On the KG-Sombor index Текст научной статьи по специальности «Математика»

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On the KG-Sombor index

Ochirbat Altangooa, Dechinpuntsag Bolormaab, Badarch Gantuyac, Tsend-Ayush Selenged

aMongolian National University of Education,

School of Mathematics and Natural Sciences, Ulaanbaatar, Mongolia, e-mail: [email protected] e)

ORCID iD: ©https://orcid.org/0009-0000-6995-9138 bMongolian National University of Education, School of Mathematics and Natural Sciences, Ulaanbaatar, Mongolia, o e-mail: [email protected], ORCID iD: ©https://orcid.org/0009-0007-2094-9220 cMongolian National University of Education,

School of Mathematics and Natural Sciences, Ulaanbaatar, Mongolia, e-mail: [email protected], °

ORCID iD: ©https://orcid.org/0000-0001-6323-6510

dNational University of Mongolia, Department of Mathematics, <

Ulaanbaatar, Mongolia,

e-mail: [email protected], corresponding author, ORCID iD: ©https://orcid.org/0000-0002-8479-6389

doi https://doi.org/10.5937/vojtehg72-49839

FIELD: mathematics ARTICLE TYPE: original scientific paper

Abstract:

Introduction/purpose: Degree-based graph invariants are a type of molecular descriptor that represent the connectivity of atoms (vertices) in a molecule through bonds (edges). They are used to model structural properties of molecules and provide valuable information for fields such as physical chemistry, pharmacology, environmental science, and material science. Recently, novel degree-based molecular structure descriptors, known as Sombor index-like graph invariants, have been explored from a geometrical perspective. These graph invariants have found applications in network science, where they are used to model dynamic effects in biological, social, and technological complex systems. There is also emerging interest in their potential military applications. Among these descriptors is the KG-Sombor index which is defined using both vertex and edge degrees.

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Methods: The study uses combinatorial graph theory to identify and analyze extremal graphs that either maximize or minimize the KG-Sombor index.

Results: The extremal graphs are characterized concerning the KG-Sombor index, with a particular focus on trees, molecular trees, and uni-cyclic graphs.

Conclusion: This research advances the theoretical understanding of

Sombor index-like graph invariants.

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E Key words: KG-Sombor index, tree, unicyclic graph, molecular tree.

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Introduction

x Let G = (V, E) be a graph with the vertex set V and the edge set E. For

w a vertex v, the degree of the vertex v, denoted by dv, is the number of edges > incident with v. The first Zagreb index M1 and the second Zagreb index M2 ft of the graph G are among the most famous and extensively studied vertex-degree-based topological indices (Gutman & Das, 2004; Horoldagva etal., 2021; Selenge & Horoldagva, 2015; Zhang & Zhang, 2006) defined as:

Mi(G) = ^ (du + dv) and M2G) = ^ dudv

CD uveE uveE

>0

For an edge uv, (du, dv) and (dv ,du) are referred to as the degree-points of the edge uv. Let O be the origin of the coordinate system, and M(du, dv) and M*(dv ,du) represent the degree-points of an edge. The distance be-o tween the points O and M is a/du + d2. Computing this for all edges in a graph and summing them yields the Sombor index (Gutman, 2021), defined as:

SO(G) = £ \OM\ = Y, VdZ + dl- (1)

uv£E uv£E

The Sombor-type indices (Dorjsembe & Horoldagva, 2022; Gutman, 2022, 2024; Tang et al., 2023) represent the latest addition to a plethora of topological indices in chemistry. The degree of an edge e = uv e E, denoted by de, is the number of edges incident to e. In (Kulli et al., 2022); a novel topological graph invariant named the KG-Sombor index is intro-

duced:

KG(G) = £ ^/dUTd*, (2) ^

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KG-Sombor index of trees and unicyclic graphs

For e = uv of a graph G, let us denote

f (du, dv) = VdU + (du + dv - 2)2 + ydiTWUTdV—2)2 (3)

and call it the weight of uv. On the other hand for an edge e = uv, there are two terms in the summation ^ue. Hence, we can reformulate the KG-Sombor index as follows.

