DOMINATION ON CACTUS CHAINS OF PENTAGONS Текст научной статьи по специальности «Математика»

DOMINATION ON CACTUS CHAINS OF PENTAGONS

Miroslava Mihajlov Carevic

Faculty for Business, Economics and Entrepeneurship,

Belgrade, Republic of Serbia,

e-mail: mm.carevic@vspep.edu.rs,

ORCID iD: https://orcid.org/0000-0001-6458-2044

DOI: 10.5937/vojtehg70-36576; https://doi.org/10.5937/vojtehg70-36576

FIELD: Materials and chemical technologies, Mathematics ARTICLE TYPE: Original scientific paper

Abstract

Introduction/purpose: A graph as a mathematical object occupies a special place in science. Graph theory is increasingly used in many spheres of business and scientific fields. This paper analyzes pentagonal cactus chains, a special type of graphs composed of pentagonal cycles in which two adjacent cycles have only one node in common. The aim of the research is to determine the dominant set and the dominance number on ortho and meta pentagonal cactus chains.

Methods: When the corresponding destinations are treated as graph nodes and the connections between them as branches in the graph, the complete structure of the graph is obtained, to which the laws of graph theory are applied. The vertices of the pentagon are treated as nodes of the graph and the sides as branches in the graph. By applying mathematical methods, the dominance was determined on one pentagon, then on two pentagons with a common node, and then on ortho and meta pentagonal cactus chains.

Results: The research has shown that the dominance number on the ortho

chain Oh of the length h > 2 is equal to the value of the expression [—1 while

2

on the meta chain Mh it is equal to the value of the expression h+1, which was proven in this paper.

Conclusion: The results show that the dominant sets and the dominance numbers on ortho and meta pentagonal cactus chains are determined and explicitly expressed by mathematical expressions. They also point to the possibility of their application in the fields of science as well as in the spheres of business in which these structures appear.

Keywords: graph, pentagonal cactus-chain, dominant set, dominance number.

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Introduction

Mathematical apparatus and mathematical methods are used in almost all fields of science, both natural (Ghergu & Radulescu, 2011; Velickovic et al, 2020) and social (Vladimirovich & Vasilyevich-Chernyaev, 2021). A graph as a mathematical object occupies a special place in science (Bakhshesh, 2022; Hajian & Rad, 2021; Hernández Mira et al, 2021). It is used in medicine, genetics, chemistry, etc. All structural formulas of covalently bound compounds are graphs. Chemical elements are represented by graphs where atoms are vertices and chemical bonds are lines in the graph (Balaban, 1985). A graphical representation of chemical structures provides a visual insight into molecular bonds and chemical properties of molecules. The QSPR study has shown that many of chemical properties of molecules are closely related to theoretical graphical invariants called molecular descriptors (Mihalic & Trinajstic, 1992). The theoretical graphical invariant is also the dominance number, which is the simplest variant of the k-dominance number that is used many times in mathematics (Zmazek & Zerovnik, 2003).

A graph is usually denoted by G, a set of its vertices (nodes) by V(G) and a set of its branches (lines) by E(G).

A set D that is a subset of the set V(G) is called a k-dominant set in the graph G if for each vertex outside the set D there is at least one vertex in the set D such that the distance between them is less than or equal to k. The number of elements of the smallest k-dominant set is called the k-dominance number and is denoted by yk. If k = 1, the 1-dominance number is called the dominance number and is denoted by y and the 1-dominant set is called the dominant set.

A cactus graph is a connected graph in which no line (branch) is in more than one cycle. The study of cactus graphs began in the middle of the 20th century. In his work (Husimi, 1950) Husimi uses these graphs in studies of cluster integrals. Riddell (Riddell, 1951) uses them in the theory of condensation. They were later used in the theory of electrical and communication networks (Zmazek & Zerovnik, 2005) as well as in chemistry (Sharma et al, 1997; Gupta et al, 2001; Gupta et al, 2002).

It is known that many chemical compounds have a pentagonal shape in their configuration. Among them are cycloalkanes, which are very common compounds in the nature. The five-membered and six-membered cycloalkanes, cyclopentane (Figure 1) and cyclohexane, which contain 5 and 6 ring carbon atoms, respectively, are very stable and their structures appear in many biological molecules.

