EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2106-2126 ISSN 1307-5543 – ejpam.com Published by New York Business Global Topological Characterization for Triangular, Regular Triangular Oxides and Silicates Networks Tarek Khalifa1, Nawaf Ali1, Muhammad Rafaqat2, Muhammad Haroon Aftab2,∗, Hassan Kanj1, Mouhammad Alakkoumi1, Kamel Jebreen3,4,5 1 College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait 2 Department of Mathematics and Statistics, The University of Lahore, Lahore 54500, Pakistan 3 Department of Mathematics, Palestine Technical University-Kadoorie, Hebron, Palestine 4 Department of Mathematics, An-Najah National University, Nablus, Palestine 5 Biostatistics and Clinical Research Department, University Hospital Lariboisière, AP-HP, Université Paris, France Abstract. Chemical graph theory can be studied with the aid of mathematical tools called m- polynomials. M-Polynomials offer a potent tool for computing different topological indices associ- ated with vertex degrees and analyzing degree-based structural information in graphs. By counting specific substructure types within them, they are able to encode information about the structure of molecules or networks. In this article, we have developed M-Polynomials with the help of differ- ent topological invariants such as first Zagreb (M1(β)), second Zagreb (M2(β)), second modified Zagreb (Mm 2 (β)), inverse sum (I(β)), harmonic index (H(β)) and Randic index (Rα0 (β)) for the molecular structures of Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangular silicate TSL(r) & Regular triangular silicate RTSL(r) networks to introduce new closed formulas to get better understanding the applications of M-Polynomials and topological indices in mathe- matical chemistry especially in the field of QSAR and QSPR study with the help of some software like MATLAB. We have also discussed the graphical behaviors of the above-mentioned structures. 2020 Mathematics Subject Classifications: 05C09, 05C10, 05C12, 05C31, 05C07 Key Words and Phrases: Topological Indices, degree, edge, M-Polynomials, TOX(r), RTOX(r), TSL(r), RTSL(r) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5324 Email addresses: tarek.khalifa@aum.edu.kw (T. Khalifa), nawaf.ali@aum.edu.kw (N. Ali), muhammad.rafaqat@math.uol.edu.pk (M. Rafaqat), haroonuet@gmail.com (M. H. Aftab), hassan.kanj@aum.edu.kw (H. Kanj), mouhammad.a@aum.edu.kw (M. Alakkoumi), k.jebreen@yahoo.com (K. Jebreen) https://www.ejpam.com 2106 © 2024 EJPAM All rights reserved. M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2107 1. Introduction A topological invariant in graph theory is a numerical or mathematical property of a graph that does not change while the graph is continuously deformed, provided that the deformation does not cause edges or vertices to break or glue together. As long as no new connections are made or old ones are severed, the graph can be deformed in many ways, such as bending, shrinking, or stretching. Consider a graph as a flexible model. The fundamental characteristics reflected by the topological invariant will not change no matter how you bend and twist it. This