CC BY
36
ESTIMATING VERTEX-DEGREE-BASED ENERGIES
Ivan Gutman
DOI: 10.5937/vojtehg70-35584;https://doi.org/10.5937/vojtehg70-35584
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University of Kragujevac, Faculty of Science, Kragujevac, Republic of Serbia, e-mail: gutman@kg.ac.rs, -o
ORCID iD: ©https://orcid.org/0000-0001-9681-1550
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FIELD: Mathematics .g
ARTICLE TYPE: Original scientific paper x
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Abstract: |
Introduction/purpose: In the current literature, several dozens of vertex-degree-based (VDB) graph invariants are being studied. To each such | invariant, a matrix can be associated. The VDB energy is the energy (= sum of the absolute values of the eigenvalues) of the respective VDB matrix. The paper examines some general properties of the VDB energy of bipartite graphs.
Results: Estimates (lower and upper bounds) are established for the o VDB energy of bipartite graphs in which there are no cycles of size divisible by 4, in terms of ordinary graph energy.
Conclusion: The results of the paper contribute to the spectral theory of VDB matrices, especially to the general theory of VDB energy.
Keywords: vertex-degree-based graph invariant, vertex-degree-based matrix, vertex-degree-based energy, energy (of graph).
Introduction
Let G be a simple graph with the vertex set V(G) and the edge set E(G).
If the vertices u,v e V(G) are adjacent, then the edge connecting them is denoted by uv. The number of edges incident to a vertex v is the degree of that vertex, and is denoted by d(v). The minimum and maximum vertex degrees are denoted by 5 and A, respectively.
Let V(G) = {v\,v2,.. .,vn}. Then the adjacency matrix A(G) = [aj] of the graph G is the symmetric matrix of order n, whose elements are (Cvetkovic et al, 2010):
O u,l3 — } KJ ViVj
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l if ViVj e E(G) al3 = { 0 if ViVje E(G) (1)
0 if i = j.
If the eigenvalues of A(G) are Xi} X2,..., Xn, then the (ordinary) energy
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E = E (G) = Y, I Xi |. (2)
yy of the graph G is defined as
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The theory of graph energy is nowadays elaborated in due detail (Li et | al, 2012; Ramane, 2020).
In the chemical and mathematical literature, a variety of vertex-degree-based (VDB) graph invariants of the form
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0 has been considered, where f is a suitably chosen function, with a pro-
o perty f (x, y) = f (y, x) (Kulli, 2020; Todeschini & Consonni, 2009).
I = I (G)= f(d(u),d(v)) (3)
uveE(G)
These are usually referred to as topological indices. Of these, we list here a few most popular and best studied ones:
f (x,y) name of index type
x + y first Zagreb t
xy second Zagreb t
x2 + y2 forgotten t
a/x2 + y2 Sombor t
y/x + y nirmala t
1/^xy Randic ;
1 Nx + y sum-connectivity ;
2/(x + y) harmonic ;
1/x2 + 1/y2 inverse degree ;
1/\fx2 + y2 modified Sombor ;
[(x + y - 2)/(xy)]l/2 atom-bond-connectivity rsj
\x - y\ Albertson rsj
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The parameters x and y (being vertex degrees) always satisfy the condition x > 1, y > 1. Bearing this in mind, we immediately recognize that most VDB indices are either monotonically increasing (t) or monotonically decreasing functions (;) of the vertex degrees. Only a few such indices do not possess such a monotonicity property (~).
It should be noted that for practically all VDB indices of type t that exist in the literature, the condition f (x,y) > 1 is satisfied for all values of x and y that occur for the edges of graphs. Analogously, for practically all VDB indices of type ;, 0 < f (x, y) < 1 holds for all values of x and y.
Taking into account Eqs. (1) and (3), we introduce the VDB matrix Ai(G) = [(ax)j via
f f (d(vi),d(v)) if ViVj e E(G) (ai)ij = I 0 if ViV3e E(G) (4)
I 0 if i = j.
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If its eigenvalues are fii,^2,■■■,yn, then the energy pertaining to the VDB invariant I, Eq. (3), is
n
Ex = Ex(G) = J2 Ifrl ■ (5)
i=i
For recent works on the investigation of this class of graph-spectral invariants see (Das et al, 2018; Gutman, 2020, 2021; Gutman et al, 2022, | 2021; Li & Wang, 2021; Shaoetal, 2021).