KG(G)=Y, f (du, dv). (4)

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Lemma 1. (Gutman & Das, 2004) Let T be a tree of order n > 2. Then M1(T) > 4n - 6 with equality if and only if T is isomorphic to Pn.

Kulli et al. (Kulli et al., 2022) stated the following theorem without proof and mentioned that the proof is analogous to the proof of Theorem 2 in (Gutman, 2021). We now give the proof of it using the well-known result of Lemma 1.

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where denotes summation over the vertices u e V and the edges £

e e E incident to u. Some fundamental properties of the KG-Sombor index x are established in (Kulli et al., 2022), along with its relationships with other

topological indices. o

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Cruz et al. (Cruz et al., 2021) and Cruz and Rada (Cruz & Rada, 2021) w investigated the extremal values of the Sombor index for chemical graphs, ^ unicyclic graphs, and bicyclic graphs. Recent studies on the Sombor index and the KG-Sombor index can be found in (Damnjanovic et al., 2023; Das et al., 2021; Horoldagva & Xu, 2021; Kosari et al., 2023; Liu et al., 2022; Rada et al., 2021; Selenge & Horoldagva, 2024) and the references cited therein. In this paper, we aim to determine the extremal graphs concerning o the KG-Sombor index for trees, unicyclic graphs, and chemical trees of a given order.

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Theorem 1. Let T be a tree of order n > 3. Then

™ 4^2(n - 3) + 2—2 + 2V5 < KG(T)

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< (n - 1)(Vn2 - 4n + 5 + V2n2 - 6n + 5) (5) with equality on the left-hand side if and only if T is isomorphic to Pn, and

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g Proof. From (3), one can easily show that

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° f (du, dv) > —= (du + du + dv - 2) + — (dv + du + dv - 2)

x v 2 v 2

" = —=(3(du + dv) - 4) (6)

^ with equality if and only if du = dv = 2.

Since T is the tree, it has at least two pendent edges. Let us denote by f (1, dx) and f (1, dy) the weights of two pendent edges e1 and e2, respec-js tively. Then dx > 2, dy > 2 and

i f (1,dx) = V1 + (dx - 1)2 + Vd2x + (dx - 1)2. (7)

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First, it is shown that for dx > 2,

3\/2

f (1, dx) dx + V5 - 2\p2 = g(dx) (8)

with equality if and only if dx = 2. If 2 < dx < 6 then from (7) and (8), one & obtains f (1, 2) = —5 + —2= g(2), —5 + —13 = f (1, 3) > g(3) = 2.5^2 + —5, V10 + 5 = f (1, 4) > g(4) = 4^2 + yfl, + V4l = f(1, 5) > g(5) = 5.5^/2 + y/5 and V26 + V6I = f (1,6) > g(6) = 7V2 + v5 On the other hand, one gets

f (1, dx) >dx - 1 + V2(dx - 1) = (1 + V2)(dx - 1)

and from this it can easily be seen that inequality (8) holds for all dx > 7. Similarly as the above there is

3V2 ~2

with equality if and only if dy = 2.

f (1,dy) > - dy + V5 - 2V2 (9)

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>— J] [(3(du + dv ) - 4)] + f (1, dx) + f (1, dy )

uv^E\{e1,e2}

^Mi(T) + f (1, dx) + f (1, dy) - —^(dx + dy + 2) - 2^2(n - 3).

Then by the definition of the KG-Sombor index and (6), one gets

KG(T) = £ f (du, dv) = J] f (du, dv) + f (1, dx) + f (1, dy) §

uv^E uv£E\{e1 ,e2}

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3 ^ "u + dv ) + f (1,dx) + f (1, dy ) f

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E (du + dv )+ f (1,dx)+ f (1,dy )

^ 2 uveE(T) OT

--3=(dx + dy + 2) --4=(n - 3) S

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Then, using (8), (9) and Lemma 1 in the above inequality gives the lower bound in (5). From (6), (8), (9) and Lemma 1, it can be easily concluded that the left-hand side inequality in (5) holds if and only if T is isomorphic S

to Pn,

To prove the right-hand side inequality, let us consider the function

0(x) = a/x2 + (n - 2)2 + y/(n - x)2 + (n - 2)2

and

1 r2 1

Vr2 + (n - 2)2 (r2 + (n - 2)2)372 y/(n - x)2 + (n - 2)2 2

x

((n - x)2 + (n - 2)2)372

(n 2) V (x2 + (n - 2)2)372 + ((n - x)2 + (n - 2)2)37V > 0'

Therefore the function is convex and if 1 < x < n-1 then the maximum of fi is attained at x = 1 or x = n - 1.