is because the connections between the vertices, or nodes, are what hold the structure together. A successful approach for computing closed-form expressions for an assortment of degree-based topological indices is to use M- polynomials. Rather than computing each index independently, the M-polynomial offers a single formula that may be used to generate various topological indices. Because of this, M-Polynomials are an effective tool for researching the connection between chemical characteristics and graph structure. Topological invariants are very effective to calculate the chemical, physical, biological properties of a chemical compound. It has so many uses in chemistry, information, biology, quantitative structure-property relationships, online networking software, industries, electronics and medicines. Definition 1. Let β be the graph of the molecular structure of the chemical compound then its M-polynomial can be computed as: M(β, x0, y0) = ∑ δ0≤i≤j≤∆0 mij(β)x i 0y j 0 (1) Where δ0 = min{dv0 : v0 ∈ V0(β)}, ∆0 = max{dv0 : v0 ∈ V0(β)}, and mij(β) the number of the edges u0v0 ∈ E0(β) such that du0 , dv0 = i, j. In 1947, Weiner [4–6] developed the formula for the boiling point of alkanes which is given by: αW (⅁) + βP3 + γ, for empirical constants α, β and γ, Weiner index W (⅁) and path’s length P3. Bollobas and Erdos [8, 9] presented the general Randic index and has been studied by both mathematicians and chemists [11, 24, 26]. For more detail, we can study the book [27]. The general Randic index is computed as Rα0(β) = ∑ u0v0∈E(β) (du0dv0) α. (2) The Randic index is a very essential index among all indices such as [27, 28, 30]. Gutman and Trinajstic [16] introduced first Zagreb and second Zagreb indices by M1(β) = ∑ u0v0∈E(β) (du0 + dv0) (3) and M2(β) = ∑ u0v0∈E(β) (du0 × dv0), (4) M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2108 respectively. For more detail [2, 14] are referred. The second modified Zagreb index is formulated by mM2(β) = ∑ u0v0∈E(β) 1 du0dv0 (5) The symmetric division index (SDD) is the one among 148 discrete Adriatic indices and is a good predictor of the total surface area for polychlorobiphenyls, see [1]. The symmetric division index of a connected graph β, is determined as SDD(β) = ∑ u0v0∈E(β) { min(du0 , dv0) max(du0 , dv0) + max(du0 , dv0) min(du0 , dv0) } (6) Harmonic index is H(β) = ∑ u0v0∈E(β) 2 du0 + dv0 (7) The inverse sum index [13, 20] is formulated as I(β) = ∑ u0v0∈E(β) du0dv0 du0 + dv0 (8) The augmented Zagreb index of β presented by Furtula et al. [19] and is computed as A(β) = ∑ u0v0∈E(β) { du0dv0 du0 + dv0 − 2 }3 (9) The above equation is also known as the minimal augmented Zagreb. 