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^ Main results
o
x A cycle of length p is a cycle consisting of (exactly) p vertices ft vi,v2, ■ ■ ■, vp, so that vi and vi+i are adjacent for i = l, 2,...,p - l, and £ also vi and vp are adjacent. As it is well known, a graph G is bipartite if and <c only if all its cycles (if any) are of even length. In this paper, we prove the results valid for bipartite graphs which do not possess cycles of a length divisible by 4. Let G be such a graph. Without loss of generality, we assume that G is connected.
Let the graph energy E and the VDB energy Ex be the quantities defined (D via Eqs. (2) and (5), and let f be the function specified in Eq. (3). Let 5 ad A be the smallest and largest vertex degrees of G.
w Theorem 1. Let G be a bipartite graph with no cycle of size divisible by 4. o Then
f (5,5) E(G) <Ex(G) < f (A, A) E(G)
holds for all VDB invariants in which the function f is monotonically increasing and f (x, y) > l for all vertex degrees x and y. Equality on both sides holds if and only if G is a regular graph, in which case 5 = A.
The examples of the VDB invariants for Theorem 1 are the above listed first and second Zagreb, forgotten, Sombor, and nirmala indices.
Theorem 2. Let G be a bipartite graph with no cycle of size divisible by 4. Then
f (A, A) E(G) <Ex(G) < f (5,5) E(G)
holds for all VDB invariants in which the function f is monotonically decreasing and 0 < f (x,y) < l for all vertex degrees x and y. Equality on both sides holds if and only if G is a regular graph.
16
The examples of the VDB invariants for Theorem 2 are the above listed " Randic, sum-connectivity, harmonic, and modified Sombor indices, as well °° as the inverse degree. ä
A tree is a connected graph with no cycles. Therefore, Theorems 1 and 2 apply to trees. For any tree 5 = 1, but Theorems 1 and 2 can be slightly strengthened.
holds for all VDB invariants in which the function f is monotonically increasing and f (x, y) > l for all x, y. Equality on the left-hand side holds if and only if n = 3.
Theorem 4. Let T be a tree with n > 3 vertices. Then
f (A, A) E(T) < Ex(T) < f (l, 2) E(T)
Theorem 3. Let T be a tree with n > 3 vertices. Then E
f (1,2) E(T) <EX(T) <f (A, A) E(T)
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holds for all VDB invariants in which the function f is monotonically de- o creasing and 0 < f (x, y) < l for all x, y. Equality on the right-hand side holds if and only if n = 3.
In addition to trees, Theorems 1 and 2 are applicable to various classes of cycle-containing graphs. Of these, of particular interest may be the hexagonal systems (molecular graphs of benzenoid hydrocarbons) (Gut-man &Cyvin, 1989). All their vertices are of degrees 2 and 3. The so-called catacondesned hexagonal systems (= hexagonal systems having no internal vertices) are known to possess only cycles of size 4p + 2. For these molecular graphs
f (2,2) E(G) < Ex(G) < f (3,3) E(G). (6)
or
f (3,3) E(G) < Ex(G) < f (2,2) E(G). (7)
depending on whether f (x, y) monotonically increases or decreases.
Hexagonal systems possessing internal vertices have cycles of size 4p , p = 3,4,..., and thus Theorems 1 and 2 are not applicable. We nevertheless conjecture that estimates (6) and (7) are valid for all hexagonal systems.
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In order to prove the above theorems, we need an auxiliary result, stated below as Lemma 3.
Energy of a weighted bipartite graph
The main part of the results outlined in this section was reported in (Gut-man et al, 2021). These are repeated here (in an abbreviated form) in order
yy to maintain completeness. Also, a few errors committed in (Gutman et al,
on
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o Let G be a bipartite graph with n vertices. Let Gw be obtained from G
< by associating weighs to its edges, so that wj is the weight of the edge ij. Then the characteristic polynomial of Gw is of the form (Cvetkovic et al, o 2010)
LU
0(Gw, X) = Xn + Y/-1)k c(Gw, k) Xn-2k (8)
>-
Ct k>1
whereas the energy of Gw satisfies the equality (Gutman, 1977, 2020; Li et al, 2012)
2 f dx
E (Gw ) = - -2
S? -V-w, - I
_J n J x
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c(Gw ,k) x2k
k1
(9)
>o Note that E(Gw) is a monotonically increasing function of any of the
x coefficients c(Gw, k).
^ According to the Sachs theorem (Cvetkovic et al, 2010)
o (-1)k c(Gw,k)= (-1)p{a) 2c(a) w(a) (10)
where Sk(Gw) is the set of all Sachs graphs of Gw possessing exactly 2k vertices, and where a is an element of S2k(Gw), containing p(a) components, of which c(a) are cycles. The weight of the Sachs graph a is equal to the product of the weights of its components. If the isolated edge ij is a component of a, then its weight is wfj. If a cycle Z is a component of a, then its weight is the product of weights of the edges contained in Z.
