Sturm's Theorem for Min matrices

Efruz Özlem Mersin; Efruz Özlem Mersin

doi:10.3934/math.2023880

AIMS Mathematics

2023, Volume 8, Issue 7: 17229-17245. doi: 10.3934/math.2023880

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Research article

Sturm's Theorem for Min matrices

Efruz Özlem Mersin ^,

Department of Mathematics, Faculty of Science and Arts, Aksaray University, Aksaray 68100, Turkey

Received: 17 February 2023 Revised: 29 April 2023 Accepted: 11 May 2023 Published: 18 May 2023
MSC : 15A18, 15B99

In the present paper, we study Min matrix $\mathcal{A}_{min} = \left[a_{min\left(i, j\right)}\right]_{i, j = 1}^n$ , where $a_s$ 's are the elements of a real sequence $\left\lbrace a_s\right\rbrace$ . We first obtain a recurrence relation for the characteristic polynomial for matrix $\mathcal{A}_{min}$ , and some relations between the coefficients of its characteristic polynomial. Next, we show that the sequence of the characteristic polynomials of the $i \times i \left(i \leq n\right)$ Min matrices satisfies the Sturm sequence properties according to different required conditions of the sequence $\left\lbrace a_s\right\rbrace$ . Using Sturm's Theorem, we get some results about the eigenvalues, such as the number of eigenvalues in an interval. Thus, we obtain the number of positive and negative eigenvalues of Min matrix $\mathcal{A}_{min}$ . Finally, we give an example to illustrate our results.

Keywords:

Citation: Efruz Özlem Mersin. Sturm's Theorem for Min matrices[J]. AIMS Mathematics, 2023, 8(7): 17229-17245. doi: 10.3934/math.2023880

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Abstract

1. Introduction

In today's world mathematics is necessary in all fields. It is an essential instrument for comprehension around us. It covers all the facts of life. Mathematics is the branch of science that deals with the reasoning of figures, numbers and order. In our daily life, we use mathematics in our routine work in various forms. There are many branches of mathematics like algebra, geometry, arithmetic, trigonometry, analysis and many other theories. Graph theory is the study of mathematical objects known as graph, which consist of vertices connected by edges. It is the mathematical theory which deals with the properties and applications of graphs. When we apply graph theory on chemistry then it is called chemical graph theory. In mathematical models graph theory is used to get a deep understanding of the physical properties of these chemical compounds. Some physical properties such as boiling point, melting point, density are associated to geometrical structure of the compound. Now a days, several ways are used in mathematical chemistry to understand chemical structure which are existing behind the chemical concepts, to create and inquire novel mathematical representation. In complete history of chemistry certain scientist, usually contemplates connections between mathematics with chemistry and the possibility of using mathematics to analyze and predict new chemical concepts. Mufti defined sanskruti and harmonic indices of certain graph structure ^[25]. Babujee calculated topological indices and new graph structures in 2012 ^[5]. Farahani worked on a new version of zagreb index of circumcoronene series of benzenoid in 2013 ^[7]. Hayat defined some degree based topological indices of certain nanotubes ^[8]. Imran worked on topological indices of certain interconnection networks in 2014 ^[14]. In 2016, Siddiqui computed zagreb indices and zagreb polynomials of some nanostars dendrimers ^[27]. Saleem computed retractions and homomorphisms on some operations of graphs ^[28]. Iqbal calculated eccentricity based topological indices of some benzenoid structures ^[12]. Yang examined two-point resistances and random walks on stellated regular graphs ^[29]. Islam defined M-polynomial and entropy of paraline graph of Napthalene in 2019 ^[11]. Iqbal worked on topological indices of subdivided and line graph of subdivided friendship graph ^[13]. In 2020, Afzal examined M-polynomial and topological indices of zigzag edge coronoid fused by starphene ^[1]. Archdeacon defined the medial graph and voltage-current duality ^[2]. Munir computed M-polynomial and degree-based topological indices of polyhex nanotubes ^[22]. Iqbal calculated ve topological indices of tickysim spinnaker model ^[15]. Azhar examined a note on valency dependence invariants of L(G(K)) Graph ^[4]. Jamil defined the first general zagreb eccentricity index ^[18]. Iqbal defined ABC4 and GA5 index of subdivided and line graph of subdivided dutch windmill graph ^[16]. Maji worked on the first entire zagreb index of various corona products and their bounds ^[23]. Imran describe computation of topological indices of NEPS of graphs ^[17]. Hayat computed topological indices for networks derived by applying graph operations from honeycomb structures ^[9], based on this idea we have computed the topological indices for transformed structures. Next we have few definitions ^[3,9]:

Definition 1:

Let K be a simple connected graph its M-polynomial is defined as;

$M(K;x, y) = \sum\limits_{S \leq a \leq b \leq T}m_{ab}(K)x^{a}y^{b}, \, \, \, \, \, \, \, \ \text{(i)}$

where: S = $Min\{d_\beta|\beta \in V(K)\}$ , T = $Max\{d_\beta|\beta \in V(K)\}$ , and $m_{ab}(K)$ is the number of edges $\gamma\beta\in E(K)$ such that $\{ d_\gamma, \ d_\beta\}$ = $\{{a}, {b}\}$ .

Definition 2:

When we place a new vertex in each face of a planar graph G, and attach that vertex with all the vertices of the respective face of G, we get the stellation of G and denote it as St(G).

Definition 3:

We introduce a node in each edge of the graph and join the nodes if their corresponding edges are adjacent and is denoted by Md(G).

Definition 4:

We introduce a node in each bounded faces of the graph and join the nodes by an edge if the faces share an edge in graph it is called bounded dual. It is represented as Bdu(G).

T. Réti introduced First and Second Zagreb indices are ^[26],

$\begin{eqnarray} M_1(K)& = &\sum\limits_{\gamma\beta \in E(K)}{(d_\gamma+d_\beta)}, \end{eqnarray}$

(1.1)

$\begin{eqnarray} M_2(K)& = &\sum\limits_{\gamma\beta \in E(K)}{(d_\gamma d_\beta)}. \end{eqnarray}$

(1.2)

Second modified Zagreb index is defined as ^[10],

$\begin{eqnarray} ^mM_2(k)& = &\sum\limits_{\gamma\beta \in E(K)}\bigg(\frac 1{d_\gamma d_\beta}\bigg). \end{eqnarray}$

(1.3)

Symmetric division and reciprocal general Randic index is ^[19],

$\begin{eqnarray} SDD(K)& = &\sum\limits_{\gamma\beta \in E(K)}\ \bigg\{\frac {min(d_\gamma, d_\beta)}{max(d_\gamma, d_\beta)}+\frac {max(d_\gamma, d_\beta)}{min(d_\gamma, d_\beta)}\bigg\}, \end{eqnarray}$

(1.4)

$\begin{eqnarray} RR_\alpha(K)& = &\sum\limits_{\gamma\beta \in E(K)}\frac 1{(d_\gamma d_\beta)^\alpha}. \end{eqnarray}$

(1.5)

Whereas, the General Randic index defined as ^[24],

$\begin{eqnarray} R_\alpha(K)& = &\sum\limits_{\gamma\beta \in E(K)}(d_\gamma d_\beta)^{\alpha}, \end{eqnarray}$

(1.6)

where $\alpha$ is an arbitrary real number.