2. Material and Methods In this study we calculate M-polynomial for Triangular oxide TOX(r), Regular trian- gular oxide RTOX(r), Triangular silicate TSL(r) & Regular triangular silicate RTSL(r) networks. M-Polynomial was invented by Klavzar and Deutsch. They also give some op- erators to find degree based topological indices directly from the M-polynomial [15]. To get different topological indices [17, 21] with their M-Polynomials we use the following table. Table-1: Shows Topological indices with their corresponding M-Polynomials Topological indices M-Polynomials First Zagreb (M1(β)) (Da +Db)(M(β; a, b))|a=b=1 Second Zagreb (M2(β)) (DaDb)(M(β; a, b))|a=b=1 Second Modified Zagreb (Mm 2 (β)) (SaSb)(M(β; a, b))|a=b=1 Inverse sum I(β) SaJDaDb(M(β; a, b))|a=b=1 Harmonic index H (β) 2SaJ(M(β; a, b))|a=b=1 Randic index Rα0(β) (Dα0 a Dα0 b )(M(β); a, b))|a=b=1 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2109 Da = a∂f(a, b) da , Db = b∂f(a, b) db , Sa = ∫ a 0 f(t, b) t dt, Sb = ∫ b 0 f(a, t) t dt, J(f(a, b)) = f(a, a). 3. Motivation In this paper, motivated by the regularity notion, we introduce a uniformity notion of graphs conceived depending on the degrees of vertices. It is natural to try to relate the regularity to uniformity of a graph. Some properties and fundamental structural characteristics of these graphs are studied. It is possible that the properties of the graphs, that we are defining in this paper may have some applications in chemistry as well as in other areas. The following results will be useful in the proof of our main results. 4. Main Results In this section of the article, we derive the closed formulas using M-Polynomials for the molecular structures of Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangular silicate TSL(r) & Regular triangular silicate RTSL(r) networks. 4.1. Triangular oxide network TOX(r) Lemma 1. The cardinalities of the graph TOX(r) are r2 + 3r + 2 2 with respect to node set and 3(r2 + r) 2 with respect to edge set. Theorem 1. For TOX(r), the M-polynomial is M(TOX (r); a, b) = 6a2b4 + 3(r − 1)a4b4 + 6(r − 2)a4b4 + 3((r − 3)2 + (r − 3)) 2 a6b6 (10) Proof : Let TOX(r) be a graph. Then we have by above lemma |V (TOX (r))| = r2 + 3r + 2 2 |E(TOX (r))| = 3(r2 + r) 2 Now, the TOX(r) has four edge partitions such as: |E1(TOX (r))| = {e = lm ∈ E(TOX (r)) : dl = 2, dm = 4} |E2(TOX (r))| = {e = lm ∈ E(TOX (r)) : dl = dm = 4} M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2110 |E3(TOX (r))| = {e = lm ∈ E(TOX (r)) : dl = 4, dm = 6} |E4(TOX (r))| = {e = lm ∈ E(TOX (r)) : dl = dm = 6} Where dl, dm are degree of edges l, m respectively. We get |E1(TOX (r))| = 6, |E2(TOX (r))| = 3(r − 1), |E3(TOX (r))| = 6(r − 2), |E4(TOX (r))| = 3((r − 3)2) + (r − 3) 2 Now using the definition of M-polynomial M(TOX (r); a, b) = ∑ r≤s mrs(TOX (r))xrys = ∑ 2≤4 m24(TOX (r))a2b4 + ∑ 4≤4 m44(TOX (r))a4b4 + ∑ 4≤6 m46(TOX (r))a4b6 + ∑ 6≤6 m66(TOX (r))a6b6 = ∑ lm∈E1 m24(TOX (r))a2b4 + ∑ lm∈E2 m44(TOX (r))a4b4 + ∑ lm∈E3 m46(TOX (r))a4b6 + ∑ lm∈E4 m66(TOX (r))a6b6 = |E1|a2b4 + |E2|a4b4 + |E3|a4b6 + |E4|a6b6 = 6a2b4 + 3(r − 1)a4b4 + 6(r − 2)a4b6 + 3((r − 3)2 + (r − 3)) 2 a6b6 Theorem 2. For triangle oxide network TOX(r) some degree based topological indices are M1(TOX (r)) = (Da +Db)(f(a, b)|a=b=1 = 18r2 − 6r M2(TOX (r)) = (DaDb)(f(a, b)|a=b=1 = 54r2 − 78r + 36 Mm 2 (TOX (r)) = (SaSb)(f(a, b)|a=b=1 = r2 24 + 11 48 r + 1 16 H(TOX (r)) = (2SaJ)(f(a, b)|a=b=1 = r2 4 + 7 10 r + 7 20 I(TOX (r)) = (SaJDaDb)(f(a, b)|a=b=1 = 9 2 r2 + 21 10 r + 1 5 Rα(TOX (r)) = (Dα aD α b )(f(a, b)|a=b=1 = 6× 8α + 3× 16α(r − 1) + 6× 24α(r − 2) + 3 2 × 