Lemma 1. (Gutman et al, 2021) If the Sachs graph a e S2k(Gw) = 0 does not contain cycles whose size is divisible by 4, then
(-1)k (-1)p(a) 2c(a) > 0 .
Proof.The Sachs graph a has p(a) components. Let among them be r0 > " 0 isolated edges, whose total number of vertices is 2r0. Let a contain r1 > 0 " cycles, whose total number of vertices is 4x+2 r1 for some integer x. Thus, £
2k = 2r0 + 4x + 2 n.
Case 1: 2k is not divisible by 4. Then (—l)k = —l whereas r0+r1 = p(a) is odd. Therefore, (—l)k (—l)p(a) > 0 and the claim of Lemma 1 holds.
» UJ i. ,,,^,,v-_L)k =+l whereas ro + ri = p(a)
is even, implying, again, (—l)k (—l)p(a) > 0. □ ^
Lemma 1 has the following noteworthy consequences:
Lemma 2.
(a) Let Gw be an edge-weighted bipartite graph whose all cycles (if any) have size not divisible by 4, and let the weights of all its edges be positive-valued. Then for any Sachs graph a e S2k(Gw) = ty,
(—l)k (—l)p(a) 2c(a) w(a) > 0 .
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(b) Therefore, because of Eq. (10), the coefficients c(Gw, k) in Eq. (8) are non-negative and are the monotonically increasing functions of the edge- ^ weights.
(c) Therefore, because of Eq. (9), the energy of the graphs Gw is a mono-tonically increasing function of the edge-weights.
From Lemma 2(c), we obtain the result needed for our proofs:
Lemma 3. Let Gw be an edge-weighted bipartite graph whose all cycles (if any) have size not divisible by 4.
(a) If for all edges ij e E(Gw), the condition wij > l holds, then E (Gw) > E(G). If wij > l for at least one edge ij, then E(Gw) > E(G).
(b) If for all edges ij e E(Gw), the condition wij < l holds, then E(Gw) < E(G). If wij < l for at least one edge ij, then E(Gw) < E(G).
(c) If in both cases (a) and (b), wij = w holds for all edges ij e E(Gw), then E (Gw )= w E (G).
Proof of Theorems 1 -4
The adjacency matrix Aj(G), Eq. (4), could be viewed as the ordinary adjacency matrix of an edge-weighted modification of the graph G. Therefore, if the condition f (dVi ,dVj ) > 1 holds, and if f (x,y) is an increasing
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function for x > 1 and y > 1, then the lower bound of Theorem 1 follows by Lemma 3 if all f (x, y) are replaced by f (5,5). The upper bound is obtained if all f (x, y) are replaced by f (A, A).
The proof of Theorem 2 is analogous.
Theorems 3 and 4 are based on the fact that no tree with n > 3 vertices is a regular graph. The only tree having two adjacent degree-one vertices is the two-vertex tree. Therefore, for trees with 3 or more vertices, the minimal (resp. maximal) value of f (x, y) is f (1,2).
References
Cvetkovic, D., Rowlinson, P. & Simic, K. 2010. An Introduction to the Theory of g Graph Spectra. Cambridge: Cambridge University Press. ISBN: 9780521134088. m Das, K.C., Gutman, I., Milovanovic, I., Milovanovic, E. & Furtula, B. 2018.
Degree-based energies of graphs. Linear Algebra and its Applications, 554, fi pp.185-204. Available at: https://doi.org/10.1016/j1aa.2018.05.027.
Gutman, 1.1977. Acyclic systems with extremal Huckel n-electron energy. The-oretica chimica acta, 45, pp.79-87. Available at: https://doi.org/10.1007/BF00552542.
CO
<5 Gutman, I. 2020. Relating graph energy with vertex-degree-based energies.
tD Vojnotehnicki glasnik/Military Technical Courier, 68(4), pp.715-725. Available at: https://doi.org/10.5937/vojtehg68-28083.
Gutman, I. 2021. Comparing degree-based energies of trees. Contributions to □ Mathematics, 4, pp.1-5. Available at: https://doi.org/10.47443/cm.2021.0030. § Gutman, I. & Cyvin, S.J. 1989. Introduction to the theory of benzenoid hydro-
o carbons. Berlin: Springer. Available at: https://doi.org/10.5860/choice.27-4521.
Gutman, I., Monsalve, J. & Rada, J. 2022. A relation between a vertex-degree-based topological index and its energy. Linear Algebra and its Applications, 636(March), pp.134-142. Available at: https://doi.org/10.1016/j1aa.2021.11.021.