In 1998, the Atomic Bond Connectivity Index is defined as ^[6],

$\begin{eqnarray} ABC(K)& = &\sum\limits_{\gamma\beta \in E(K)}\sqrt{\frac{d_{\gamma}+d_{\beta}-2}{d_\gamma d_\beta}}. \end{eqnarray}$

(1.7)

Geometric-Arithmetic index is ^[30],

$\begin{eqnarray} GA(K)& = &\sum\limits_{\gamma\beta\in E(K)}\frac{2\sqrt{d_{\gamma}d_{\beta}}}{d_{\gamma}+d_{\beta}}. \end{eqnarray}$

(1.8)

In 2015, the General Version of Harmonic Index is defined ^[31],

$\begin{eqnarray} H_k(K)& = &\sum\limits_{\gamma\beta \in E(K)}\bigg(\frac 2{d_{\gamma}+d_{\beta}}\bigg)^k. \end{eqnarray}$

(1.9)

First and Second Gourava Indices were introduced by Kalli which is defined as, respectively ^[20],

$\begin{eqnarray} GO_1(K)& = &\sum\limits_{\gamma\beta \in E(K)}\big[(d_{\gamma}+d_{\beta})+(d_\gamma d_\beta)\big], \end{eqnarray}$

(1.10)

$\begin{eqnarray} GO_2(K)& = &\sum\limits_{\gamma\beta \in E(K)}\big[(d_{\gamma}+d_{\beta})(d_\gamma d_\beta)\big]. \end{eqnarray}$

(1.11)

Kalli introduced the First and Second Hyper-Gourava Indices which is defined as, respectively ^[21],

$\begin{eqnarray} HGO_1(K)& = &\sum\limits_{\gamma\beta\in E(K)}\big[(d_{\gamma}+d_{\beta})+(d_\gamma d_\beta)\big]^2, \end{eqnarray}$

(1.12)

$\begin{eqnarray} HGO_2(K)& = &\sum\limits_{\gamma\beta\in E(K)}\big[(d_{\gamma}+d_{\beta})(d_\gamma d_\beta)\big]^2. \end{eqnarray}$

(1.13)

2. Methodology

At first, we obtain transformed pattern of molecular structures zigzag and triangular benzenoid named as $T_1$ and $T_2$ . Next we define M-polynomial and topological indices of these networks by applying the stellation, medial and bounded dual. Further we define edge partition depending upon degree based vertices of $T_1$ and $T_2$ and then calculate the indices.

3. Results and discussion

3.1. M-polynomial and degree based topological indices of $T_1$ structure

Now we will construct the stellation, medial and bounded dual operations on simple and undirected zigzag benzenoid structure. We get a new transformed network (say) $T_1$ , shown in for n = 2. After this we will compute M-polynomial and degree based topological indices of our newly obtained network. Where, blue is stellation, red is medial, green is bounded dual operations applied on $T_1$ .

Figure 1. Transformed structure

$T_1$ .

DownLoad: Full-Size Img PowerPoint

Following shows the types of edges and their count for network $T_1$ ; $n\geq2$ . Now, we will calculate the M-polynomial and vertex degree based topological indices namely first Zagreb index, second Zagreb index, second modified index, symmetric division index, general Randic index, reciprocal general Randic index, atomic bond connectivity index, geometric arithmetic index, general version of harmonic index, first and second gourava indices, first and second hyper gourava indices.

3.1.1. Theorem

Let $T_1$ , be the transformed network then its M-polynomial is

$M\big(T_1;x, y\big) = \big(8n+8\big)x^3y^4+8x^3y^7+\big(4n-4\big)x^3y^8+\big(4n+4\big)x^4y^4+\big(8n-4\big)x^4y^5+\big(8n-4\big)x^4y^6 \\ +\big(4n-2\big)x^5y^6+4x^5y^7+\big(8n-8\big)x^5y^8+2x^7y^8+\big(2n-3\big)x^8y^8.\\$

Proof:

Using definition of M-polynomial and information from Table 3, we have

$M(T_1;x, y) = \sum\limits_{a \leq b}m_{ab}x^{a}y^{b} = \sum\limits_{3\leq 4}m_{34}x^3y^4+\sum\limits_{3\leq 7}m_{37}x^3y^7+\sum\limits_{3\leq 8}m_{38}x^3y^8+\sum\limits_{4\leq4}m_{44}x^4y^4+\sum\limits_{4\leq5}m_{45}x^4y^5 \\ +\sum\limits_{4\leq6}m_{46}x^4y^6+\sum\limits_{5\leq6}m_{56}x^5y^6+\sum\limits_{5\leq7}m_{57}x^5y^7+\sum\limits_{5\leq8}m_{58}x^5y^8 +\sum\limits_{7\leq8}m_{78}x^7y^8+\sum\limits_{8\leq8}m_{88}x^8y^8\\ = |E_{3, 4}|x^3y^4+|E_{3, 7}|x^3y^7+|E_{3, 8}|x^3y^8+|E_{4, 4}|x^4y^4+|E_{4, 5}|x^4y^5+|E_{4, 6}|x^4y^6+|E_{5, 6}|x^5y^6 \\ +|E_{5, 7}|x^5y^7+|E_{5, 8}|x^5y^8+|E_{7, 8}|x^7y^8+|E_{8, 8}|x^8y^8\\ = \big(8n+8\big)x^3y^4+8x^3y^7+\big(4n-4\big)x^3y^8+\big(4n+4\big)x^4y^4+\big(8n-4\big)x^4y^5+\big(8n-4\big)x^4y^6 \\ +\big(4n-2\big)x^5y^6+4x^5y^7+\big(8n-8\big)x^5y^8+2x^7y^8+\big(2n-3\big)x^8y^8.$

Figure 2. M-polynomial of

$T_1$ .

DownLoad: Full-Size Img PowerPoint

3.1.2. Theorem

Let $T_1$ , be the transformed network and

be its M-polynomial. Then, the first Zagreb index $M_1\big(T_1\big)$ , the second Zagreb index $M_2\big(T_1\big)$ , the second modified Zagreb index $^mM_2\big(T_1\big)$ , the general Randic index $R_\alpha(T_1)$ , where $\alpha \ \epsilon \ N$ , reciprocal general Randic index $RR_\alpha(T_1)$ , where $\alpha \ \epsilon \ N$ , and the symmetric division degree index $SDD(T_1)$ obtained from M-polynomial are as follows:

$M_1(T_1) = 464n-48, \\ M_2(T_1) = 1176n-264, \\ ^mM_2(T_1) = \frac{11}{12}\big(n+1\big)+\frac{11}{30}\big(n-1\big)+\frac{13}{30}\big(2n-1\big)+\frac{1}{64}\big(2n-3\big) +\frac{223}{420}, \\ R_\alpha(T_1) = 12^\alpha(8n+8)+21^\alpha(8)+24^\alpha(4n-4)+16^\alpha(4n+4)+20^\alpha(8n-4)+24^\alpha(8n-4) \\ +30^\alpha(4n-2)+35^\alpha(4)+40^\alpha(8n-8)+56^\alpha(2)+64^\alpha(2n-3), \\ RR_\alpha(T_1) = \frac{8}{12^\alpha}\big(n+1\big)+\frac{8}{21^\alpha}+\frac{4}{24^\alpha}\big(n-1\big)+\frac{4}{4^{2\alpha}}\big(n+1\big) +\frac{4}{20^\alpha}\big(2n-1\big)+\frac{4}{24^\alpha}\big(2n-1\big) \\ +\frac{2}{30^\alpha}\big(2n-1\big)+\frac{4}{35^\alpha}+\frac{8}{40^\alpha}\big(n-1\big) +\frac{2}{56^\alpha}+\frac{1}{64^\alpha}\big(2n-3\big), \\ SDD(T_1) = \frac{74}{3}\big(n+1\big)+\frac{899}{30}\big(n-1\big)+\frac{314}{15}\big(2n-1\big)+2\big(2n-3\big)+\frac{14527}{420}.$

Proof: Let $f(x, y)$ = $M\big(T_1;x, y\big)$ be the M-polynomial of the transformed network $T_1$ . Then