Figure 1- Cyclopentane Рис. 1 - Циклопентан Слика 1 - Циклопентан

Their ring structures are also included in the composition of steroids. A large number of steroids are synthesized in laboratories and used in the treatment of cancer, arthritis, various allergies and other diseases (Balaban & Zeljkovic, 2021). Pentagonal forms in combination with hexagonal forms are present in many compounds, among which are heterocyclic compounds: morphine, benzofuran, dibenzothiophene and others.

In this paper, we analyze the k-dominance of pentagonal cactus chains. Hexagonal cactus chains were investigated in papers (Farrell, 1987; Vukicevic & Klobucar, 2007). Afterwards, the papers (Majstorovic et al, 2012; Klobucar & Klobucar, 2019) determined the dominance number on a uniform hexagonal cactus chain, the dominance number on an arbitrary hexagonal network, and the total and double total dominance number on a hexagonal network. The K-dominance on rhomboidal cactus chains (Carevic et al, 2020) as well as on the icosahedral-hexagonal network (Carevic, 2021) was also investigated.

Pentagonal cactus-chains

The pentagonal cactus-chain G is a graph consisting of a cycle with 5 vertices. A vertex that is common to two or three pentagons is called a cutvertex. If each pentagon in the graph G has at most 2 cut-vertices and each cut-vertex is divided between exactly 2 pentagons, the graph G is called a pentagonal cactus-chain.

With Gh we will denote a pentagonal cactus-chain of the length h and Gh = РгР2...Ph where Pl are successive pentagons in the chain (Figure 2).

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Figure 2 - Pentagonal cactus-chain of the length 7 Рис. 2 - Пятиугольная кактус-цепочка длиной 7 Слика 2 - Петоугаони кактус-ланац дужине 7

Denote by х and y the vertices in the graph G and by d(x, y) the distance between them, where the distance between two vertices is equal to the number of branches located from one vertex to another. Denote by Pi the minimum distance between the pentagons Pl and Pl+2: Pi = min{d(x,y): хеР1луеР1+2, i = 1,2,...,h- 2} Then pi is the distance between the pentagons Pl and Pl+2. With the exception of the first and last pentagons in the cactus chain, which have one cut-vertex, all other pentagons have two cut-vertices, and they are called inner pentagons.

In the pentagonal cactus chain Gh, we distinguish between ortho and meta inner pentagons. An inner pentagon is called an ortho pentagon if its cut-vertices are adjacent, and a meta pentagon if the distance between its cut-vertices is d = 2.

A pentagonal cactus chain is uniform if all its inner pentagons are of the same type. A chain Gh is called an ortho-chain, and is denoted by 0h if all its inner pentagons are ortho-pentagons (Figure 3).

Figure 3 - Ortho cactus-chain 05 Рис. 3 - Орто-кактус-цепочка 05 Слика 3 - Орто кактус-ланац О5

Analogously, a chain Gh is called a meta-chain, and is denoted by Mh if all its inner pentagons are meta-pentagons (Figure 4).

Figure 4 - Meta cactus-chain M5 Рис. 4 - Мета кактус-цепочка М5 Слика 4 - Мета кактус-ланац М5

To determine the dominant set on the uniform pentagonal cactus chains Oh and Mh, it will be necessary to point out certain vertices in the cactus chain. That is why it is necessary to mark them. In the ortho pentagon Pl the cut-vertices are adjacent and we will denote them by Vt and Vi+1. The other vertices in Pl it will be denoted by x[, xl2andx^ (Figure 5):

Figure 5 - Marking vertices in the ortho pentagon Рис. 5 - Обозначение вершин в ортогональном пятиугольнике Слика 5 - Означава^е чворова у орто петоуглу

In the meta pentagon Pl the cut-vertices are at a distance d = 2 and we will denote them by V2i-1 and V2i+1. With V2iwe will denote the vertex to which it applies d(V2i-1, V2i) = d(V2i, V2i+1) = 1. The other two nodes in the pentagon Pl will be denoted x[andxl2 (Figure 6):

Figure 6 - Marking vertices in the meta pentagon Рис. 6 - Обозначение вершин в мета-пятиугольнике Слика 6 - Означава^е чворова у мета петоуглу

Research results

In this section, we consider 1-dominance on ortho and meta pentagonal cactus chains. We will first consider the dominance of one pentagon and two adjacent pentagons in the ortho and meta chain of cacti. Lemma 3.1. The dominance number for the pentagon is y = 2. Proof: Let us denote the vertices of the pentagon by хъ x2, x3, x4, x5 (Figure 7):