Gutman, I., Redzepovic, I. & Rada, J. 2021. Relating energy and Sombor energy. Contributions to Mathematics, 4, pp.41-44. Available at: https://doi.org/10.47443/cm.2021.0054.
Kulli, V.R. 2020. Graph indices. In: Pal, M., Samanta, S. & Pal, A. (Eds.), Handbook of Research of Advanced Applications of Graph Theory in Modern Society, pp.66-91. Hershey, USA: IGI Global. Available at: https://doi.org/10.4018/978-1-5225-9380-5.ch003.
Li, X., Shi, Y. & Gutman, I. 2012. Introduction. In: Graph Energy, pp.1-9. New York, NY: Springer. Available at: https://doi.org/10.1007/978-1-4614-4220-2_1.
Li, X. & Wang, Z. 2021. Trees with extremal spectral radius of weighted adjacency matrices among trees weighted by degree-based indices. Linear Algebra and Its Applications, 620, pp.61-75. Available at: https://doi.org/10.1016/j.laa.2021.02.023.
Ramane, H.S. 2020. Energy of graphs. In: Pal, M., Samanta, S., & Pal, A. (Eds.) Handbook of Research on Advanced Applications of Graph Theory in Modern Society, pp.267-296. Hershey, PA, USA: IGI Global. Available at: https://doi.org/10.4018/978-1-5225-9380-5.ch011.
Shao, Y., Gao, Y., Gao, W. & Zhao, X. 2021. Degree-based energies of trees. Linear Algebra and Its Applications, 621, pp.18-28. Available at: https://doi.org/10.1016/jJaa.2021.03.009.
Todeschini, R. & Consonni, V. 2009. Molecular Descriptors for Chemoinforma-tics. Weinheim: Wiley-VCH. ISBN: 978-3-527-31852-0.
ОЦЕНКА ЭНЕРГИЙ, ОНСОВАННЫХ НА СТЕПЕНИ ВЕРШИН
Иван Гутман
Крагуевацкий университет, естественно-математический факультет, г Крагуевац, Республика Сербия
РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА:
ВИД СТАТЬИ: оригинальная научная статья Резюме:
Введение / цель: В новейшей литературе изучаются десятки инвариантов графов, основанных на степени вершин (VDB). К каждому такому инварианту может присоединиться матрица. Энергия VDB - это энергия (= сумма абсолютных значений собственных значений) соответствующей матрицы VDB. В данной статье исследуются некоторые общие свойства VDB-энергии двудольных графов.
Результаты: Получены оценки (нижней и верхней границы) по энергии VDB двудольных графов, не имеющих циклов величины, кратной 4, в зависимости от обычной энергии графа.
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27.29.19 Краевые задачи и задачи на
собственные значения для обыкновенных дифференциальных уравнений и систем уравнений
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Выводы: Результаты статьи вносят вклад в спектральную теорию матриц VDB, а особенно в общую теорию энергии VDB.
Ключевые слова: инвариант графа основанный на степени вершины, матрица основанная на степени вершины, энергия основанная на степени вершины, энергия (графа).
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^ ПРОЦЕНА ЕНЕРГША ЗАСНОВАНИХ НА СТЕПЕНИМА
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Иван Гутман
Универзитету Крагу]евцу, Природно-математички факултет,
I Крагу]евац, Република Срби]а
ОБЛАСТ: математика
^ ВРСТА ЧЛАНКА: оригинални научни рад
Сажетак:
Увод/цил>: У Hoeujoj литератури npoy4aeajy се 6pojHe графовске инварианте засноване на степенима чворова ^ (VDB). Сващ од ових инвар^анти може се придружити
^ матрица. VDB енерг^а jе енерг^а (= збир апсолутних
q вредности сопствених вредности) одговараjуfiе VDB мат-
рице. Рад истражуjе неке опште особине VDB енерг^е би-ш партитних графова.
° Резултати: Доб^ене су процене (док>е и горъе границе)
о за VDB енерг^у бипартитних графова ко\и немаjу циколве
величине деъиве са 4, а у зависности од обичне графовске енерг^е.
Закъучак: Резултати овог рада доприносе спектралноj теории VDM матрица, а посебно општоj теории VDB енерг^е.
Къучне речи: инварианта заснована на степенима чворова, матрица заснована на степенима чворова, енерг^а заснована на степенима чворова, енерг^а (графа).
Paper received on / Дата получения работы / Датум приема чланка: 27.12.2021. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 04.01.2022.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 05.01.2022.
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© 2022 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier <m
(http://vtg.mod.gov.rs, http://BTr.MO.ynp.cp6). This article is an open access article distributed under co the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
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