Now, the required partial derivatives and integrals are obtained as:

By using the information given in Table 3 and formulas for Table 1;

$D_xf\big(x, y\big) = 3\big(8n+8\big)x^3y^4+24x^3y^7+3\big(4n-4\big)x^3y^8+4\big(4n+4\big)x^4y^4+4\big(8n-4\big)x^4y^5 \\ +4\big(8n-4\big)x^4y^6+5\big(4n-2\big)x^5y^6+20x^5y^7+5\big(8n-8\big)x^5y^8+14x^7y^8+8\big(2n-3\big)x^8y^8, \\ D_yf\big(x, y\big) = 4\big(8n+8\big)x^3y^4+56x^3y^7+8\big(4n-4\big)x^3y^8+4\big(4n+4\big)x^4y^4+5\big(8n-4\big)x^4y^5 \\ +6\big(8n-4\big)x^4y^6+6\big(4n-2\big)x^5y^6+28x^5y^7+ 8\big(8n-8\big)x^5y^8+16x^7y^8+8\big(2n-3\big)x^8y^8, \\ D_xD_yf\big(x, y\big) = 12\big(8n+8\big)x^3y^4+168x^3y^7+24\big(4n-4\big)x^3y^8+16\big(4n+4\big)x^4y^4+20\big(8n-4\big)x^4y^5 \\ +24\big(8n-4\big)x^4y^6+30\big(4n-2\big)x^5y^6+140x^5y^7+40\big(8n-8\big)x^5y^8+112x^7y^8 \\ +64\big(2n-3\big)x^8y^8, \\ S_xS_yf\big(x, y\big) = \frac{2}{3}\big(n+1\big)x^3y^4+\frac{8}{21}x^3y^7+\frac{1}{6}\big(n-1\big)x^3y^8+\frac{1}{4}\big(n+1\big)x^4y^4 +\frac{1}{5}\big(2n-1\big)x^4y^5 \\ +\frac{1}{6}\big(2n-1\big)x^4y^6+\frac{1}{15}\big(2n-1\big)x^5y^6+\frac{4}{35}x^5y^7+\frac{1}{5}\big(n-1\big)x^5y^8 \\ +\frac{1}{28}x^7y^8+\frac{1}{64}\big(2n-3\big)x^8y^8, \\ D_x^{\alpha}D_y^{\alpha}f(x, y) = 12^{\alpha}(8n+8)x^3y^4+21^{\alpha}(8)x^3y^7+24^{\alpha}(4n-4)x^3y^8+4^{2\alpha}(4n+4)x^4y^4+20^{\alpha}(8n-4)x^4y^5 \\ +24^{\alpha}(8n-4)x^4y^6+30^{\alpha}(4n-2)x^5y^6+35^{\alpha}(4)x^5y^7+40^{\alpha}(8n-8)x^5y^8+56^{\alpha}(2)x^7y^8 \\ +46^{\alpha}(2n-3)x^8y^8, \\ S_x^{\alpha}S_y^{\alpha}f\big(x, y\big) = \frac{8}{12^\alpha}\big(n+1\big)x^3y^4+\frac{8}{21^\alpha}x^3y^7 +\frac{4}{24^\alpha}\big(n-1\big)x^3y^8+\frac{4}{4^{2\alpha}}\big(n+1\big)x^4y^4 \\ +\frac{4}{20^\alpha}\big(2n-1\big)x^4y^5+\frac{4}{24^\alpha}\big(2n-1\big)x^4y^6+\frac{2}{30^\alpha}\big(2n-1\big)x^5y^6+\frac{4}{35^\alpha}x^5y^7 \\ +\frac{8}{40^{\alpha}}\big(n-1\big)x^5y^8+\frac{2}{56^\alpha}x^7y^8+\frac{1}{64^\alpha}\big(2n-3\big)x^8y^8, \\ S_yD_xf\big(x, y\big) = 6\big(n+1\big)x^3y^4+\frac{24}{7}x^3y^7+\frac{3}{2}\big(n-1\big)x^3y^8+4\big(n+1\big)x^4y^4+\frac{16}{5}\big(2n-1\big)x^4y^5 \\ +\frac{8}{3}\big(2n-1\big)x^4y^6+\frac{5}{3}\big(2n-1\big)x^5y^6+\frac{20}{7}x^5y^7+5\big(n-1\big)x^5y^8 \\ +\frac{7}{4}x^7y^8+(2n-3)x^8y^8, \\ S_xD_yf\big(x, y\big) = \frac{32}{3}\big(n+1\big)x^3y^4+\frac{56}{3}x^3y^7+\frac{32}{3}\big(n-1\big)x^3y^8+4\big(n+1\big)x^4y^4+5\big(2n-1\big)x^4y^5 \\ +6\big(2n-1\big)x^4y^6+\frac{12}{5}\big(2n-1\big)x^5y^6+\frac{28}{5}x^5y^7+\frac{64}{5}\big(n-1\big)x^5y^8 \\ +\frac{16}{7}x^7y^8+(2n-3)x^8y^8.$

Table 1. Derivation of some degree-based topological indices from M-polynomial ^[3].

$Topological$ $Index$	Derivation from M (K; x, y)
$First$ $Zagreb$	$(D_x+D_y)(M(K; x, y))\|_{x=1=y}$	(ii)
$Second$ $Zagreb$	$(D_xD_y)(M(K; x, y))\|_{x=1=y}$	(iii)
$Second$ $Modified$ $Zagreb$	$(S_xS_y)(M(K; x, y))\|_{x=1=y}$	(iv)
$general$ $Randic$	$(D_x^{\alpha}D_y^{\alpha})(M(K; x, y))\|_{x=1=y}$	(v)
$Reciprocal$ $general$ $Randic$	$(S_x^{\alpha}S_y^{\alpha})(M(K; x, y))\|_{x=1=y}$	(vi)
$Symmetric$ $Division$ $Index$	$(S_yD_x)+(S_xD_y)(M(K; x, y))\|_{x=1=y}$	(vii)

| Show Table

DownLoad: CSV

Table 2. Planar and non-planar graph.

$Graph$	$Planar$ / $Non-Planar$
$St(G_1, G_2)$	$Planar$
$Md(G_1, G_2)$	$Planar$
$Bdu(G_1, G_2)$	$Non-Planar$
$T_1$	$Non-Planar$
$T_2$	$Non-Planar$

| Show Table

DownLoad: CSV

Consequently,

$M_1(T_1) = (D_x+D_y)f(x, y)|_{x = 1 = y} = 464n-48, \\ M_2(T_1) = D_xD_yf(x, y)|_{x = 1 = y} = 1176n-264, \\ ^mM_2(T_1) = S_xS_yf(x, y)|_{x = 1 = y} = \frac{11}{12}\big(n+1\big)+\frac{11}{30}\big(n-1\big)+\frac{13}{30}\big(2n-1\big) +\frac{1}{64}\big(2n-3\big)+\frac{223}{420}, \\ R_\alpha(T_1) = D_x^{\alpha} D_y^{\alpha}f(x, y)|_{x = 1 = y} = 12^\alpha(8n+8)+(21)^\alpha8+24^\alpha(4n-4)+16^\alpha(4n+4) +20^\alpha(8n-4)\\ +24^\alpha(8n-4)+30^\alpha(4n-2)+(35)^\alpha4 +40^\alpha(8n-8)+(56)^\alpha2+64^\alpha(2n-3), \\ RR_\alpha(T_1) = S_x^{\alpha}S_y^{\alpha}f(x, y)|_{x = 1 = y} = \frac{8}{12^\alpha}\big(n+1\big)+\frac{8}{21^\alpha} +\frac{4}{24^\alpha}\big(n-1\big)+\frac{4}{4^{2\alpha}}\big(n+1\big) +\frac{4}{20^\alpha}\big(2n-1\big) \\ +\frac{4}{24^\alpha}\big(2n-1\big)+\frac{2}{30^\alpha}\big(2n-1\big)+\frac{4}{35^\alpha}+\frac{8}{40^\alpha}\big(n-1\big) +\frac{2}{56^\alpha}+\frac{1}{64^\alpha}\big(2n-3\big), \\ SDD(T_1) = (S_yD_x+S_xD_y)(x, y)|_{x = 1 = y} = \frac{74}{3}\big(n+1\big)+\frac{899}{30}\big(n-1\big)+\frac{314}{15}\big(2n-1\big)+2\big(2n-3\big)+\frac{14527}{420}.$

3.1.3. Theorem

For $T_1$ , the Atomic Bond Connectivity, Geometric Arithematic Index and General Harmonic Index are as follows, respectively.