Figure 7 - Dominant set on a pentagon Рис. 7 - Доминирующее множество на пятиугольнике Слика 7 - Доминантни скуп на петоуглу

One pentagon vertex dominates two adjacent vertices. Let us take the vertex хг. It dominates the vertices x2 and x5. As the pentagon has 5 vertices, domination over the other two vertices x3 and x4 is necessary. We conclude that one of the remaining two vertices must belong to the dominant set on the pentagon. Let it be the vertex x3. Thus, the set D = [xx, x3} is the dominant set for a given pentagon but it is not the only

dominant set whose cardinality is equal to 2. They are also sets that contain any two non-adjacent pentagon vertices. Let us prove that any of the mentioned two-membered sets is the minimum dominant set on the pentagon. Assuming that there is a dominant set of less cardinality D', it would have to contain only one vertex and one vertex cannot dominate the remaining 4 vertices of the pentagon. Thus, the minimum dominant set on a pentagon is a two-membered set, so the dominance number for the pentagon is y = 2.

Lemma 3.2. The dominance number for two pentagons with one cutvertex is y = 3.

Proof: Let us denote the vertices of two pentagons by one common vertex with x1, x2, . . ., x9 (Figure 8):

Figure 8 - Dominant set for two pentagons with a cut-vertex Рис. 8 - Доминирующее множество для двух пятиугольников с пересекающейся

вершиной

Слика 8 - Доминантни скуп за два петоугла са пресеченим чвором

Let x1 be the cut-vertex of the given pentagons P1 and P2.Based on Lemma 3.1. the pentagon P1 excluding the vertex x1 must have another dominant vertex that is not adjacent to the vertex x1.Let it be the vertex x3. Also by applying Lemma 3.1. the pentagon P2 excluding the vertex x1 must have another dominant vertex that is not adjacent to the vertex x1. Let it be the vertex x7. Thus the nodes x1, x3 and x7 dominate over the nodes x2, x4, x5, x6, x8 andxg so the dominant set for the pentagons P1P2 is the set D = {x1,x3, x7}. Analogous to the consideration in Lemma 3.1. the set D is not the only three-membered set that is dominant on P1P2but there is no dominant set of less cardinality. Suppose that there is a dominant set D' whose cardinality is equal to 2. Let D' contain one vertex from each

pentagon, for example D' = {x1,x3}.The vertices x1 and x3 would then dominate over the remaining 7 vertices in P1P2 and this is impossible.The vertex x1 as a common vertex for both pentagons dominates over two neighboring vertices in both pentagons, so it dominates over 4 vertices in P1P2.The vertex x3, or any other vertex not adjacent to the vertex x1 dominates two adjacent vertices.So, the total sum of vertices covered by dominance is 4 + 2 = 6 and that is less than 7.Thus, 2 vertices cannot dominate the remaining 7 vertices in P1P2. We conclude that the minimum dominant set for P1P2 is a three-membered set and y = 3.

Let us consider the dominance on pentagonal ortho and meta cactus chains of arbitrary length.

Theorem 3.1. y( Oh) = y for each h> 2 Ah eN.

Proof: We observe a pentagonal ortho cactus-chain 0h = P1P2... Ph (Figure 9) and a set:

D0h = {4, i = 1, h} и {V2i, i = 1, -J

%9K

Figure 9 - Minimum dominant set for 0h Рис. 9 - Минимально доминирующее множество для 0h Слика 9 - Минимални доминантни скуп за 0h

Let us prove that D0 is the dominant set of minimum cardinality for a

pentagonal ortho cactus-chain 0h= P1P2... Ph.

Let us divide the ortho-chain 0h into subchains P21 1P21, i = 1, 2, ... , (Figure 10) and the last pentagon Ph if h is an odd number.