108α(r2 − 5r + 6) Proof : As the M-polynomial of TOX(r) is M(TOX (r); a, b) = 6a2b4 + 3(r − 1)a4b4 + 6(r − 2)a4b6 + 3((r − 3)2 + (r − 3)) 2 a6b6 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2111 = f(a, b) then Da(f(a, b)) = 12a2b4 + 12(r − 1)a4b4 + 24(r − 2)a4b6 + 9((r − 3)2 + (r − 3))a6b6 Db(f(a, b)) = 24a2b4 + 12(r − 1)a4b4 + 36(r − 2)a4b6 + 9((r − 3)2 + (r − 3))a6b6 (Dα aD α b ) = 6× 8αa2b4 + 3× 16αa4b4 + 6× 24α(r − 2)a4b6 + 3 2 × 108α((r − 3)2 + (r − 3))a6b6 Sa(f(a, b)) = 3a2b4 + 3 4 (r − 1)a4b4 + 3 2 (r − 2)a4b6 + ((r − 3)2 + (r − 3)) 4 a6b6 Sb(f(a, b)) = 3 2 a2b4 + 3 4 (r − 1)a4b4 + (r − 2)a4b6 + ((r − 3)2 + (r − 3)) 4 a6b6 SaSb(f(a, b)) = 1 2 a2b4 + 3 16 (r − 1)a4b4 + 1 4 (r − 2)a4b6 + ((r − 3)2 + (r − 3)) 24 a6b6 J(f(a, b)) = 6a6 + 3(r − 1)a8 + 6(r − 2)a10 + 3((r − 3)2 + (r − 3)) 2 a12 SaJ(f(a, b)) = a6 + 3 8 (r − 1)a8 + 3 5 (r − 2)a10 + 1 8 ((r − 3)2 + (r − 3))a12 SaJDaDb(f(a, b)) = 8a6 + 6(r − 1)a8 + 72 5 (r − 2)a10 + 18 4 ((r − 3)2 + (r − 3)) Now, by using the operators of table (1) First Zagreb index M1(TOX (r)) = (Dx +Db)(f(a, b)|a=b=1 = 18r2 − 6r (2) Second Zagreb index M2(TOX (r)) = (DaDb)(f(a, b)|a=b=1 = 54r2 − 78r + 36 (3) Second Modified Zagreb index Mm 2 (TOX (r)) = (SaSb)(f(a, b)|a=b=1 = r2 24 + 11 48 r + 1 16 (4) Harmonic index H(TOX (r)) = (2SaJ)(f(a, b)|a=b=1 = r2 4 + 7 10 r + 7 20 (5) Inverse sum I(TOX (r)) = (SaJDaDb)(f(a, b)|a=b=1 = 9 2 r2 + 21 10 r + 1 5 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2112 (6) Randic index Rα(TOX (r)) = (Dα aD α b )(f(a, b)|a=b=1 = 6× 8α + 3× 16α(r − 1) + 6× 24α(r − 2) + 3 2 × 108α(r2 − 5r + 6) 4.2. Regular triangular oxide network RTOX(r) Lemma 2. The regular triangular oxide network RTOX(r) has 3r2 +6r edges and 3 2 r2 + 9 2 r + 1 vertices. Theorem 3. The M-polynomial for a regular triangular oxide network RTOX(r) is M(RTOX (r); a, b) = 2a2b2 + 6ra2b4 + (3r2 − 2)a4b4 (11) Proof : From the above lemma |V (RTOX (r)| = 3 2 r2 + 9 2 r + 1 |E(RTOX (r)| = 3r2 + 6r RTOX(r) has three edge partitions as |E1(RTOX (r))| = {e = lm ∈ E(RTOX (r) : dl = dm = 2} |E2(RTOX (r))| = {e = lm ∈ E(RTOX (r) : dl = 2, dm = 4} |E3(RTOX (r))| = {e = lm ∈ E(RTOX (r) : dl = dm = 4} Where, |E1(RTOX (r))| = 2, |E2(RTOX (r))| = 6r, |E3(RTOX (r))| = 3r2 − 2 From the definition of M-polynomial M(RTOX (r); a, b) = ∑ r≤s mrs(RTOX (r))arbs = ∑ 2≤2 m22(RTOX (r))a2b2 + ∑ 2≤4 m24(RTOX (r))a2b4 + ∑ 4≤4 m44(RTOX (r))a4b4 = ∑ lm∈E1 m22(RTOX (r))a2b2 + ∑ lm∈E2 m24(RTOX (r))a2b4 + ∑ lm∈E3 m44(RTOX (r))a4b4 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2113 = |E1|a2b2 + |E2|a2b4 + |E3|a4b4 = 2a2b2 + 6ra2b4 + (3r2 − 2)a4b4 Theorem 4. For a regular triangular oxide network RTOX(r) some degree based topolog- ical indices are M1(RTOX (r)) = (Da +Db)(f(a, b)|a=b=1 = 24r2 + 36r − 8 M2(RTOX (r)) = (DaDb)(f(a, b)|a=b=1 = 48r2 + 48r − 24 Mm 2 (RTOX (r)) = (SaSb)(f(a, b)|a=b=1 = 3 16 r2 + 3 4 r + 3 8 H(RTOX (r)) = (2SaJ)(f(a, b)|a=b=1 = 3 4 r2 + 2r + 1 2 I(RTOX (r)) = SaJDaDb(M(G; a, b))|a=b=1 = 6r2 + 8r − 2 Rα(RTOX (r)) = (Dα aD α b )(f(a, b))|a=b=1 = 2× 4α + 6× 8αr + (3r2 − 2)× 16α Proof : From the previous theorem M(RTOX (r); a, b) = f(a, b) = 2a2b2 + 6ra2b4 + (3r2 − 2)a4b4 then, Da(f(a, b)) = 4a2b2 + 12ra2b4 + 4(3r2 − 2)a4b4 Db(f(a, b)) = 4a2b2 + 24ra2b4 + 4(3r2 − 2)a4b4 DaDb(f(a, b)) = 8a2b2 + 48ra2b4 + 16(3r2 − 2)a4b4 Sa(f(a, b)) = a2b2 + 3ra2b4 + (3r2 − 2) 4 a4b4 Sb(f(a, b)) = a2b2 + 3 2 ra2b4 + (3r2 − 2) 4 a4b4 SaSb(f(a, b)) = 1 2 a2b2 + 3 4 ra2b4 + (3r2 − 2) 16 a4b4 Dα aD α b (f(a, b)) = 2× 4α + 6× 8αr + (3r2 − 2)× 16α J(f(a, b)) = f(a, a) = 2a4 + 6ra6 + (3r2 − 2)a8 SaJ(f(a, b)) = 1 2 a4 + ra6 + (3r2 − 2) 8 a8 SaJDaDb(f(a, b)) = 2a4 + 8ra6 + 2(3r2 − 2)a8 Now using the operators given in table (1) First Zagreb index M1(RTOX (r)) = (Da +Db)(f(a, b))|a=b=1 = 24r2 + 36r − 8 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2114 (2) Second Zagreb index M2(RTOX (r)) = (DaDb)(f(a, b))|a=b=1 = 48r2 + 48r − 24 (3) Second Modified Zagreb index Mm 2 (RTOX (r)) = (SaSb)(f(a, b))|a=b=1 = 3 16 r2 + 3 4 r + 3 8 (4) Harmonic index H(RTOX (r)) = (2SaJ)(f(a, b))|a=b=1 = 3 4 r2 + 2r + 1 2 (5) Inverse sum I(RTOX (r)) = (SaJDaDb)(f(a, b))|a=b=16r 2 + 8r − 2 (6) Randic index Rα(RTOX (r)) = (Dα aD α b )(f(a, b))|a=b=1 = 2 × 4α + 6 × 8αr + (3r2 − 2) × 16α 4.3. Triangular silicate network TSL(r) Lemma 3. The triangular silicate network TSL(r) has 3(r2 + r) edges and r2 + 2r + 1 vertices. Theorem 5. The M-polynomial of triangular silicate network TSL(r) for r ≥ 4 is: M(TSL(r); a, b) = 3a3b3 + 6ra3b6 + 3(r − 1)a6b6 + 3 4 (r2 − 3r + 2)a3b9 +6(r − 2)a6b9 + 3 4 (r2 − 5r + 6)a9b9 Proof From the above lemma we have |V (TSL(r))| = r2 + 2r + 1 |E(TSL(r))| = 3(r2 + r) We know that TSL(r) has six edge partitions |E1(TSL(r))| = {e = lm ∈ E(TSL(r) : dl = dm = 3} |E2(TSL(r))| = {e = lm ∈ E(TSL(r) : dl = 3, dm = 6} M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2115 |E3(TSL(r))| = {e = lm ∈ E(TSL(r) : dl = dm = 6} |E4(TSL(r))| = {e = lm ∈ E(TSL(r) : dl = 3, dm = 9} |E5(TSL(r))| = {e = lm ∈ E(TSL(r) : dl = 6, dm = 9} |E6(TSL(r))| = {e = lm ∈ E(TSL(r) : dl = dm = 9} |E1(TSL(r))| = 3 , |E2(TSL(r))| =6r, |E3(TSL(r))| = 3(r − 1), |E4(TSL(r))| =3 2(r 2 − 3r + 2), |E5(TSL(r))| = 6(r − 2) , |E6(TSL(r))| =3 2(r 2 − 5r + 6) Now putting the values in the definition of M-polynomials as M(TSL(r); a, b) = ∑ r≤s mrs(TSL(r))a rbs = ∑ 3≤3 m33(TSL(r))a 3b3 + ∑ 3≤6 m36(TSL(r))a 3b6 + ∑ 6≤6 m66(TSL(r))a 6b6 + ∑ 3≤9 m39(TSL(r))a 3b9 + ∑ 6≤9 m69(TSL(r))a 6b9 + ∑ 9≤9 m99(TSL(r))a 9b9 = ∑ lm∈E1 m33(TSL(r))a 3b3 + ∑ lm∈E2 m36(TSL(r))a 3b6 + ∑ lm∈E3 m66(TSL(r))a 6b6 + ∑ lm∈E4 m39(TSL(r))a 3b9 + ∑ lm∈E5 m69(TSL(r))a 6b9 + ∑ lm∈E6 m99(TSL(r))a 9b9 = |E1|a3b3 + |E2|a3b6 + |E3|a6b6 + |E4|a3b9 + |E5|a6b9 + |E6|a9b9 = 3a3b3 + 6ra3b6 + 3(r − 1)a6b6 + 3 2 (r2 − 3r + 2)a3b9 + 6(r − 2)a6b9 + 3 2 (r2 − 5r + 6)a9b9 Theorem 6. Some degree based topological indices of triangular silicate network TSL(r) are M1(TSL(r)) = (Da +Db)(f(a, b)|a=b=1 = 45r2 − 9r M2(TSL(r)) = (DaDb)(f(a, b)|a=b=1 = 162r2 − 189r + 81 Mm 2 (TSL(r)) = (SaSb)(f(a, b)|a=b=1 = 2 27 r2 + 29 108 r + 5 36 H(TSL(r)) = (2SaJ)(f(a, b)|a=b=1 = 5 12 r2 + 21 20 r + 2 5 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2116 I(TSL(r)) = (SaJDaDb)(f(a, b)|a=b=1 = 81 8 r2 − 51 40 r − 9 20 Rα(TSL(r)) = (Dα aD α b )(f(a, b)|a=b=1 = 3× 9α + 6× 18αr + 3× 36α(r − 1) + 3 2 × 27α(r2 − 3r + 2) + 6× 54α(r − 2) + 3 2 × 81α(r2 − 5r + 6) Proof : From the above theorem we have M(TSL(r); a, b) = 3a3b3 + 6ra3b6 + 3(r − 1)a6b6 + 3 2 (r2 − 3r + 2)a3b9 + 6(r − 2)a6b9 + 3 2 (r2 − 5r + 6)a9b9 = f(a, b) Now, Da(f(a, b)) = 9a3b3 + 18ra3b6 + 18(r − 1)a6b6 + 9 2 (r2 − 3r + 2)a3b9 + 36(r − 2)a6b9 + 27 2 (r2 − 5r + 6)a9b9 (12) Db(f(a, b)) = 9a3b3 + 36ra3b6 + 18(r − 1)a6b6 + 27 2 (r2 − 3r + 2)a3b9 + 54(r − 2)a6b9 + 27 2 (r2 − 5r + 6)a9b9 (13) DaDb(f(a, b)) = 27a3b3 + 108ra3b6 + 108(r − 1)a6b6 + 81 2 (r2 − 3r + 2)a3b9 + 324(r − 2)a6b9 + 243 2 (r2 − 5r + 6)a9b9 (14) Dα aD α b = 3× 9αa3b3 + 6× 18αra3b6 + 3× 36α(r − 1)a6b6 + 3 2 × 27α(r2 − 3r + 2)a3b9 + 6× 54α(r − 2)a6b9 + 3 2 × 81α(r2 − 5r + 6)a9b9 (15) Sa(f(a, b)) = a3b3 + 2ra3b6 + (r − 1) 2 a6b6 + (r2 − 3r + 2) 2 a3b9 + (r − 2)a6b9 (r2 − 5r + 6) 6 a9b9 (16) M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2117 Sb(f(a, b)) = a3b3 + ra3b6 + (r − 1) 2 a6b6 + (r2 − 3r + 2) 6 a3b9 + 2 3 (r − 2)a6b9 + (r2 − 5r + 6) 6 a9b9 (17) SaSb(f(a, b)) = 1 3 a3b3 + r 3 a3b6 + (r − 1) 12 a6b6 + (r2 − 3r + 2) 18 a3b9 + (r − 2) 9 a6b9 + (r2 − 5r + 6) 54 a9b9 (18) J(f(a, b)) = 3a6 + 6ra9 + 3 2 (r2 − r)a12 + 6(r − 2)a15 + 3 2 (r2 − 5r + 6)a18 (19) SaJ(f(a, b)) = 1 2 a6 + 2 3 ra9 + (r2 − r) 8 a12 + 6(r − 2) 15 a15 + (r2 − 5r + 6) 12 a18 (20) SaJDaDb(f(a, b)) = 9 2 a6+12ra9+ 9 8 (r2−3r+2)a12+ 108 5 (r−2)a15+ 27 4 (r2−5r+6)a18 (21) Now, using the formula given in the table (1) First Zagreb index (1)M1(TSL(r)) = (Da +Db)(f(a, b)|a=b=1 = 45r2 − 9r (2) Second Zagreb index (2)M2(TSL(r)) = (DaDb)(f(a, b)|a=b=1 = 162r2 − 189r + 81 (3) Second Modified Zagreb index (3)Mm 2 (TSL(r)) = (SaSb)(f(a, b)|a=b=1 = 2 27 r2 + 29 108 r + 5 36 (4) Harmonic index (4)H(TSL(r)) = (2SaJ)(f(a, b)|a=b=1 = 5 12 r2 + 21 20 r + 2 5 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2118 (5) Inverse sum (5)I(TSL(r)) = (SaJDaDb)(f(a, b)|a=b=1 = 81 8 r2 − 51 40 r − 9 20 (6) Randic indea (6)Rα(TSL(r)) = (Dα aD α b )(f(a, b)|a=b=1 = 3× 9α + 6× 18αr + 3× 36α(r − 1) + 3 2 × 27α(r2 − 3r + 2) + 6× 54α(r − 2) + 3 2 × 81α(r2 − 5r + 6) 4.4. Regular triangular silicate network RTSL(r) Lemma 4. The regular triangular silicate network RTSL(r) has 6r2 + 12r edges and 5 2 r2 + 13 2 r + 1 vertices. Theorem 7. The M-polynomial of regular triangular silicate network RTSL(r) is: M(RTSL(r); a, b) = (3r + 4)a3b3 + (3r2 + 9r − 2)a3b6 + (3r2 − 2)a6b6 (22) Proof : As lemma we have |V (RTSL(r))| = 5 2 r2 + 13 2 r + 1 |E(RTSL(r))| = 6r2 + 12r We know RTSL(r) has three edge partitions |E1(RTSL(r))| = {e = lm ∈ E(RTSL(r) : dl = dm = 3} |E2(RTSL(r)| = {e = lm ∈ E(RTSL(r) : dl = 3, dm = 6} |E3(RTSL(r))| = {e = lm ∈ E(RTSL(r) : dl = dm = 6} Such that: |E1(RTSL(r))|= 3r+4, |E2(RTSL(r))| = 3r2 + 9r − 2, |E3(RTSL(r))|=3r2 − 2 Now using the definition of M-polynomial M(RTSL(r); a, b) = ∑ r≤s mrs(RTSL(r))a rbs = ∑ 3≤3 m33(RTSL(r))a 3b3 + ∑ 3≤6 m36(RTSL(r))a 3b6 + ∑ 6≤6 m66(RTSL(r))a 6b6 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2119 = ∑ lm∈E1 m33(RTSL(r))a 3b3 + ∑ lm∈E2 m36(RTSL(r))a 3b6 + ∑ lm∈E3 m66(RTSL(r))a 6b6 = |E1|a3b3 + |E2|a3b6 + |E3|a6b6 = (3r + 4)a3b3 + (3r2 + 9r − 