$1) \ \ ABC(T_1) = 2n\sqrt{6}+(n+1)\frac{4\sqrt{15}}{3}+(14n+11)\frac{\sqrt14}{56}+(n-1)\frac{2\sqrt110}{5} \\ +(2n-1)\bigg(2\sqrt{\frac{7}{5}}+\frac{4}{\sqrt3}+\sqrt{\frac{6}{5}}\bigg)+\frac{16\sqrt{42}}{21}+\frac{\sqrt182}{14}, \\ 2) \ \ GA(T_1) = (n+1)\bigg(\frac{32\sqrt3}{7}+4\bigg)+(n-1)\frac{32\sqrt10}{13}+(32n-21)\frac{8\sqrt6}{55}+(2n-1)\bigg(\frac{16\sqrt5}{9}+\frac{4\sqrt30}{11}\bigg) \\ +(2n-3)+\frac{8\sqrt21}{5}+\frac{2\sqrt35}{3}+\frac{8\sqrt14}{15}, \\ 3) \ \ H_k(T_1) = (8n+8)\big(\frac2{7}\big)^k+(8n+4)\big(\frac{1}{5}\big)^k+(4n-4)\big(\frac2{11}\big)^k +(4n+4)\big(\frac1{4}\big)^k+(8n-4)\big(\frac2{9}\big)^k \\ +(4n-2)(\frac2{11})^k+4(\frac1{6})^k+(8n-8)(\frac2{13})^k+2(\frac2{15})^k+(2n-3)(\frac1{8})^k.$

Proof:

1). According to Eq (1.7)

$ABC(T_1) = \sum\limits_{\gamma\beta \in E(K)}\sqrt{\frac{d_{\gamma}+d_{\beta}-2}{d_\gamma d_\beta}}.$

By using the information given in Table 3.

$= (8n+8)\bigg(\sqrt{\frac{3+4-2}{3\times4}}\bigg)+8\bigg(\sqrt{\frac{3+7-2}{3\times7}}\bigg)+(4n-4)\bigg(\sqrt{\frac{3+8-2}{3\times8}}\bigg) \\ +(4n+4)\bigg(\sqrt{\frac{4+4-2}{4\times4}}\bigg)+(8n-4)\bigg(\sqrt{\frac{4+5-2}{4\times5}}\bigg)+(8n-4)\bigg(\sqrt{\frac{4+6-2}{4\times6}}\bigg) \\ +(4n-2)\bigg(\sqrt{\frac{5+6-2}{5\times6}}\bigg)+4\bigg(\sqrt{\frac{5+7-2}{5\times7}}\bigg)+(8n-8)\bigg(\sqrt{\frac{5+8-2}{5\times8}}\bigg) \\ +2\bigg(\sqrt{\frac{7+8-2}{7\times8}}\bigg)+(2n-3)\bigg(\sqrt{\frac{8+8-2}{8\times8}}\bigg)\\ = 2n\sqrt{6}+(n+1)\frac{4\sqrt{15}}{3}+(14n+11)\frac{\sqrt14}{56}+(n-1)\frac{2\sqrt110}{5} \\ +(2n-1)\bigg(2\sqrt{\frac{7}{5}}+\frac{4}{\sqrt3}+\sqrt{\frac{6}{5}}\bigg)+\frac{16\sqrt{42}}{21}+\frac{\sqrt182}{14}.\\$

Table 3. Types and count of edges for

$T_1$ .

$Types$ $of$ $edges$	$Count$ $of$ $edges$
$(3, 4)$	$8n+8$
$(3, 7)$	$8$
$(3, 8)$	$4n-4$
$(4, 4)$	$4n+4$
$(4, 5)$	$8n-4$
$(4, 6)$	$8n-4$
$(5, 6)$	$4n-2$
$(5, 7)$	$4$
$(5, 8)$	$8n-8$
$(7, 8)$	$2$
$(8, 8)$	$2n-3$

| Show Table

DownLoad: CSV

2). According to Eq (1.8)

$GA(T_1) = \sum\limits_{\gamma\beta\in E(K)}\frac{2\sqrt{d_{\gamma}d_{\beta}}}{d_{\gamma}+d_{\beta}}.$

By using the information given in Table 3.

$= (8n+8)\bigg(\frac{2\sqrt{3\times4}}{3+4}\bigg)+8\bigg(\frac{2\sqrt{3\times7}}{3+7}\bigg)+(4n-4)\bigg(\frac{2\sqrt{3\times8}}{3+8}\bigg) +(4n+4)\bigg(\frac{2\sqrt{4\times4}}{4+4}\bigg)\\ +(8n-4)\bigg(\frac{2\sqrt{4\times5}}{4+5}\bigg)+(8n-4)\bigg(\frac{2\sqrt{4\times6}}{4+6}\bigg) +(4n-2)\bigg(\frac{2\sqrt{5\times6}}{5+6}\bigg)+4\bigg(\frac{2\sqrt{5\times7}}{5+7}\bigg)\\ +(8n-8)\bigg(\frac{2\sqrt{5\times8}}{5+8}\bigg) +2\bigg(\frac{2\sqrt{7\times8}}{7+8}\bigg)+(2n-3)\bigg(\frac{2\sqrt{8\times8}}{8+8}\bigg)\\ = (n+1)\bigg(\frac{32\sqrt3}{7}+4\bigg)+(n-1)\frac{32\sqrt10}{13}+(32n-21)\frac{8\sqrt6}{55}+(2n-1)\bigg(\frac{16\sqrt5}{9}+\frac{4\sqrt30}{11}\bigg) \\ +(2n-3)+\frac{8\sqrt21}{5}+\frac{2\sqrt35}{3}+\frac{8\sqrt14}{15}.\\$

3). According to Eq (1.9)

$H_k(T_1) = \sum\limits_{\gamma\beta \in E(K)}\bigg(\frac 2{d_{\gamma}+d_{\beta}}\bigg)^k.$

By using the information given in Table 3.

$= (8n+8)\bigg(\frac{2}{3+4}\bigg)^{k}+8\bigg(\frac{2}{3+7}\bigg)^{k}+(4n-4)\bigg(\frac{2}{3+8}\bigg)^{k} +(4n+4)\bigg(\frac{2}{4+4}\bigg)^{k}+(8n-4)\bigg(\frac{2}{4+5}\bigg)^{k} \\ +(8n-4)\bigg(\frac{2}{4+6}\bigg)^{k}+(4n-2)\bigg(\frac{2}{5+6}\bigg)^{k}+4\bigg(\frac{2}{5+7}\bigg)^{k}+(8n-8)\bigg(\frac{2}{5+8}\bigg)^{k} +2\bigg(\frac{2}{7+8}\bigg)^{k} +(2n-3)\bigg(\frac{2}{8+8}\bigg)^{k}\\ = (8n+8)\big(\frac2{7}\big)^k+(8n+4)\big(\frac{1}{5}\big)^k+(4n-4)\big(\frac2{11}\big)^k +(4n+4)\big(\frac1{4}\big)^k+(8n-4)\big(\frac2{9}\big)^k \\ +(4n-2)(\frac2{11})^k+4(\frac1{6})^k+(8n-8)(\frac2{13})^k+2(\frac2{15})^k+(2n-3)(\frac1{8})^k.$

3.1.4. Theorem

For $T_1$ , the First, Second Gourava Indices and the First, Second Hyper Gourava Indices are as follows, respectively.

