph G is the legth of a shortest path coectig u ad v [4]. The eccetricity of a vertex v V(G) is e(v) max{d(u, v) : u V(G)}. The radius of G is r(g) mi{e(v) : v V(G)}, whereas the diameter of G is d(g) max{e(v) : v V(G)}. Hece r(g) e(v) d(g), for every v V(G). I this paper, we deote by K, K a,b, K 1,a, C, ad P the complete graph, complete bipartite graph, star, cycle, ad path, respectively.

3 Laplacia Sum-Eccetricity Eergy of a Graph 211 The sum-eccetricity matrix of a graph G is deoted by S e (G) ad defied as S e (G) (s ij ) [22, 26], where s ij { e(vi ) + e(v j ) if v i v j E 0 otherwise. If µ 1, µ 2,..., µ, are the eigevalues of S e (G), the the sum-eccetricity eergy is ES e (G) µ i. Defiitio 1.1. Let G be a graph of order ad size m. The Laplacia sumeccetricity matrix of G, deoted by LS e (G) (l ij ), is defied as LS e (G) D(G) S e (G). The Laplacia sum-eccetricity spectrum of G, cosistig of ζ 1, ζ 2,..., ζ, is the spectrum of the Laplacia sum-eccetricity matrix. This leads us to defie the Laplacia sum-eccetricity eergy of a graph G as LS e (G) ζ i 2m. (3) If, i additio, we defie the auxiliary quatity η i as η i ζ i 2m the LS e E(G) η i. Lemma 1.2. Let G be a (, m)-graph. The Proof. ζ i 2m. ζ i trace(ls e (G)) l ii d i 2m. Theorem 1.3. The Laplacia sum-eccetricity eergy of the complete graph K is LS e E(K ) 4( 1).

4 212 B. Sharada, M. I. Sowaity ad I. Gutma Proof. Recallig that the eccetricity of ay vertex of K is uity, directly from the defiitio of the Laplacia sum-eccetricity matrix, we calculate that [ ] d d + 2 Spec(LS e (K )) 1 1 where d 1 is the degree of ay vertex of K. Usig the fact that 2m 1 d, we get by Eq. (3) LS e E(K ) d d + d + 2 d + + d + 2 d 2d + 2( 1) 4( 1). 2. Bouds for Laplacia Sum-Eccetricity Eergy Theorem 2.1. Let G be a (, m)-graph. The LS e E(G) l 2 ij 4m2. (4) j1 Proof. We have ηi 2 ( ζ i 2m ) 2 (ζ 2i 4m ) ζ i + 4m2 2 By Lemma 1.2, ηi 2 ζi 2 4m ζ 2 i 8m2 ζ i + 4m2. + 4m2 Usig the Cauchy Schwarz iequality η i η 2 i ζ 2 i 4m2. we get ( ) η i ζi 2 4m2.

5 Laplacia Sum-Eccetricity Eergy of a Graph 213 O the other had, ζi 2 trace(ls 2 e(g)) j1 l 2 ij ad iequality (4) follows. It should be oted that iequality (4) is just a variat of the classical McClellad s upper boud for ordiary graph eergy [16, 17]. Corollary 2.2. Let G be a r-regular graph. The ηi 2 Example 2.3. If G K, the If G K a,b. The j1 l 2 ij r 2. ηi 2 4( 1). ( ηi 2 ab a + b ab ). a + b I particular, for G K 1,a, with a + 1: ( ηi 2 a ) + 19a. a + 1 I what follows we derive aother upper boud for the Laplacia sum-eccetricity eergy usig Weyl s iequality for matrices. Theorem 2.4. (Weyl s iequality) [19] Let X ad Y be Hermitia matrices. If for 1 i, λ i (X), λ i (Y), λ i (X+Y) are the eigevalues of X, Y, ad X+Y, respectively, the λ i (X) + λ (Y) λ i (X + Y) λ i (X) + λ 1 (Y). The matrices LS e (G), S e (G), ad D(G) are all Hermitia matrices. I additio, we use the facts that the eigevalues of the diagoal matrix are the etries i the diagoal, ad that the eergy of a matrix X is equal to the eergy of X. We thus arrive at: Theorem 2.5. Let G be a (, m)-graph with maximal vertex degree. The LS e E(G) ES e (G) + k + 2m ( k) (5) where k {ζ i : ζ i 2m/}.