2)a3b6 + (3r2 − 2)a6b6 Theorem 8. Some well-known degree based topological indices of RTSL(r) are: M1(RTSL(r)) = (Da +Db)(f(a, b))|a=b=1 = 36r2 + 99r − 18 M2(RTSL(r)) = (DaDb)(f(a, b))|a=b=1 = 162r2 + 189r − 72 Mm 2 (RTSL(r)) = (SaSb)(f(a, b))|a=b=1 = r2 4 + 5 6 r − 5 18 H(RTSL(r)) = (2SaJ)(f(a, b))|a=b=1 = 7 6 r2 + 3r + 5 18 I(RTSL(r)) = (SaJDaDb)(f(a, b))|a=b=1 = 15r2 + 21r − 6 Rα(RTSL(r)) = (Dα aD α b )(f(a, b))|a=b=1 = (3r + 4)× 9α + (3r2 + 9r − 2)× 18αr + (3r2 − 2)× 36α Proof : From the above theorem we have M(RTSL(r); a, b) = (3r + 4)3a3b3 + (3r2 + 9r − 2)a3b6 + (3r2 − 2)a6b6 Now, Da(f(a, b)) = 3(3r + 4)3a3b3 + 3(3r2 + 9r − 2)a3b6 + 6(3r2 − 2)a6b6 Db(f(a, b)) = 3(3r + 4)3a3b3 + 6(3r2 + 9r − 2)a3b6 + 6(3r2 − 2)a6b6 DaDb(f(a, b)) = 9(3r + 4)3a3b3 + 18(3r2 + 9r − 2)a3b6 + 36(3r2 − 2)a6b6 Dα aD α b = (3r + 4)× 9αa3b3 + (3r2 + 9r − 2)× 18αa3b6 + (3r2 − 2)× 36αa6b6 Sa(f(a, b)) = 3r + 4 3 a3b3 + 3r2 + 9r − 2 3 a3b6 + 3r2 − 2 6 a6b6 Sb(f(a, b)) = 3r + 4 3 a3b3 + 3r2 + 9r − 2 6 a3b6 + 3r2 − 2 6 a6b6 SaSb(f(a, b)) = (3r + 4)a3b3 + 3r2 + 9r − 2 18 a3b6 + 3r2 − 2 36 a6b6 J(f(a, b)) = (3r + 4)a6 + (3r2 + 9r − 2)a9 + (3r2 − 2)a12 SaJ(f(a, b)) = (3r + 4) 6 a6 + (3r2 + 9r − 2) 9 a9 + (3r2 − 2) 12 a12 SaJDaDb(f(a, b)) = (3r + 4)a6 + 2(3r2 + 9r − 2)a9 + 3(3r2 − 2)a12 M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2120 Now by the operators of table (1) First Zagreb index M1(RTSL(r)) = (Da +Db)(f(a, b))|a=b=1 = 36r2 + 99r − 18 (2) Second Zagreb index M2(RTSL(r)) = (DaDb)(f(a, b))|a=b=1 = 162r2 + 189r − 72 (3) Second Modified Zagreb index Mm 2 (RTSL(r)) = (SaSb)(f(a, b))|a=b=1 = r2 4 + 5 6 r − 5 18 (4) Harmonic index H(RTSL(r)) = (2SaJ)(f(a, b))|a=b=1 = 7 6 r2 + 3r + 5 18 (5) Inverse sum I(RTSL(r)) = (SaJDaDb)(f(a, b))|a=b=1 = 15r2 + 21r − 6 (6) Randic index Rα(RTSL(r)) = (Dα aD α b )(f(a, b))|a=b=1 = (3r+4)×9α+(3r2+9r−2)×18αr+(3r2−2)×36α. 5. Graphical Representation M-polynomials’ graphical behavior for different networks may provide insight into the structural features they have and how they transform when the size or connectedness of the network varies. Let’s examine the M-polynomials’ graphical behaviors for the networks given below in Figures (1-4): • First Zagreb index (FZI). • Second Zagreb index (SZI). • Second modified Zagreb indices (SMZI). M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2121 5.1. First Zagreb index for TOX(r), RTOX(r), TSL(r) & RTSL(r) −4 −2 0 2 4 −5 0 5 0 1,000 FZI-TOX(r) FZI-RTOX(r) FZI-TSL(r) FZI-RTSL(r) Figure 1: First Zagreb index for TOX(r), RTOX(r), TSL(r) & RTSL(r) 5.2. Second Zagreb index for TOX(r), RTOX(r), TSL(r) & RTSL(r) −4 −2 0 2 4 −5 0 5 0 2,000 4,000 SZI-TOX(r) SZI-RTOX(r) SZI-TSL(r) SZI-RTSL(r) Figure 2: Second Zagreb index for TOX(r), RTOX(r), TSL(r) & RTSL(r) M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2122 5.3. Second Modified Zagreb index for TOX(r), RTOX(r), TSL(r) −4 −2 0 2 4 −5 0 5 0 5 SMZI-TOX(r) SZMI-RTOX(r) SMZI-TSL(r) Figure 3: Second Modified Zagreb index for TOX(r), RTOX(r), TSL(r) 5.4. Second Modified Zagreb index for RTSL(r) −4 −2 0 2 4 −5 0 50 5 10 SMZI-TOX(r) Figure 4: Second Modified Zagreb index for RTSL(r) 