$1) \ \ GO_1(T_1) = 1640n-312, \\ 2) \ \ GO_2(T_1) = 13128n-4404, \\ 3) \ \ HGO_1(T_1) = 68064n-26124, \\ 4) \ \ HGO_2(T_1) = 5816720n-3573928.$

Proof:

1). According to Eq (1.10)

$\ \ \ GO_1(T_1) = \sum\limits_{\gamma\beta \in E(K)}[(d_{\gamma}+d_{\beta})+(d_\gamma d_\beta)].$

By using the information given in Table 3.

$= (8n+8)\big[(3+4)+(3\times4)\big]+8\big[(3+7)+(3\times7)\big]+(4n-4)\big[(3+8)+(3\times8)\big] +(4n+4)\big[(4+4)\\ +(4\times4)\big]+(8n-4)\big[(4+5)+(4\times5)\big]+(8n-4)\big[(4+6)+(4\times6)\big] +(4n-2)\big[(5+6)+(5\times6)\big]\\ +4\big[(5+7)+(5\times7)\big]+(8n-8)\big[(5+8)+(5\times8)\big] +2\big[(7+8)+(7\times8)\big]+(2n-3)\big[(8+8)+(8\times8)\big]\\ = 1640n-312.$

2). According to Eq (1.11)

$\ \ \ GO_2(T_1) = \sum\limits_{\gamma\beta \in E(K)}[(d_{\gamma}+d_{\beta})(d_\gamma d_\beta)].$

By using the information given in Table 3.

$= (8n+8)\big[(3+4)(3\times4)\big]+8\big[(3+7)(3\times7)\big]+(4n-4)\big[(3+8)(3\times8)\big] +(4n+4)\big[(4+4)(4\times4)\big]\\ +(8n-4)\big[(4+5)(4\times5)\big]+(8n-4)\big[(4+6)(4\times6)\big] +(4n-2)\big[(5+6)(5\times6)\big]+4\big[(5+7)(5\times7)\big]\\ +(8n-8)\big[(5+8)(5\times8)\big] +2\big[(7+8)(7\times8)\big]+(2n-3)\big[(8+8)(8\times8)\big]\\ = 13128n-4404.$

3). According to Eq (1.12)

$\ \ \ HGO_1(T_1) = \sum\limits_{\gamma\beta\in E(K)}[(d_{\gamma}+d_{\beta})+(d_\gamma d_\beta)]^2.$

By using the information given in Table 3.

$= (8n+8)[(3+4)+(3\times4)]^{2}+8[(3+7)+(3\times7)]^{2}+(4n-4)[(3+8)+(3\times8)]^{2} \\ +(4n+4)[(4+4)+(4\times4)]^{2}+(8n-4)[(4+5)+(4\times5)]^{2}+(8n-4)[(4+6)+(4\times6)]^{2} \\ +(4n-2)[(5+6)+(5\times6)]^{2}+4[(5+7)+(5\times7)]^{2}+(8n-8)[(5+8)+(5\times8)]^{2} \\ +2[(7+8)+(7\times8)]^{2}+(2n-3)[(8+8)+(8\times8)]^{2}\\ = 68064n-26124.$

4). According to Eq (1.13)

$\ \ \ HGO_2(T_1) = \sum\limits_{\gamma\beta\in E(K)}[(d_{\gamma}+d_{\beta})(d_\gamma d_\beta)]^2.$

By using the information given in Table 3.

$= (8n+8)[(3+4)(3\times4)]^{2}+8[(3+7)(3\times7)]^{2}+(4n-4)[(3+8)(3\times8)]^{2} +(4n+4)[(4+4)(4\times4)]^{2} \\ +(8n-4)[(4+5)(4\times5)]^{2}+(8n-4)[(4+6)(4\times6)]^{2}+(4n-2)[(5+6)(5\times6)]^{2} \\ +4[(5+7)(5\times7)]^{2}+(8n-8)[(5+8)(5\times8)]^{2}+2[(7+8)(7\times8)]^{2}+(2n-3)[(8+8)(8\times8)]^{2}\\ = 5816720n-3573928.$

3.2. M-polynomial and degree based topological indices of $T_2$ structure

Now we will construct the stellation, medial and bounded dual operations on simple and undirected triangular benzenoid structure. We get a new transformed network (say) $T_2$ , shown in for n = 5. After this we will compute M-polynomial and degree based topological indices of our newly obtained network. Where, blue is stellation, red is medial, green is bounded dual operations applied on $T_2$ .

Figure 3. Transformed structure

$T_2$ .

DownLoad: Full-Size Img PowerPoint

Following shows the types of edges and their count for network $T_2$ ; $n\geq5$ . Now, we will calculate the different M-polynomial and vertex degree based topological indices namely first Zagreb index, second Zagreb index, second modified index, symmetric division index, general Randic index, reciprocal general Randic index, atomic bond connectivity index, geometric arithmetic index, general version of harmonic index, first and second gourava indices, first and second hyper gourava indices.

Table 4. Types and count of edges for

$T_2$ .

$Types$ $of$ $edges$	$Count$ $of$ $edges$
$(3, 4)$	$6n+6$
$(3, 8)$	$9$
$(3, 10)$	$3n-6$
$(4, 4)$	$3n+3$
$(4, 5)$	$6n-6$
$(4, 6)$	$6n-6$
$(5, 6)$	$3n-3$
$(5, 8)$	$6$
$(5, 10)$	$6n-12$
$(6, 6)$	$6(n-1)^2$
$(6, 8)$	$3$
$(6, 10)$	$9n-18$
$(6, 12)$	$6[(n-(n-1))+(n-(n-2))+...$
	$+(n-4)+(n-3)]$
$(8, 10)$	$6$
$(10, 10)$	$3n-6$
$(10, 12)$	$6n-18$
$(12, 12)$	$3[(n-4)+(n-5)+...$
	$+(n-(n-1))]$

| Show Table

DownLoad: CSV

3.2.1. Theorem

Let $T_2$ , be the transformed network then its M-polynomial is

$M(T_2;x, y) = (6n+6)x^3y^4+9x^3y^8+(3n-6)x^3y^{10}+(3n+3)x^4y^4+(6n-6)x^4y^5+(6n-6)x^4y^6 \\ +(3n-3)x^5y^6+6x^5y^8+(6n-12)x^5y^{10}+6(n-1)^2x^6y^6+3x^6y^8+(9n-18)x^6y^{10} \\ +6[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]x^6y^{12}+6x^8y^{10}+(3n-6)x^{10}y^{10} \\ +(6n-18)x^{10}y^{12}+3[(n-4)+(n-5)+...+(n-(n-1))]x^{12}y^{12}.$