6 214 B. Sharada, M. I. Sowaity ad I. Gutma Proof. Let ζ 1 ζ 2 ζ be the Laplacia sum-eccetricity eigevalues, µ 1 µ 2 µ be the sum-eccetricity eigevalues ad ρ 1 ρ 2 ρ be the eigevalues of the degree matrix. We assume that 1 k r. Usig Theorem 2.4 we get µ i + ρ ζ i µ i + ρ 1. Sice ρ 0, Sice 2m/ 0, µ i ζ i µ i + ρ 1. µ i 2m ζ i 2m µ i + ρ 1. Now we have to distiguish betwee two cases. Case 1: If ζ i 2m 0, the ζ i 2m µ i + ρ 1. If there are k ζ i s, satisfy this coditio, the ζ i 2m (µ i + ρ 1 ) µ i + kρ 1. (6) Case 2: If ζ i 2m 0, the ζ i 2m µ i 2m. If we have, µ i 0 for i k + 1,..., r ad µ i 0 for i r + 1,...,. The ζ i 2m µ i 2m + µ i 2m ir+1 2m + µ i + ir+1 2m ir+1 µ i. (7) Combiig the relatios (6) ad (7), we get ζ i 2m µ i + kρ 1 + µ i µ i + 2m ir+1 ( k) ES e (G) + kρ 1 + 2m ( k) from which (5) follows straightforwardly.

7 Laplacia Sum-Eccetricity Eergy of a Graph 215 Corollary 2.6. If the graph G is r-regular, the Proof. From Theorem 2.5, we have LS e E(G) ES e (G) + r. (8) LS e E(G) ES e (G) + kr + 2m ( k). Sice, i additio, for a r-regular graph, 2m/ r, ad iequality (8) follows. LS e E(G) ES e (G) + r(k + k) Lemma 2.7. [24] For the complete bipartite graph K a,b, the sum-eccetricity eergy is ES e (K a,b ) 8 ab. Corollary 2.8. For the complete bipartite graph K a,b, LS e E(K a,b ) 8 ab + k max{a, b} + 2ab (a + b k). (9) a + b Proof. For K a,b, 2m/ 2ab/(a + b). Usig Lemma 2.7, we get (9) from (5). 3. Twieergetic Graphs I this sectio, we poit out a remarkable feature of Laplacia sum-eccetricity eergy. Defiitio 3.1. Let G be a graph of order, ad let ζ i, i 1, 2,...,, be its Laplacia sum-eccetricity eigevalues. We say that G is twieergetic if LS e E(G) ζ i. The above defiitio meas that ζ i 2m ζ i. (10) The umber of positive eigevalues ad egative eigevalues (icludig their multiplicities) are deoted by ζ + (G) ad ζ (G), respectively. For the sake of simplicity, we assume that there are o zero Laplacia sum-eccetricity eigevalues, i.e., that ζ + (G) + ζ (G).

8 216 B. Sharada, M. I. Sowaity ad I. Gutma Theorem 3.2. A graph G is a Laplacia sum-eccetricity twieergetic if it satisfies the followig two coditios: ζ i (G) 2m, i 1, 2,..., ζ+ i (G). (11) ζ + (G) ζ (G). (12) Proof. Let ζ + (G) r, where 1 r. The ζ i 2m ζ i 2m + ζ i 2m + ir+1 ir+1 ζ i 2m ζ i + 2m ( r). (13) Let k be the umber of eigevalues satisfyig the coditio ζ i (G) 2m. The, 1 k r, ad ζ i 2m ( ζ i 2m ) + ( ) 2m ζ i ζ i 2m k ζ i + 2m (r k). (14) Substitutig (14) back ito (13) yields ζ i 2m ζ i + ζ i ζ i + 2m ir+1 ( 2k) ζ i 2 ζ i + 2m ( 2k). If the coditio (11) is obeyed, i.e., if k r, the ζ i 0. If, i additio, also the coditio (12) is obeyed, i.e., 2r, the 2m ( 2k) 0. Thus, if both coditios (11) ad (12) are satisfied, the the relatio (10) holds, i.e., the graph G is twieergetic.