6. Applications M-Polynomials are mostly used in graph theory because of their effectiveness in analysing degree-based topological indices. M-Polynomials provide as a consistent source for the col- lection of the various topological indices connected with a graph’s vertex degrees. These numerical descriptors, known as indices, are particularly helpful in chemical graph theory, where the graph is used to represent a molecule. Chemical Graph theory plays important role in the everyday life applications such as image processing unit, bio sensors, math- ematical chemistry, computer science, artificial intelligence, social science and medicine. M.H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 2106-2126 2123 With the help of quantitative structure property relationship study (QSPRs) and quantita- tive structure activity relationship study (QSARs), the topology [23, 29] of the molecular structures Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangular sil- icate TSL(r) and Regular triangular silicate RTSL(r) obtained from the given chemical compound can be correlated and further discussed for the latest research works being used by many pharmacists, chemists and researchers to get better understanding in their fields. 7. Conclusions and Novelty In conclusion, because of their straightforward counting schemes, the M-polynomials for the networks of triangle oxide and triangle silicate exhibit linear behaviors on the graph. Regular versions of these networks may show more ordered patterns in their M- polynomial graphs, but more specific information about their structures and counting techniques would be needed to create accurate graphical representations. In analysing the relationship between a graph’s characteristics and structure (as represented by vertex degrees), M-polynomials are crucial. Researchers can examine the correlation between different topological indices and distinct chemical or physical properties of the molecule represented by the graph, as M-Polynomials provide a practical method for obtaining these indices. For example, research could look into the relationship between a molecule’s boiling point and the Zagreb index, which is derived from an M-Polynomial. This may offer insightful information about the relationship between a molecule’s structure and behavior. Consideration of the molecular structures of Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangular silicate TSL(r) & Regular triangular silicate RTSL(r) networks. • Association of the molecular structures with their corresponding mathematical graphs. • Application of M-Polynomials on the above-mentioned molecular graphs to get new and closed generalized formulas. • The new developed formulas can be applied in mathematical chemistry and can also be used by chemists, pharmacists and researchers for more scientific experiments or lab works. For the molecular structure of the networks of triangular oxide (TOX(r), regular triangular oxide (RTOX(r), triangular silicate (TSL(r), and regular triangular silicate (RTSL(r)), we have calculated the M-Polynomials [17]. 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