Proof: Using definition of M-polynomial and information from Table 4, we have

$M(T_2;x, y) = \sum\limits_{a \leq b}m_{ab}x^{a}y^{b} = \sum\limits_{3\leq 4}m_{34}x^3y^4+\sum\limits_{3\leq 8}m_{38}x^3y^8+\sum\limits_{3\leq 10}m_{3 \ 10}x^3y^{10}+\sum\limits_{4\leq4}m_{44}x^4y^4 \\ +\sum\limits_{4\leq5}m_{45}x^4y^5+\sum\limits_{4\leq6}m_{46}x^4y^6+\sum\limits_{5\leq6}m_{56}x^5y^6+\sum\limits_{5\leq8}m_{58}x^5y^8+\sum\limits_{5\leq10}m_{5 \ 10}x^5y^{10} \\ +\sum\limits_{6\leq6}m_{66}x^6y^6+\sum\limits_{6\leq8}m_{68}x^6y^8+\sum\limits_{6\leq10}m_{6 \ 10}x^6y^{10} +\sum\limits_{6\leq12}m_{6 \ 12}x^6y^{12}+\sum\limits_{8\leq10}m_{8 \ 10}x^8y^{10} \\ +\sum\limits_{10\leq10}m_{10 \ 10}x^{10}y^{10}+\sum\limits_{10\leq12}m_{10 \ 12}x^{10}y^{12}+\sum\limits_{12\leq12}m_{12 \ 12}x^{12}y^{12}\\ = |E_{3, 4}|x^3y^4+|E_{3, 8}|x^3y^8+|E_{3, 10}|x^3y^{10}+|E_{4, 4}|x^4y^4+|E_{4, 5}|x^4y^5+|E_{4, 6}|x^4y^6+|E_{5, 6}|x^5y^6 \\ +|E_{5, 8}|x^5y^{8}+|E_{5, 10}|x^5y^{10}+|E_{6, 6}|x^6y^6+|E_{6, 8}|x^6y^8+|E_{6, 10}|x^6y^{10}+|E_{6, 12}|x^6y^{12} \\ +|E_{8, 10}|x^8y^{10}+|E_{10, 10}|x^{10}y^{10}+|E_{10, 12}|x^{10}y^{12}+|E_{12, 12}|x^{12}y^{12}\\ = (6n+6)x^3y^4+9x^3y^8+(3n-6)x^3y^{10}+(3n+3)x^4y^4+(6n-6)x^4y^5+(6n-6)x^4y^6 \\ +(3n-3)x^5y^6+6x^5y^8+(6n-12)x^5y^{10}+6(n-1)^2x^6y^6+3x^6y^8+(9n-18)x^6y^{10} \\ +6\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big]x^6y^{12}+6x^8y^{10}+(3n-6)x^{10}y^{10} \\ +(6n-18)x^{10}y^{12}+3\big[(n-4)+(n-5)+...+(n-(n-1))\big]x^{12}y^{12}.$

Figure 4. M-polynomial of

$T_2$ for n = 6.

DownLoad: Full-Size Img PowerPoint

Figure 5. M-polynomial of

$T_2$ .

DownLoad: Full-Size Img PowerPoint

3.2.2. Theorem

Let $T_2$ , be the transformed network and

$M(T_2;x, y) = (6n+6)x^3y^4+9x^3y^8+(3n-6)x^3y^{10}+(3n+3)x^4y^4+(6n-6)x^4y^5+(6n-6)x^4y^6 \\ +(3n-3)x^5y^6+6x^5y^8+(6n-12)x^5y^{10}+6(n-1)^2x^6y^6+3x^6y^8+(9n-18)x^6y^{10} \\ +6\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big]x^6y^{12}+6x^8y^{10}+(3n-6)x^{10}y^{10} \\ +(6n-18)x^{10}y^{12}+3\big[(n-4)+(n-5)+...+(n-(n-1))\big]x^{12}y^{12},$

be its M-polynomial. Then, the first Zagreb index $M_1\big(T_2\big)$ , the second Zagreb index $M_2\big(T_2\big)$ , the second modified Zagreb $^mM_2(T_2\big)$ , the general Randic index $R_\alpha(T_2)$ , where $\alpha \ \epsilon \ N$ , reciprocal general Randic index $RR_\alpha(T_2)$ , where $\alpha \ \epsilon \ N$ , and the symmetric division degree index $SDD(T_2)$ obtained from M-polynomial are as follows:

$M_1(T_2) = 678n-816+72(n-1)^2+108\big[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)\big]+72\big[(n-4)+(n-5)+...+(n-(n-1))\big], \\ M_2(T_2) = 2424n-3774+216(n-1)^2+432\big[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)\big]+432\big[(n-4)+(n-5)+...+(n-(n-1))\big], \\ ^mM_2(T_2) = \frac{11}{16}\big(n+1\big)+\frac{2}{5}\big(n-2\big)+\frac{13}{20}\big(n-1\big)+\frac{1}{20}\big(n-3\big)+\frac{53}{80} +\frac{1}{6}\big(n-1\big)^2\\ +\frac{1}{12}\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big] \\ +\frac{1}{48}\big[(n-4)+(n-5)+...+(n-(n-1))\big], \\ R_\alpha(T_2) = 12^\alpha(6n+6)+24^\alpha(9)+30^\alpha(3n-6)+4^{2\alpha}(3n+3)+20^\alpha(6n-6) +24^\alpha(6n-6) \\ +30^\alpha(3n-3)+40^\alpha(6)+50^\alpha(6n-12)+6(n-1)^2(36)^\alpha+48^\alpha(3)+60^\alpha(9n-18) \\ +72^\alpha[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]6+80^\alpha(6)+100^\alpha(3n-6) \\ +120^\alpha(6n-18)+144^\alpha[(n-4)+(n-5)+...+(n-(n-1))]3, \\ RR_\alpha(T_2) = \frac{6}{12^\alpha}\big(n+1\big)+\frac{9}{24^\alpha}+\frac{3}{30^\alpha}\big(n-2\big)+\frac{3}{4^{2\alpha}}\big(n+1\big) +\frac{6}{20^\alpha}\big(n-1\big)+\frac{6}{24^\alpha}\big(n-1\big) \\ +\frac{3}{30^\alpha}\big(n-1\big)+\frac{6}{40^\alpha}+\frac{6}{50^\alpha}\big(n-2\big)+\frac{6}{36^\alpha}\big(n-1\big)^2 +\frac{3}{48^\alpha}+\frac{9}{60^\alpha}\big(n-2\big)+\frac{6}{80^\alpha} \\ +\frac{3}{100^\alpha}\big(n-2\big)+\frac1{72^\alpha}\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big]6 \\ +\frac{6}{120^\alpha}\big(n-3\big)+\frac{1}{144^\alpha}\big[(n-4)+(n-5)+...+(n-(n-1))\big]3, \\ SDD(T_2) = \frac{37}{2}\big(n+1\big)+\frac{157}{5}\big(n-1\big)+\frac{523}{10}\big(n-2\big)+\frac{61}{5}\big(n-3\big)+\frac{2371}{40}+12(n-1)^2 \\ +15[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)] \\ +6[(n-4)+(n-5)+...+(n-(n-1))].$

Proof: Let $f(x, y)$ = $M(T_2;x, y)$ be the M-polynomial of the transformed network $T_2$ . Then

Now, the required partial derivatives and integrals are obtained.