Figure 10 - Subchain of the ortho-chain Oh Рис. 10 - Подцепочка орто-цепочки Oh Слика 10 - Подланац орто ланца Oh

Based on Lemma 3.2. the set A{ = {x^i-1,x^i,V2i} for i = 1, 2,...,

the dominant set of minimum cardinality for the subchain P21 1P21. ortho-chain of the length h for h = 2k, k £ N is composed of J subchains

is An

P

2i-1P2i, i= 1, 2,... (Figure 9A), so the set

D1= \jhAi,

h

for k = -

2

is a dominant set for the ortho-chain 0h. Therefore, it is Y(0h) < card(Di) = h- 3 = y

where we have marked the cardinality of the set D1 with card(D^). If h is an odd number (Figure 9B ), then the set

D2=u!=1Aiu{xh,Vh+J, for k = is a dominant set for the ortho-chain 0 and then is

/(Oh) < card(D2) =

3 + 2 =

Note that the set D1 for k = j if h an even number is equal to the following expression:

D1= \JhAt = U

k r Y2i-1 v2i у д _ i=1l x2 >x2 > v2l} =

= {x1,x2,V2} U {xi,x4,V4]U... U {xh-1,xh,Vh} = = { 4, ¿ = 1,2.....h}U {V2i,i = l,2.....h}

2

Also for k =

and h is an odd number, the set D2 is equal to the

following expression:

D2= Uf=1Ai и (x-, Vh+i)= -1

= Ul=i{xii-1,xii,V2i} и (x*, Vh+i}=

= (xl x%, V2} U (x3, x4, ^}U . . . U (х--1, xh, Vh} и (xh, V-+i}=

= (4, ¿ = 1,2.....h}U (^,¿ = 1,2.....[hl}

In case h is an even number, - =

2

then we conclude that it is D* = D,

So, the set D0 = (xl2, i = 1, h} и (V2i, i = 1, -J is the dominant set

for the ortho-chain Oh when h is even or odd number.

3 h

. So, y(Oh) <

Also, in the case where h is an even number, — =

' 2

when h is even or odd number. Prove that the set D0h is the dominant

set of minimal cardinality. Each subchain p2l-1p21 contains 3 dominant nodes based on Lemma 3.2. Based on this, we conclude that each dominant set on the chain Oh contains more than 3 or exactly 3 dominant nodes in each subchain P2l-1P2land more than 2 or exactly 2 dominant nodes in the last pentagon if h is an odd number, based on Lemma 3.1. So, we conclude that it is y(Oh) > h ■ 3 in case h is an even number, and

y(Oh) > h ■ 3 + 2 in case h is an odd number. When we combine both

cases, we get that y(Oh) > It follows from y(Oh) < 3h

2

that it is y(Oh) =

3h 2 .

and y(Oh) > Corollary 3.1 .D0h c D0h+1 for each h > 2 a h eN. Theorem 3.2. y(Mh) = h + 1 for each h > 2 a h eN. Proof: We observe a pentagonal meta cactus-chain Mh = P1P2. Ph (Figure 11) and set:

DMh = {V2i-1, i = 1, h + 1}

Figure 11 - Minimum dominant set for Mh Рис. 11 - Минимально доминирующее множество для Mh Слика 11 - Минимални доминантни скуп за Mh

Let us prove that DMft is the dominant set of minimum cardinality for a pentagonal meta cactus-chain Mh = P1?2... Ph. Based on Lemma 3.1. each pentagon has a dominant set made up of two non-adjacent vertices. Thus, the set {V_2i_1, V2i+1) is dominant for the pentagon P( for each i = 1, h. By merging the dominant sets of all pentagons in the chain, we get a set that is dominant for the whole chain. But, each pentagon P( has a common vertex with the pentagon Pi+1 for each i = 1, h - 1. Common vertices should not be repeated in the dominant set. So, the set

= UjU V2i_i, ^+1} \ Uf=_i1{ ^+1} is the dominant set for the meta-chain Mh. Note that it is

Uj=1{ V2i_1, ^¿+1} \ Uf=_11{ ^¿+1} =

{{V1, Fs} U {F3, ^5} U {F5, ^7} U ... U {V2h-1, ^2h+1}}\ {^3, .....^-1} =

{V1T3T5.....V2h_1,V2h+1} = {V2i_1, i = 1, h + 1}

Thus, the set DMft = { V2(_1, i = 1, h + 1} is the dominant set for the meta-chain Mh for each heN and h >2. Let us prove that DMft is the dominant set of minimal cardinality. Suppose that there is a set S of less cardinality that is dominant on the meta-chain Mh. The set S would then have one node less than the set DMft. Let it be a vertex y2j+1 for any i = 1, h. Then the pentagon P( would have only one dominant node V2j_1. Based on Lemma 3.1. that is not possible. We conclude that DMft is the minimum dominant set for Mh so it is y(Mh) = h + 1.

Corollary 3.2._DMft c DMft+1 for each h > 2 a h eN.

