New bounds for Laplacian energy Текст научной статьи по специальности «Математика»

QPI/in/lHAflHI/1 HAyMHM PAflOBM OPI/in/lHAflbHblE HAyMHblE CTATbll ORIGINAL SCIENTIFIC PAPERS

NEW BOUNDS FOR LAPLACIAN ENERGY

Ivan Gutman

University of Kragujevac, Faculty of Science, Kragujevac, Republic of Serbia, e-mail: [email protected],

ORCID iD: http://orcid.org/0000-0001-9681-1550

DOI: 10.5937/vojtehg68-24257; https://doi.org/10.5937/vojtehg68-24257

FIELD: Mathematics (Mathematics Subject Classification: primary 05C50,

secondary 05C90) ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English

Abstract:

Introduction/purpose: The Laplacian energy (LE) is the sum of absolute values of the terms p-2m/n, where p,, i=1,2,..,n, are the eigenvalues of the Laplacian matrix of the graph G with n vertices and m edges. The basic results of the theory of LE are outlined, and some new obtained. Methods: Spectral theory of Laplacian matrices is applied. Results: A new class of lower bounds for LE is derived. Conclusion: The paper contributes to the Laplacian spectral theory and tp the theory of graph energies.

Keywords: spectral graph theory, Laplacian spectrum (of graph), Laplacian energy.

Introduction

Throughout this paper, we are concerned with simple graphs, i.e. graphs without directed, multiple, or weighted edges, and without loops. Let G be such a graph, possessing n vertices and m edges. For details of the graph theory see (Harary, 1969), (Cvetkovic, 1981).

Let the vertices of the graph G be labeled by v1, v2, ..., vn . Let deg(vi) be the degree (= number of first neighbors) of the vertex vi. Then the Laplacian matrix of G, denoted by L(G), is the square matrix of the order n, whose (i,j)-element is equal to -1 if the vertices vi and vj are adjacent, it is 0 when the vertices vi and vj are not adjacent, and deg(vi) if i=j. The eigenvalues of L(G), denoted bu p, , i=1,2,...,n, form the Laplacian spectrum of the graph G. For details of the theory of Laplacian matrices and their spectra see (Grone et al, 1990), (Merris, 1994).

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The (ordinary) energy of a graph is defined as the sum of absolute values of the eigenvalues of the adjacency matrix (Li et al, 2012), (Gutman & Furtula, 2019). Extending this concept to Laplacian eigenvalues, the Laplacian energy was defined as (Gutman & Zhou, 2006):

LE = X

t=i

A

2m

n

(1)

For details on the mathematical properties of the Laplacian energy see (Andriantiana, 2016), (Gutman & Furtula, 2019).

Preparations and the main results

The Laplacian eigenvalues p , i=1,2,...,n, are non-negative real numbers. If the underlying graph G is connected, then exactly one of these eigenvalues is equal to zero (and the other n-1 eigenvalues are positive-valued). The following relations

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Xa = 2m

(2)

i=1

and

XA = 2m +£ deg(Vi )2

i= 1

i=1

(3)

are well known (Grone et al, 1990). At this point, we note that the sum of squares of vertex degrees is the much studied first Zagreb index; for details see (Borovicanin et al, 2017) and the references cited therein. Combining Eqs. (2) and (3), we directly obtain

2m 2 = 2 M

where

X

i=1

M

A

n

(4)

1

m _ 2m_ + deg(v; )2 n 21?

Numerous upper and lower bounds for the Laplacian energy are known (Gutman & Zhou, 2006), (Andriantiana, 2016), of which we mention here

24M < LE < 2M and the McClelland-type upper bound

LE < V 2 Mn .

(5)

In this paper, we offer four new lower bounds for LE, namely

r ^ abn + 2M

LE >--(6)

a + b

2m LE >-+

ab(n -1) + 2M -

2m

v n j

n

LE > ^JM a + b

T ^ 2m 2\[ab LE >-+ ■

a + b

n

a + b

2 M

2m

n

(n -1)

(7)

(8)

(9)

For connected graphs with at least four vertices, all bounds (6)-(9) are strict.

The meaning of the parameters a and b is explained in the subsequent section. Observe that the multiplier in (8) and (9) is the ratio between the geometric and arithmetic means of a and b.

Proofs of bounds (6)-(9)

In order to avoid trivialities, we assume that the graphs considered are connected and have more than three vertices. Let

Mi

2m

n

X..

and label the Laplacian eigenvalues of the considered graph so that

X1 > X2 > - > Xn.

In addition, let X1 = a and Xn = b. Then

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(a - Xi )(b - Xi) = ab- (a + b) X; + X2 < 0

(10)

holds for all /=1,2,...,n, and is strictly negative for at least one value of i. Summing (10) over all /, and bearing in mind Eqs. (1) and (4), we get

abn - (a + b)LE + 2M < 0

from which inequality (6) directly follows.

For connected graphs, exactly one Laplacian eigenvalue is equal to zero. Therefore, one X-value is equal to 2m/n. Let this be X#.

If we sum (10) over all i, except i = # , then we arrive at

ab(n -1) - (a + b)

LE

2m

n

+

2 M

2m

n

< 0

which implies inequality (7).

Applying the relation p + Q > 2^/PQ , which holds for any positive real

numbers P,Q, with equality if and only if P=Q, we can transfer inequality (6) into inequality (8). In the very same manner, bound (9) is obtained from (7).

It is worth noting that the lower bound (8) has a similar algebraic form as the upper bound (5). Thus (5) and (8) estimate the Laplacian energy from both sides in an analogous, McClelland-type manner.

It can be shown that among the lower bounds (6)-(9), bound (7) is the best. In addition, (6) is better than (8), whereas (7) is better than (9). Numerical testing shows that bound (6) is sharper than {9), However, to verify this by exact mathematical methods seems to be a tough task and remains an open problem.