By using the information given in Table 4 and formulas for Table 1:

$D_xf(x, y) = 3(6n+6)x^3y^4+27x^3y^8+3(3n-6)x^3y^{10}+4(3n+3)x^4y^4+4(6n-6)x^4y^5 \\ +4(6n-6)x^4y^6+5(3n-3)x^5y^6+30x^5y^8+5(6n-12)x^5y^{10}+36(n-1)^2x^6y^6+18x^6y^8 \\ +6(9n-18)x^6y^{10}+36[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]x^6y^{12} \\ +48x^8y^{10}+10(3n-6)x^{10}y^{10}+10(6n-18)x^{10}y^{12} +36[(n-4)+(n-5)+...\\ +(n-(n-1))]x^{12}y^{12}, \\ D_yf(x, y) = 4(6n+6)x^3y^4+72x^3y^8+10(3n-6)x^3y^{10}+4(3n+3)x^4y^4+5(6n-6)x^4y^5 \\ +6(6n-6)x^4y^6+6(3n-3)x^5y^6+48x^5y^8+10(6n-12)x^5y^{10}+36(n-1)^2x^6y^6+24x^6y^8 \\ +10(9n-18)x^6y^{10}+72[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]x^6y^{12} \\ +60x^8y^{10}+10(3n-6)x^{10}y^{10}+12(6n-18)x^{10}y^{12} +36[(n-4)+(n-5)+...\\ +(n-(n-1))]x^{12}y^{12}, \\ D_xD_yf(x, y) = 12(6n+6)x^3y^4+216x^3y^8+30(3n-6)x^3y^{10}+16(3n+3)x^4y^4+20(6n-6)x^4y^5 \\ +24(6n-6)x^4y^6+30(3n-3)x^5y^6+240x^5y^8+50(6n-12)x^5y^{10}+216(n-1)^2x^6y^6 \\ +144x^6y^8+60(9n-18)x^6y^{10}+432[(n-(n-1))+(n-(n-2))+... \\ +(n-4)+(n-3)]x^6y^{12}+480x^8y^{10}+100(3n-6)x^{10}y^{10}+120(6n-18)x^{10}y^{12} \\ +432[(n-4)+(n-5)+...+(n-(n-1))]x^{12}y^{12}, \\ S_xS_yf(x, y) = \frac{1}{2}\big(n+1\big)x^3y^4+\frac{3}{8}x^3y^8+\frac{1}{10}\big(n-2\big)x^3y^{10} +\frac{3}{16}\big(n+1\big)x^4y^4+\frac{3}{10}\big(n-1\big)x^4y^5 \\ +\frac{1}{4}\big(n-1\big)x^4y^6+\frac{1}{10}\big(n-1\big)x^5y^6+\frac{3}{20}x^5y^8 +\frac{3}{25}\big(n-2\big)x^5y^{10}+\frac{1}{6}\big(n-1\big)^2x^6y^6+\frac{1}{16}x^6y^8 \\ +\frac{3}{20}\big(n-2\big)x^6y^{10} +\frac{1}{12}\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big]x^6y^{12} \\ +\frac{3}{40}x^8y^{10}+\frac{3}{100}\big(n-2\big)x^{10}y^{10}+\frac{1}{20}\big(n-3\big)x^{10}y^{12} \\ +\frac{1}{48}\big[(n-4)+(n-5)+...+(n-(n-1))\big]x^{12}y^{12}, \\ D_x^{\alpha}D_y^{\alpha}f(x, y) = 12^\alpha(6n+6)x^3y^4+24^\alpha(9)x^3y^8+30^\alpha(3n-6)x^3y^{10}+4^{2\alpha}(3n+3)x^4y^4+20^\alpha(6n-6)x^4y^5 \\ +24^\alpha(6n-6)x^4y^6+30^\alpha(3n-3)x^5y^6+40^\alpha(6)x^5y^8+50^\alpha(6n-12)x^5y^{10} \\ +(36)^{\alpha}(n-1)^26x^6y^6+48^\alpha(3)x^6y^8+60^\alpha(9n-18)x^6y^{10}+72^\alpha[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)]6x^6y^{12}+80^\alpha(6)x^8y^{10}+100^\alpha(3n-6)x^{10}y^{10}+120^\alpha(6n-18)x^{10}y^{12} \\ +144^\alpha[(n-4)+(n-5)+...+(n-(n-1))]3x^{12}y^{12}, \\ S_x^{\alpha}S_y^{\alpha}f(x, y) = \frac{6}{12^\alpha}\big(n+1\big)x^3y^4+\frac{9}{24^\alpha}x^3y^8 +\frac{3}{30^\alpha}\big(n-2\big)x^3y^{10} +\frac{3}{4^{2\alpha}}\big(n+1\big)x^4y^4\\ +\frac{6}{20^\alpha}\big(n-1\big)x^4y^5 +\frac{6}{24^\alpha}\big(n-1\big)x^4y^6 +\frac{3}{30^\alpha}\big(n-1\big(n-1\big)x^5y^6 +\frac{6}{40^\alpha}x^5y^8\\ +\frac{6}{50^\alpha}\big(n-2\big)x^5y^{10} +\frac{6}{36^\alpha}\big(n-1\big)^2x^6y^6 +\frac{3}{48^\alpha}x^6y^8+\frac{9}{60^\alpha}\big(n-2\big)x^6y^{10} \\ +\frac1{72^\alpha}\bigg[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\bigg]6x^6y^{12} \\ +\frac{6}{80^\alpha}x^8y^{10}+\frac{3}{100^\alpha}\big(n-2\big)x^{10}y^{10} +\frac{6}{120^\alpha}\big(n-3\big)x^{10}y^{12} \\ +\frac1{144^\alpha}\bigg[(n-4)+(n-5)+...+(n-(n-1))\bigg]3x^{12}y^{12}, \\ S_yD_xf(x, y) = \frac{9}{2}(n+1)x^3y^4+\frac{27}{8}x^3y^8+\frac{9}{10}(n-2)x^3y^{10}+3(n+1)x^4y^4+\frac{24}{5}(n-1)x^4y^5 \\ +4(n-1)x^4y^6+\frac{5}{2}(n-1)x^5y^6+\frac{15}{4}x^5y^8+3(n-2)x^5y^{10}+6(n-1)^2x^6y^6 +\frac{9}{4}x^6y^8\\ +\frac{27}{5}(n-2)x^6y^{10} +3[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]x^6y^{12} \\ +\frac{24}{5}x^8y^{10}+3(n-2)x^{10}y^{10}+5(n-3)x^{10}y^{12} +3[(n-4)+(n-5)+...+(n-(n-1))]x^{12}y^{12}, \\ S_xD_yf(x, y) = 8(n+1)x^3y^4+24x^3y^8+10(n-2)x^3y^{10}+3(n+1)x^4y^4+\frac{15}{2}(n-1)x^4y^5 \\ +9(n-1)x^4y^6+\frac{18}{5}(n-1)x^5y^6+\frac{48}{5}x^5y^8+12(n-2)x^5y^{10}+6(n-1)^2x^6y^6 +4x^6y^8\\ +15(n-2)x^6y^{10}+12[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]x^6y^{12} \\ +\frac{15}{2}x^8y^{10}+3(n-2)x^{10}y^{10}+\frac{36}{5}(n-3)x^{10}y^{12} \\ +3[(n-4)+(n-5)+...\\ +(n-(n-1))]x^{12}y^{12}.$