Let uv e E. Then since T is a tree, one obtains

2(n - 1) = 2|EI = dw > du + dv + n - 2

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that is, du + dv < n' Therefore,

f (du, dv) < Vd2u + (n - 2)2 + Vd2 + (n - 2)2 < fi(du) < 0(1) = 0(n - 1)

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from is convex. Then, by the definition of the KG-Sombor index, one gets

KG(T) = £ f (dud) < E = (n - WW

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that is our required upper bound. Suppose now that the right-hand side equality holds in (5). Then du = 1 and dv = n - 1 for all uv e E. Hence T is a star Sn and one can see easily that the right-hand side equality holds

g in (5) for Sn. This completes the proof. □

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m Theorem 2. Let G be a unicyclic graph of order n > 3. Then

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ft KG(G) > 4nV2

with equality if and only if G is isomorphic to Cn.

w Proof.Let uv be an edge of G. Then, similarly as in the proof of the previous

^ theorem, one gets

° f (du,dv) > ~^(3(du + dv) - 4)

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o and Lemma 2, one obtains

& KG(G) = Y f (du, dv) > -^Y. (du + dv) - 2nV2

uv&E uv&E

= — Mi(G) - 2nV2 > 4nV2 2

with equality if and only if G is isomorphic to Cn. □

KG-Sombor index of molecular trees

By Theorem 1 it is evident that the path Pn has a minimal KG-value among molecular trees of order n. Therefore, this section determines the extremal graphs with the maximal KG-value among molecular trees of order n. For n = 3k + 2, k > 1, we denote by Tn the set of trees of order

n such that the degree of every vertex is either one or four. For n = 3k, k > 3, we denote by Tn the set of trees of order n such that only one vertex has degree two, and its neighbors have degree four, while the remaining vertices have degree one or four. For n = 3k + 1, k > 4, we denote by Tn a. the set of trees of order n such that only one vertex has degree three, and its neighbors have degree four, while the remaining vertices have degree one or four. o

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If n < 3 then there is only one tree of order n. If n = 4, then there are $ two trees that are P4 and S4, thus KG(P4) < KG(S4). Hence, we assume ^ that n > 5. For n = 6, n = 7, n = 10, we have determined the graphs with the maximum KG-Sombor index by using SageMath.

Figure 1 - Graphs with the maximum KG-Sombor index for n = 6,7,10.

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The values of the KG-Sombor index of the above graphs are 3\/i0 + 3^5 + 5^2 + 15, V41 + V34 + 2^13 + 3^10 + 2^5 + 15 and 2^41 + 2^34 + ^/13 + 6\/10 + \/5 + 30, respectively. For the remaining values of n, the following theorem holds.

Theorem 3. Let T be a molecular tree of order n.

(i) If n = 3k + 2, k > 1 then

Tsmm\ ^ 10 + 2^10 4^13, r,

KG(T) < -(n + 1) + -^(n - 5).

(ii) If n = 3k, k > 3 then

KG(T) < (10 + 2^I°>" + ^ +8^2 + 4^5 - 12^13.

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(iii) If n = 3k + 1, k > 4 then

KG(T) < (10±M«)2 + M3! +3V55 + 3V41+ - 52^.

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The equalities hold if and only if T e Tn.

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Proof. Let ni (i = 1,2,3,4) be the number of vertices of the degree i in T. Also let m,ij (1 < i < j < 4) be the number of edges of T connecting the vertices of the degree i and j. Then there is mi,i = 0 and

U\ + U2 + u + u = n

mi,2 + mi,3 + mi,4 + m.2,2 + m2,3 + m2,4 + m3,3 + m3,4 + m4,4 = n - 1 mi,2 + mi,3 + mi,4 = ni mi,2 + 2m2,2 + m2,3 + m2,4 = 2n2 mi,3 + m2,3 + 2m3,3 + m3,4 = 3n3 mi 4 + m2 4 + m3 4 + 2m4 4 = 4n4.