A special case: regular graphs

A graph is said to be regular if all its vertices have equal degrees. Let, thus, the considered graph G be regular, and let deg(vi)=r for all i=1,2,...,n. Then

X deg(Vi) = nr, X deg(Vi )2 = nr2

2m

= r

M = m

i=1

i=1

n

to:

Bearing this in mind, for regular graphs, inequalities (6)-(9) reduce

LE >

(ab + r)n a + b

ab(n -1) + r(n - r)

LE > r +

E > 2jab r

LE >-Wr

a + b

2\[ab

a + b

LE > r h--^I r(n - r)(n -1)

a + b

(6a) (7a) (8a) (9a)

For regular graphs, the Laplacian and ordinary energies coincide. Therefore, bounds (6a)-(9a) hold also for the ordinary energy of regular graphs. Bounds of this kind (for ordinary graph energy) were recently communicated (Oboudi, 2019), (Gutman, 2019).

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References

Andriantiana, E.O.D. 2016. Laplacian energy. In: Gutman, I. & Li, X. (Eds.), Graph Energies - Theory and Applications.Kragujevac: University of Kragujevac, pp.49-80.

Borovicanin, B., Das, K.C., Furtula, B., & Gutman, I. 2017. Bounds for Zagreb indices. MATCH Communications in Mathematical and in Computer Chemistry, 78(1), pp.17-100 [online]. Available at:

http://match.pmf.kg.ac.rs/electronic_versions/Match78/n1/match78n1_17-100.pdf. [Accessed: 30 November 2019]

Cvetkovic, D. 1981. Teorija grafova i njene primene.Belgrade: Naucna knjiga (in Serbian).

Grone, R., Merris, R., & Sunder, V.S. 1990. The Laplacian Spectrum of a Graph. SIAM Journal on Matrix Analysis and Applications, 11(2), pp.218-238. Available at: https://doi.org/10.1137/0611016.

Gutman, I. 2019. Oboudi-type bounds for graph energy. Mathematics Interdisciplinary Research, 4(2), pp.151-155 [online]. Available at: http://mir.kashanu.ac.ir/article_96938_35e2a0e7a15dcdd1e796325b33542469.p df. [Accessed: 30 November 2019]

Gutman, I., & Furtula, B. 2019. Graph Energies: Survey, Census, Bibliography.Kragujevac: Centar SANU. Bibliography.

Gutman, I., & Zhou, B. 2006. Laplacian energy of a graph. Linear Algebra and its Applications, 414(1), pp.29-37. Available at: https://doi.org/10.1016/jJaa.2005.09.008.

Harary, F. 1969. Graph Theory.Addison-Wesley. Reading.

Li, X., Shi, Y., & Gutman, I. 2012. Introduction. In: Graph Energy.New York, NY: Springer Science and Business Media LLC., pp.1-9. Available at: https://doi.org/10.1007/978-1-4614-4220-2_1. jg Merris, R. 1994. Laplacian matrices of graphs: A survey. Linear Algebra

0 and its Applications, 197-198, pp. 143-176. Available at: > https://doi.org/10.1016/0024-3795(94)90486-3. ° Oboudi, M.R. 2019. A new lower bound for the energy of graphs. Linear

° Algebra and its Applications, 580, pp.384-395. Available at: of https://doi.org/10.1016/j1aa.2019.06.026.

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0 НОВЫЕ ОГРАНИЧЕНИЯ ЛАПЛАСОВОИ ЭНЕРГИИ

^ Иван Гутман

° Университет в г. Крагуевац, Естественно-математический факультет,

1 г. Крагуевац, Республика Сербия ш

ВИД СТАТЬИ: оригинальная научная статья

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА _ ВИД СТАТЬИ: оригинальна:

< ЯЗЫК СТАТЬИ: английский

Резюме:

Введение/цель: Лапласова энергия (ЛЭ) графа является суммой абсолютных величин термина р, -2т/п, при чем р,, ¡=1,2,..,п, < представляют собственные значения матрицы Лапласа графа О

о с п узлами и т ветвями. Кроме основных результатов теории

2 Лапласа, в работе приведены и некоторые нововыявленные.

Методы: В работе применялась спектральная теория матриц ^ Лапласа.

н Результаты: Выявлен новый класс предельных значений энергии

Лапласа.

о Выводы: Данная работа делает вклад в развитие спектральной

> теории Лапласа и теории энергии графов.

Ключевые слова: спектральная теория графов, лапласовский спектр (графа), энергия Лапласа.

НОВА ОГРАНИЧЕНА ЗА ЛАПЛАСОВУ ЕНЕРГШУ

Иван Гутман

Универзитет у Крагу]евцу, Природно-математички факултет, Крагу]евац, Република Срби]а

ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни рад

иЕЗИК ЧЛАНКА: енглески

Сажетак:

Увод/сврха: Лапласова енерг^а (1Б) графа jе сума апсолутних £ вредности израза р-2т/п, где р,, ¡=1,2,...,п, представъаjу сопствене вредности Лапласове матрице графа О са п чворова и т грана. Поред основних резултата теорбе Лапласове енерг^е дати су и неки новодоб^ени. Методе: КоришЯена jе спектрална теорба Лапласових матрица. ^ Резултати: Изводи се нова класа доних ограничена за Лапласову енерг^у.

Закъучак: Рад даjе допринос Лапласовоj спектралноj теории као и теории енерг^а графа.

Къучне речи: спектрална теорба графова, Лапласов спектар ^ (графа), Лапласова енерг^а.

Paper received on / Дата получения работы / Датум приема чланка: 01.12.2019. |

Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 20.12.2019. О

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 22.12.2019.

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