Consequently,

$M_1(T_2) = (D_x+D_y)f(x, y)|_{x = 1 = y} = 678n-816+72(n-1)^2+108\big[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)\big]+72\big[(n-4)+(n-5)+...+(n-(n-1))\big], \\ M_2(T_2) = D_xD_yf(x, y)|_{x = 1 = y} = 2424n-3774+216(n-1)^2+432\big[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)\big]+432\big[(n-4)+(n-5)+...+(n-(n-1))\big], \\ ^mM_2(T_2) = S_xS_yf(x, y)|_{x = 1 = y} = \frac{11}{16}\big(n+1\big)+\frac{2}{5}\big(n-2\big)+\frac{13}{20}\big(n-1\big)+\frac{1}{20}\big(n-3\big)+\frac{7}{10} +\frac{1}{6}\big(n-1\big)^2\\ +\frac{1}{12}\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big] \\ +\frac{1}{48}\big[(n-4)+(n-5)+...+(n-(n-1))\big], \\ R_\alpha(T_2) = D_x^{\alpha} D_y^{\alpha}f(x, y)|_{x = 1 = y} = 12^\alpha(6n+6)+24^\alpha(9)+30^\alpha(3n-6)+4^{2\alpha}(3n+3)+20^\alpha(6n-6) \\ +24^\alpha(6n-6)+30^\alpha(3n-3)+40^\alpha(6)+50^\alpha(6n-12)+6(n-1)^2(36)^\alpha+48^\alpha(3) \\ +60^\alpha(9n-18)+72^\alpha[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]6 \\ +80^\alpha(6)+100^\alpha(3n-6)+120^\alpha(6n-18)+144^\alpha[(n-4)+(n-5)+...+(n-(n-1))]3, \\ RR_\alpha(T_2) = S_x^{\alpha}S_y^{\alpha}f(x, y)|_{x = 1 = y} = \frac{6}{12^\alpha}\big(n+1\big)+\frac{9}{24^\alpha}+\frac{3}{30^\alpha}\big(n-2\big)+\frac{3}{4^{2\alpha}}\big(n+1\big) +\frac{6}{20^\alpha}\big(n-1\big)+\frac{6}{24^\alpha}\big(n-1\big) \\ +\frac{3}{30^\alpha}\big(n-1\big)+\frac{6}{40^\alpha}+\frac{6}{50^\alpha}\big(n-2\big)+\frac{6}{36^\alpha}\big(n-1\big)^2 +\frac{3}{48^\alpha}+\frac{9}{60^\alpha}\big(n-2\big)+\frac{6}{80^\alpha} \\ +\frac{3}{100^\alpha}\big(n-2\big)+\frac1{72^\alpha}\big[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)\big]6 \\ +\frac{6}{120^\alpha}\big(n-3\big)+\frac{1}{144^\alpha}\big[(n-4)+(n-5)+...+(n-(n-1))\big]3, \\ SDD(T_2) = (S_yD_x+S_xD_y)(x, y)|_{x = 1 = y} = \frac{37}{2}\big(n+1\big)+\frac{157}{5}\big(n-1\big)+\frac{523}{10}\big(n-2\big)+\frac{61}{5}\big(n-3\big) +\frac{2371}{40} \\ +12(n-1)^2+15[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)] \\ +6[(n-4)+(n-5)+...+(n-(n-1))].$

3.2.3. Theorem

For $T_2$ , the Atomic Bond Connectivity, Geometric Arithmetic Index and General Harmonic Index are as follows, respectively.

$1) \ \ \ \ ABC(T_2) = (7n)\frac{\sqrt6}{4}+(n+1){\sqrt15}+(n-1)\bigg(3\sqrt{\frac{7}{5}}+{2\sqrt3}+3\sqrt{\frac{3}{10}}\bigg) \\ (n-2)\bigg(\frac{\sqrt330}{10}+\frac{3\sqrt26}{5}+\frac{3\sqrt210}{10}+\frac{9\sqrt2}{10}\bigg)+\frac{3\sqrt110}{10}+\frac3{2}+\frac{6}{\sqrt5} \\ +(n-1)^2{\sqrt10}+[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]{2\sqrt2} \\ +[(n-4)+(n-5)+...+(n-(n-1))]\frac{\sqrt22}{4}, \\ 2) \ \ \ \ GA(T_2) = (24n+36)\frac{\sqrt3}{7}+(132n+48)\frac{\sqrt6}{55}+(50n-113)\frac{6\sqrt30}{143}+3(n+1) \\ +(8n)\frac{\sqrt5}{3}+(n-2)\bigg({4\sqrt2}+\frac{9\sqrt15}{4}+3\bigg)+\frac{24\sqrt10}{13}+6(n-1)^2 \\ +[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)]{4\sqrt2} \\ +3[(n-4)+(n-5)+...+(n-(n-1))], \\ 3) \ \ \ H_k(T_2) = (6n+6)(\frac2{7})^k+9(\frac2{11})^k+(3n-6)(\frac2{13})^k+(3n+3)(\frac1{4})^k+(6n-6) \\ (\frac2{9})^k+(6n-6)(\frac1{5})^k+(3n-3)(\frac2{11})^k+6(\frac2{13})^k+(6n-12)(\frac2{15})^k \\ +6(n-1)^2)(\frac1{6})^k+3(\frac1{7})^k+(9n-18)(\frac1{8})^k+6[(n-(n-1)) \\ +(n-(n-2))+...+(n-4)+(n-3)](\frac1{9})^k+6(\frac1{9})^k+(3n-6)(\frac1{10})^k \\ +(6n-18)(\frac1{11})^k+3[(n-4)+(n-5)+...+(n-(n-1))](\frac1{12})^k.$

Proof: Using Eqs 1.7–1.9 and Table 4, we get the results 1–3.

3.2.4. Theorem

For $T_2$ , the First, Second Gourava Indices and the First, Second Hyper Gourava Indices are as follows, respectively.

$1) \ \ \ GO_1(T_2) = 3102n-4590+288(n-1)^2+540[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)]+504[(n-4)+(n-5)+...+(n-(n-1))], \\ 2) \ \ \ GO_2(T_2) = 40548n-74610+2592(n-1)^2+7776[(n-(n-1))+(n-(n-2)) \\ +...+(n-4)+(n-3)]+10368[(n-4)+(n-5)+...+(n-(n-1))], \\ 3) \ \ \ HGO_1(T_2) = 267984n-531210+13824(n-1)^2 \\ +48600[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)] \\ +84672[(n-4)+(n-5)+...+(n-(n-1))], \\ 4) \ \ \ HGO_2(T_2) = 66901488n-158433396+1119744(n-1)^2 \\ +10077696[(n-(n-1))+(n-(n-2))+...+(n-4)+(n-3)] \\ +35831808[(n-4)+(n-5)+...+(n-(n-1))].$

Proof: Using Eqs 1.10–1.13 and Table 4 we get the above results 1–4.

4. Conclusions

In this paper, we defined M-polynomial of the transformed zigzag benzenoid and transformed triangular benzenoid structures by applying stellation, medial and bounded dual operations to get networks named as $T_1$ and $T_2$ . With the help of M-polynomial, we computed certain degree-based topological indices such as first Zagreb index, second Zagreb index, second modified Zagreb index, general Randic index, reciprocal general Randic index, symmetric division degree index. We also computed atomic bound connectivity index, geometric arithmetic index, general harmonic index, first and second gourava indices, first and second hyper gourava indices. M-polynomial is used to calculate the certain degree based topological indices as a latest developed instrument in the chemical graph theory. In future, we can compute other indices on these structures and some additional transformed structures can be studied for a variety of topological indices to have an insight about their properties.

Acknowledgments

This work was supported by the Key teaching research project of quality engineering in Colleges and universities in Anhui Province. Anhui Vocational college of Electronics and Information Technology Guangzhou Zhuoya Education Investment Co., Ltd. Practical Education Base (subject No: 2020sjjd093). The authors are very thankful for this research funding. We are also very grateful to the reviewers for their valuable suggestions.

Conflict of interest

The authors declare no conflict of interest.

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2. Methodology

3. Results and discussion

3.1. M-polynomial and degree based topological indices of $T_1$ structure

3.1.1. Theorem

3.1.2. Theorem

3.1.3. Theorem

3.1.4. Theorem

3.2. M-polynomial and degree based topological indices of $T_2$ structure

3.2.1. Theorem

3.2.2. Theorem

3.2.3. Theorem

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3.1.1. Theorem

3.1.2. Theorem

3.1.3. Theorem

3.1.4. Theorem

3.2. M-polynomial and degree based topological indices of T2 T_2 structure

3.2.1. Theorem

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3.2.3. Theorem

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3.1. M-polynomial and degree based topological indices of $T_1$ structure

3.2. M-polynomial and degree based topological indices of $T_2$ structure