(10)

From the above, the following equations are easily obtained:

3 4 5 5 3

n =-mi , 2 + - mi , 3 + - mi , 4 + m-2 , 2 + 77 m2 , 3 + 7 m-2 ,4 2 3 4 b 4

2 7 1

+ 3 m3,3 + 12 m-3,4 + ^ mi,4,

2

mi,4 =^(n + 1)

1

2 5 2 1 1 - - 2mi,2 + -mi,3 + m2,2 + -m2,3 + -m2,4 + -m3,3 + -m3,4

3 \ 3 3 2 3 b

1 , . 1 1 1 5 2 7

m.4,4 =^(n - 5) + - mi,2 + - mi,3 - - m2,2 -7: m2,3 -7 m2,4 -77 m3,3

3 3 9 3 9 3 9

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(11)

Then, using (11), one obtains

KG(T ) = £ f (du,dv)

uveE(G)

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ij

£ |Vi2 + (i + j - 2)2 + V j2 + (i + j - 2)2

i<i<j<4

--(y/2 + y/E)mit2 + (V5 + VT3)mi,3 + (5 + VT0)mi,4 + 4\/2m2,2 + (3V2 + Vr3)m2,3 + (4V2 + 2y/5)m2 ,4 + 10m3,3 + (V34 + + 4\iY3m44

=-5-(n + 1) +---—(n - 5) + Ci2mi,2 + ci3mi,3

+ C22 m2,2 + C23m2,3 + C24m2,4 + C33 m3,3 + c34m3,4,

(12)

where C12 = V2 + - 4(5+/I°) + ^ - -2-425, C13 = ^5 + Vl3 -10(5+^ + 4e913 — -1.625, C22 = 4^2 - ^^^^ - 4e/H - -4.592, C23 = 3^2 + - - e13 - -3.792, C24 = 4^2 + 2^5 - 5+^0 - ^ -

-2.206, C33 = 10 - 2(5+/^) - ^^ - -3.031 and C34 = ^34 + V41 -

5+^/10 32^/13 9 9

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-1.492. Note that

C22 < C23 < C33 < C12 < C13 < C34 < C24 < 0.

Now the following three cases are distinguished. (i) If n = 3k + 2, k > 1 then from (12) and (13), one obtains

KGT) < 10+o2^V + 1) + i^(n - 5)

(13)

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with equality holding if and only if

m1,2 = m1,3 = m.2,2 = m2,3 = m2,4 = m3,3 = m3,4 = 0. Hence one gets n1 = m1)4 = 2(n + 1)/3 and m4,4 = (n - 5)/3. Also, there

is n2 = n3 = 0.

(ii) If n = 3k, k > 3 then one can easily see that n2 = 0 or n3 = 0. If n2 > 1 then

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m1,2 + 2m2,2 + m2,3 + m2,4 = 2n2 > 2 from (10). Therefore, one gets

KG(T) < 10±M°(n + 1) + ^(n - 5)

+ C24(m1,2 + 2m2,2 + m2,3 + m2,4)

< 10±M5(n + 1) + ^(n - 5) + 2C24

= (10+2^ + 4^ + ^ + 4V5 _ ^ 33

since C12 < C23 < C24 and C22 < 2C24.

(14)

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If n3 > 1 then

mi,3 + m2,3 + 2m3,3 + m,3A = 3«3 > 3 from (10). Hence, one obtains

„ KG(T) < T0±MV + 1) + ^(n - 5)

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^ + C34(mi,3 + m2,3 + 2m3,3 + m-3,4)

<-5-(n + 1) + ^^(n - 5)+3C34

< 3 3

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since c23 < c13 < c34 and c33 < 2c34. From (14) and (15), one gets the required result because 3c34 < 2c24. Equality holds in (14) if and only if

n2 = 1, n3 = 0 and m2 4 = 2.