9 Laplacia Sum-Eccetricity Eergy of a Graph 217 It ow remais to see if twieergetic graphs exist at all. That such graphs do exist is verified by the followig examples. Example 3.3. The paths P 2, P 4, P 8, ad P 16 are twieergetic graphs. For P 2, direct calculatios gives ζ 1 3, ζ 2 1, ad 2m/ 1. Therefore, 2 ζ i 4 ad 2 ζ i 2m 4. For P 4 we get ζ , ζ , ζ , ζ , ad 2m/ 6/4. Therefore, 4 ζ i 4 ζ i 2m The cases P 8 ad P 16 are verified aalogously. Fidig more examples of twieergetic graphs, as well as their complete structural characterizatio remais a task for the future. Refereces [1] P. G. Bhat, S. D Souza, Color sigless Laplacia eergy of graphs, AKCE It. J. Graphs Comb. 14 (2017) [2] Ş. B. Bozkurt, A. D. Gügör, I. Gutma, Radić spectral radius ad Radić eergy, MATCH Commu. Math. Comput. Chem. 64 (2010) [3] Ş. B. Bozkurt, A. D. Gügör, I. Gutma, A. S Çevik, Radić matrix ad Radić eergy, MATCH Commu. Math. Comput. Chem. 64 (2010) [4] F. Buckley, F. Harary, Distace i Graphs, Addiso Wesley, Redwood, [5] A. Cafure, D. A. Jaume, L. N. Grippo, A. Pastie, M. D. Safe, V. Trevisa, I. Gutma, Some results for the (sigless) Laplacia resolvet, MATCH Commu. Math. Comput. Chem. 77 (2017) [6] D. Cvetković, P. Rowliso, S. Simić, A Itroductio to the Theory of Graph Spectra, Cambridge Uiv. Press, Cambridge, [7] K. C. Das, S. Sorgu, K. Xu, O Radić eergy of graphs, i: I. Gutma, X. Li (Eds.), Graph Eergies Theory ad Applicatios, Uiv. Kragujevac, Kragujevac, 2016, pp [8] N. De, O eccetricity versio of Laplacia eergy of a graph, Math. Iterdisc. Res. 2 (2017)

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11 Laplacia Sum-Eccetricity Eergy of a Graph 219 [25] Y. Z. Sog, P. Arbelaez, P. Hall, C. Li, A. Balikai, Fidig sematic structures i image hierarchies usig Laplacia graph eergy, i: K. Daiilidis, P. Maragos, N. Paragios (Eds.), Computer Visio CECV 2010 (Europea Coferece o Computer Visio, 2010), Part IV, Spriger, Berli, 2010, pp [26] M. Sowaity ad B. Sharada, The sum-eccetricity eergy of a graph, It. J. Rec. Iovat. Treds Comput. Commu. 5 (2017) [27] H. Zhag, X. Bai, H. Zheg, H. Zhao, J, Zhou, J. Cheg, H. Lu, Hierarchical remote sesig image aalysis via graph Laplacia eergy, IEEE Geosci. Remote Sesig Lett. 10 (2013) Biligiriragaiah Sharada Departmet of Studies i Computer Sciece, Uiversity of Mysore, Maasagagotri, Mysuru , Idia sharadab21@gmail.com Mohammad Issa Sowaity Departmet of Studies i Mathematics, Uiversity of Mysore, Maasagagotri, Mysuru , Idia mohammad_d2007@hotmail.com Iva Gutma Faculty of Sciece, Uiversity of Kragujevac, P.O. Box 60, Kragujevac, Serbia gutma@kg.ac.rs