Conclusion

In this paper, we have shown the arrangement of vertices in dominant sets on uniform ortho and meta pentagonal cactus chains that appear in molecule structures of numerous compounds. We also proved that the dominance number for a pentagonal ortho-chain of the length h is equal to

the value of the expression y while for a pentagonal meta-chain it is

equal to h + 1.

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ДОМИНИРОВАНИЕ НА ПЯТИУГОЛЬНЫХ КАКТУС-ЦЕПЯХ

Мирослава Михайлов Царевич

Высшая школа экономики и предпринимательства,

г. Белград, Республика Сербия

РУБРИКА ГРНТИ: 27.45.17 Теория графов,

61.13.21 Химические процессы ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: Граф как математический объект занимает особое место в науке. Теория графов все чаще используется во многих видах деятельности и различных научных областях. В данной статье анализируются пятиугольные кактус-цепочки, как особый вид графов, состоящих из пятиугольных циклов, в которых два соседних цикла имеют только один общий узел. Цель исследования заключалась в определении доминирующего множества и доминируещего числа в орто- и мета-пятиугольных куктус-цепочках.

Методы: Когда соответствующие положения рассматриваются как узлы графа, а связи между ними - как ветви графа, получается полная структура графа, к которой применяются законы теории графов. Вершины пятиугольника рассматриваются как узлы графа, а стороны - как ветви графа. С помощью математических методов, было определено доминирование на одном пятиугольнике, затем на двух пятиугольниках с общим узлом, а затем на орто- и мета-пятиугольных кактус-цепочек.

Результаты: Исследование показало, что число доминирования на орто-цепи Oh с длиной h > 2 равно значению выражения в то время как на мета-цепи Mh оно равно значению выражения h+1, что и следовалось доказать в данной статье.

Выводы: Результаты исследования показали, что доминирующие множества и числа доминирования в орто- и мета-пятиугольных кактус-цепочках определяются и эксплицитно исчисляются математическими выражениями. Они также указывают на возможность их применения как в области науки, так и в сферах бизнеса, в которых присутствуют эти структуры. Ключевые слова: граф, пятиугольная кактус-цепочка, доминирующее множество, число доминирования.

ДОМИНАЦША НА ПЕТОУГАОНИМ КАКТУС-ЛАНЦИМА

Мирослава Миха]лов Цареви^

Висока школа за пословну економи]у и предузетништво, Београд, Република Срби]а

ОБЛАСТ: матери]али и хеми]ске технологи]е, математика ВРСТА ЧЛАНКА: оригинални научни рад

Сажетак:

Увод/цил>: Гоаф као математички оЩекат заузима посебно место у науци. Теорба графова налази све веЬу примену у многоброjним

сферама пословаъа, као и научним областима. У овом раду анализирани су петоугаони кактус-ланци щи представъа}у посебну врсту графа саставъеног од петоугаоних циклуса у ко}има два суседна циклуса има]у за}еднички само }едан чвор. Цил> истраживаъа }есте одре^иваъе доминантног скупа и доминацирког бро}а на орто и мета петоугаоним кактус-ланцима.

Методе: Када се одговара]уПа одредишта третира]у као чворови графа, а везе ме^у ъима као гране у графу, доби}а се потпуна структура графа на ко}у се примеру законитости теорбе графова. Темена петоугла су третирана као чворови графа, а странице као гране у графу. Применом математичких метода одре^ена jе доминаци}а на ]едном петоуглу, затим на два петоугла са за}едничким чвором, а након тога на орто и мета петоугаоним кактус-ланцима.

Резултати: Истраживаъа су показала да jе доминацирки бро] на

орто ланцу Oh дужине h >2 ¡еднак вредности израза |у] , док¡е на

мета ланцу Mh ¡еднак вредности израза h + 1, што ¡е доказано у раду.

Закъучак: Резултати показу]у да су доминантни скупови и доминацирки брорви на орто и мета петоугаоним кактус-ланцима одре^ени и експлицитно исказани математичким изразима. Тако^е, упуПу]у на могуЬност ъихове примене у областима науке, као и у сферама пословаъа у ко]има се порвъу}у ове структуре. Къучне речи: граф, петоугаони кактус-ланац, доминантни скуп, доминацирки бро].

Paper received on / Дата получения работы / Датум приема чланка: 22.02.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 22.06.2022.

Paper accepted for publishingon / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 24.06.2022.

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