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. KG(T) < (10 + ^'/T5>n + 4^ +3/54 + 3^41+5^1»

O 3 3 °

^ (16)

Let now n3 = 0. If n2 = 1, then the system of equations n1 + n4 = n - 1 and n1 + 4n4 = 2(n - 2) has no integer solution. Thus n2 > 2 and, similarly as in (ii), one also gets

KG(T) <T0±M5(„ + 1> + M!(„ - 5)

+ C24(mi,2 + 2m2,2 + m2,3 + m2,4) <T0±M«(n + 1) + MI(„ - 5) + 4C24

< (10 + 2^0)2 + 4Vnn +3^34 ++5±VTo - ^

3 3 3 3

from 4c24 < 3c34. Equality holds in (16) if and only if n2 = 0, n3 = 1 and m3,4 = 3. On the other hand, in each case it can be easily concluded that the equality holds if and only if T eTn. □

Conclusions

Topological indices play a vital role in conducting quantitative structure-activity relationship and quantitative structure-property relationship studies. Numerous topological indices have been defined in the literature and several of them are applied as a means to model physical, chemical, pharmaceutical, and other properties of molecules. Gutman pioneered the introduction of SO-type indices within the field of mathematical chemistry. Our study determined the minimal and the maximal KG-Sombor index for the class of trees and chemical trees. Moreover, we proved that Cn is the unique graph with the minimal KG-Sombor index among all unicyclic graphs of order n. However, the problems of determining the graphs with the maximal KG-Sombor index in the class of unicyclic graphs, and finding the extremal KG-Sombor index in the class of bicyclic graphs remain open.

References

Cruz, R., Gutman, I. & Rada, J. 2021. Sombor index of chemical graphs. Applied Mathematics and Computation, 399, art.number:126018. Available at: https://doi.org/10.1016/j.amc.2021.126018.

Cruz, R. & Rada, J. 2021. Extremal values of the Sombor index in unicyclic and bicyclic graphs. Journal of Mathematical Chemistry, 59, pp.1098-1116. Available at: https://doi.org/10.1007/s10910-021-01232-8.

Damnjanovic, I., Milosevic, M. & Stevanovic, D. 2023. A Note on Extremal Sombor Indices of Trees with a Given Degree Sequence. MATCH Communications in Mathematical and in Computer Chemistry, 90(1), pp.197-202. Available at: https://doi.org/10.46793/match.90-1.197D.

Das, K.C., Cevik, A.S., Cangul, I.N. & Shang, Y. 2021. On Sombor Index. Symmetry, 13(1), art.number:140. Available at: https://doi.org/10.3390/sym13010140.

Dorjsembe, Sh. & Horoldagva, B. 2022. Reduced Sombor index of bicyclic graphs. Asian-European Journal of Mathematics, 15(07), art.number:2250128. Available at: https://doi.org/10.1142/S1793557122501285.

Gutman, I. 2021. Geometric Approach to Degree-Based Topological Indices: Sombor Indices. MATCH Communications in Mathematical and in Computer Chemistry, 86(1), pp.11-16 [online]. Available at:

https://match.pmf.kg.ac.rs/electronic_versions/Match86/n1/match86n1_11-16.pdf [Accessed: 14 March 2024].

"J Gutman, I. 2022. Sombor indices - back to geometry. Open Journal of Discrete

Applied Mathematics, 5(2), pp.1-5. Available at: https://doi.org/10.30538/psrp-odam2022.0072.

Gutman, I. 2024. Relating Sombor and Euler indices. Vojnotehnicki glas-nik/Military Technical Courier, 72(1), pp.1-12. Available at: https://doi.org/10.5937/vojtehg72-48818.

Gutman, I. & Das, K.Ch. 2004. The first Zagreb index 30 years after. MATCH cd Communications in Mathematical and in Computer Chemistry, 50, pp.83-92 [on-^ line]. Available at:

g https://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf o [Accessed: 14 March 2024]. < Horoldagva, B., Selenge, T.-A., Buyantogtokh, L. & Dorjsembe, Sh. 2021. Up-

per bounds for the reduced second Zagreb index of graphs. Transactions on Com-g binatorics, 10(3), pp.137-148. Available at: w https://doi.org/10.22108/toc.2020.125478.1774. £ Horoldagva, B. & Xu, C. 2021. On Sombor index of graphs. MATCH Commu-

ai nications in Mathematical and in Computer Chemistry, 86(3), pp.703-713 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match86/n3/match8 6n3_703-713.pdf [Accessed: 14 March 2024].

Kosari, S., Dehgardi, N. & Khan, A. 2023. Lower bound on the KG-Sombor in-^ dex. Communications in Combinatorics and Optimization, 8(4), pp.751-757. Avail-§ able at: https://doi.org/10.22049/cco.2023.28666.1662.

g Kulli, V.R., Harish, N., Chaluvaraju, B. & Gutman, I. 2022. Mathematical prop-

erties of KG-Sombor index. Bulletin of International Mathematical Virtual Institute, 12(2), pp.379-386 [online]. Available at: http://www.imvibl.org/buletin/bulletin_im vi_12_2_22/bulletin_imvi_12_2_22_379_386.pdf [Accessed: 14 March 2024].

Liu, H., Gutman, I., You, H. & Huang, Y. 2022. Sombor index: review of extremal results and bounds. Journal of Mathematical Chemistry, 60, pp.771-798. Available at: https://doi.org/10.1007/s10910-022-01333-y.

Rada, J., Rodriguez, J.M. & Sigarreta, J.M. 2021. General properties on Sombor indices. Discrete Applied Mathematics, 299, pp.87-97. Available at: https://doi.org/10.1016/j.dam.2021.04.014.

Selenge, T.-A. & Horoldagva, B. 2024. Extremal Kragujevac trees with respect to Sombor indices. Communications in Combinatorics and Optimization, 9(1), pp.177-183. Available at: https://doi.org/10.22049/cco.2023.28058.1430.

Selenge, T.-A. & Horoldagva, B. 2015. Maximum Zagreb indices in the class of k-apex trees. Korean Journal of Mathematics, 23(3), pp.401-408. Available at: https://doi.org/10.11568/kjm.2015.23.3.401.

Tang, Z., Li, Q. & Deng, H. 2023. Trees with Extremal Values of the Sombor-Index-Like Graph Invariants. MATCH Communications in Mathematical and in Computer Chemistry, 90(1), pp.203-222. Available at: https://doi.org/10.46793/match.90-1.203T.

Zhang, H. & Zhang, S. 2006. Uncyclic graphs with the first three smallest and largest first general Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry, 55(2), pp.427-438 [online]. Available at: https://match.pm f.kg.ac.rs/electronic_versions/Match55/n2/match55n2_427-438.pdf [Accessed: 14 March 2024].

En el índice KG-Sombor

Ochirbat Altangooa, Dechinpuntsag Bolormaaa, Badarch Gantuyaa, Tsend-Ayush Selengeb

a Universidad Nacional de Educación de Mongolia, Facultad de Matemáticas y Ciencias Naturales, Ulaanbaatar, Mongolia

b Universidad Nacional de Mongolia, Departamento de Matemáticas, Ulaanbaatar, Mongolia, autor de correspondencia

CAMPO: matemáticas

TIPO DE ARTÍCULO: artículo científico original Resumen:

Introducción/objetivo: Los invariantes de gráficos basados en grados son un tipo de descriptores moleculares que representan la conectividad de los átomos (vértices) en una molécula a través de enlaces (bordes). Se utilizan para modelar propiedades estructurales de moléculas y proporcionar información valiosa para campos como la química física, la farmacología, las ciencias ambientales y las ciencias de materiales. Recientemente, se han explorado desde una perspectiva geométrica nuevos descriptores de estructuras moleculares basados en grados, conocidos como invariantes de gráficos tipo índice de Sombor. Estas invariantes gráficas han encontrado aplicaciones en la ciencia de redes, donde se utilizan para modelar efectos dinámicos en sistemas complejos biológicos, sociales y tecnológicos. También está surgiendo un interés en sus posibles aplicaciones militares. Entre estos descriptores se encuentra el índice KG-Sombor, que se define utilizando grados de vértice y borde.

Métodos: El estudio utiliza la teoría de grafos combinatoria para identificar y analizar gráficos extremos que maximizan o minimizan el índice KG-Sombor.

Resultados: Los gráficos extremos se caracterizan en relación con el índice KG-Sombor, con especial atención a árboles, árboles moleculares y gráficos unicíclicos.

Conclusión: Esta investigación avanza en la comprensión teórica de las invariantes gráficas similares al índice de Sombor.

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Palabras claves: índice KG-Sombor, árbol, gráfico unicíclico, árbol molecular.

Индекс КГ-Сомбор

Очирбат Алтангооа, Дичинпунтсаг Болормааа, Бадарч Гантуяа, Тсенд-Аюш Селенге6

а Монгольский национальный педагогический университет, факультет математики и естественных наук, г Улан-Батор, Монголия

6 Монгольский национальный университет, математический факультет, г Улан-Батор, Монголия, корреспондент

РУБРИКА ГРНТИ: 27.29.19 Краевые задачи и задачи на

собственные значения для обыкновенных

дифференциальных уравнений и систем уравнений ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: Недавно с геометрической точки зрения были исследованы новые дискрипторы молекулярной структуры на основе степеней, известные как инварианты графов, подобные Сомборскому индексу. Эти инварианты графа нашли применение в сетевой науке, где они используются для моделирования динамических эффектов в биологических, социальных и сложных технологических системах. Также растет интерес к их потенциальному применению в военных целях. Среди этих дескрипторов и ^-Сомборский индекс, который определяется с использованием степеней как вершины, так и ребра. Методы: В данном исследовании используется комбинаторная теория графов для выявления и анализа экстремальных графов, которые либо максимизируют, либо минимизируют КГ-Сомборский индекс. Результаты: Экстремальные графы характеризуются КГ-Сомборским индексом, при этом особое внимание уделяется деревьям, молекулярным деревьям и одноцикличе-ским графам.

Выводы: Данное исследование вносит вклад в расширение теоретического понимания инвариантов графов, подобных Сомборскому индексу.

Ключевые слова: КГ-Сомборский индекс, дерево, одноцик-лический граф, молекулярное дерево.

КГ-Сомборски индекс

Окирбат Аптангу3, Дичинпантсаг Боломаа, Бадарч Гантуи|аа, ЦендАуш Селенге6

а Монгопски национапни универзитет за образовав, Факуптет математике и природних наука, Улан Батор, Монголка

б Национални универзитет Монголке, Одсекза математику, Улан Батор, Монголка, ауторза преписку

ОБЛАСТ: математика

КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:

Увод/цил>: На степенима засноване графовске инварианте тип су молекуларних дескриптора ко\и представъа}у повезаност атома (чворова) у молекулу путем веза (грана). Користе се за моделоваше структурних сво}става молекула и пружа}у драгоцене информаци]е у областима по-пут физичке хеми}е, фармакологи]е, науке о животно] сре-дини, као и науке о матери}алима. Из геометри}ске перспективе недавно су проучавани нови дескриптори молеку-ларне структуре на бази степена, познати као графовске инварианте сродне Сомборском индексу. Ове графовске инварианте нашле су примену у науци о мрежама где се користе за моделоваше динамичких утица]а у биолошким, друштвеним и сложеним технолошким системима. Тако-посто}и и интересоваше за потенци]алне примене у во}-сци. Ме^у овим дескрипторима налази се КГ-Сомборски индекс ко\и се дефинише коришЯеъем степенова и чворова и грана.

Методе: У истраживашу се користи комбинаторна тео-ри}а графова за идентификаци}у и анализу екстремалних графова ко\и или максимизу]у или минимизу]у КГ-Сомборски индекс.

Резултати: Екстремални графови се карактеришу у од-носу на КГ-Сомборски индекс, са посебним освртом на ста-бла, молекуларна стабла и уницикличне графове.

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Закъучак: Овим истражива^ем унапре^у}е се теорирко разумева^е графовских инвари]анти сродних Сомборском индексу.

Къучне речи: КГ-Сомборски индекс, стабло, унициклични граф, стабло молекула.

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yy Paper received on: 22.03.2024.

Manuscript corrections submitted on: 14.11.2024. O Paper accepted for publishing on: 16.11.2024.

^ © 2024 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier O (http://vtg.mod.gov.rs, http://BTr.M0.ynp.cp6). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).