# On the Sum of Degree-Based Topological Indices of Rhombus-Type Silicate and Oxide Structures

Authors: Rong Qi; Haidar Ali; Usman Babar; Jia-Bao Liu; Parvez Ali
Journal: Journal of Mathematics
 (2021)

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## Body

## 1. Introduction and Preliminary Results

The representation of a graph is expressed by numbers, polynomials, and matrices. Graphs have their own characteristics that may be calculated by topological indices, and under graph automorphism, the topology of graphs remains unchanged. Degree-based topological indices are exceptionally important in different classes of indices and take on a vital role in graphic theory and in particular in science.Silicate is a chemical compound and has many commercial uses. It is used for the manufacture of different glass and ceramics organic compounds in large scale due to its cheapness and availability everywhere in the world. Silicates can be obtained from the Earth’s crust. In general, solid silicates are well-characterized and stable. Silicates like sodium orthosilicate and metasilicate, which have alkali cations and tiny or chain-like anions, are water soluble. When crystallised from a solution, they generate multiple solid hydrates. Water glass, which is made up of soluble sodium silicates and combinations, is a significant industrial and home chemical. For the construction of networks rhombus oxide and silicate, we refer the readers to 10. Rhombus silicate networkRHSLt and rhombus oxide network RHOXt are shown in Figures 1 and 2, respectively.Figure 1
Graph of rhombus silicate network (RHSLt).Figure 2
Graph of rhombus oxide network (RHOXt).In this article,G is considered a network with a VG vertex set and an edge set of EG and dr is the degree of vertex r∈VG. Let SGr denote the sum of the degrees of all vertices adjacent to a vertex r. Graovac et al. defined fifth M-Zagreb indices as polynomials for a molecular graph [1], and these are characterized as follows.LetG be a graph. Then,(1)M1G5G=∑rs∈EGSGr+SGs,(2)M2G5G=∑rs∈EGSGr+SGs.V. R. Kulli [2], motivated by the above indices, described some new topological indices, and he named them as the fifth M-Zagreb indices of first and second type and fifth hyper-M-Zagreb indices of first and second type of a graph G. They are defined as(3)M1aG5G=∑rs∈EGSGr+SGsa,(4)M2aG5G=∑rs∈EGSGr+SGsa,(5)HM1G5G=∑rs∈EGSGr+SGs2,(6)HM2G5G=∑rs∈EGSGr+SGs2.They also define a new version of Zagreb index which is called as the third Zagreb index or fifthM3-Zagreb [3].(7)M3G5G=∑rs∈EGSGr−SGs.Corresponding to the above indices, he defined the general fifthM1-Zagreb polynomial and the general fifth M2-Zagreb polynomial of a molecular graph G as(8)M1aG5G,x=∏rs∈EGxSGr+SGsa,(9)M2aG5G,x=∏rs∈EGxSGr+SGsa.The fifthM1- and M2- Zagreb polynomials of a graph are defined as(10)M1G5G,x=∏rs∈EGxSGr+SGs,(11)M2G5G,x=∏rs∈EGxSGr+SGs.The fifthHM1 and HM2 Zagreb polynomials of the graph are defined as(12)HM1G5G,x=∏rs∈EGxSGr+SGs2,(13)HM2G5G,x=∏rs∈EGxSGr+SGs2.


## 2. Main Results

We have studied the topological indices introduced by Kulli [2, 4] named as fifth M-Zagreb indices, fifth M-Zagreb polynomials, and M3−Zagreb index and computed exact formulae of these indices for rhombus-type silicate and oxide networks. Ali et al. studied degree-based topological indices for various networks [5–8]. For the basic notations and definitions, see [9–11].


### 2.1. Results for the Rhombus Type of Silicate Networks

In this section, we calculate degree-based topological indices of the dimensiont for rhombus-type silicate networks. In the following theorems, we compute M-Zagreb indices and polynomials.Theorem 2.1.1.
LetG1≅RHSLt be the rhombus-type silicate network; then, the first and second fifth M-Zagreb indices are equal to(14)M1G5G1=361−10t+18t2,M2G5G1=18119−490t+480t2.Proof.
The outcome can be obtained by using the edge partition in Table1.
By using equation [5],(15)M1G5G1=∑rs∈EG1SGr+SGs,M1G5G1=12+12|E1G1t|+12+24|E2G1t|+15+15|E3G1t|+15+24|E4G1t|+15+27|E5G1t|+18+24|E6G1t|+18+27|E7G1t|+18+30|E8G1t|+24+27|E9G1t|+27+27|E10G1t|+27+30|E11G1t|+30+30|E12G1t|,=12+126+12+246+15+154t−4+15+248+15+2716t−24+18+242+18+278t−12+18+306t2−20t+16+24+278+27+278t−14+27+308t−16+30+306t2−24t+24.
By doing some calculations, we obtain(16)M1G5G1=361−10t+18t2.
Thus, from [6],(17)M2G5G=∑rs∈EG1SGr+SGs,M2G5G1=12×12|E1G1t|+12×24|E2G1t|+15×15|E3G1t|+15×24|E4G1t|+15×27|E5G1t|+18×24|E6G1t|+18×27|E7G1t|+18×30|E8G1t|+24×27|E9G1t|+27×27|E10G1t|+27×30|E11G1t|+30×30|E12G1t|,=12×126+12×246+15×154t−4+15×248+15×2716t−24+18×242+18×278t−12+18×306t2−20t+16+24×278+27×278t−14+27×308t−16+30×306t2−24t+24.
By doing some calculations, we obtain(18)M2G5G1=18119−490t+480t2.Table 1
Edge partition of rhombus-type silicate network (RHSLt) based on sum of degrees of end vertices of each edge.
Sr,SsNumber of edgesWherers∈EG112,12612,24615,154t−415,24815,2716t−2418,242Wherers∈EG118,278t−218,306t2−20t+1624,27827,278t−1427,308t−1630,306t2−24t+24Theorem 2.1.2.
Consider the rhombus-type silicate networkG1≅RHSLt for t∈ℕ. Then, the first and second general fifth M-Zagreb indices are equal to(19)M1aG5G1=21+3a31+a+23+2a31+a5a−4×31+2a5a+61+2a−23+a31+a7a−22+a15a+3a161+a+21+a21a−7×21+a27a+8×39a+8×51a−16×57a+t−5×3a41+2a−23+2a31+a5a+22+a15a+24+a21a+23+a27a+8×45a+8×57a+t221+4a3+21+2a31+a5a,M2aG5G1=21+4a31+2a+21+5a31+2a−22+a31+5a−8×31+4a5a+23+2a31+2a25a+21+4a27a+81+a45a+81+a81a+42+a135a−4×225a−24+a405a−14×729a+t−23+2a31+2a25a−201+a27a+4×225a+23+a243a+16×405a+23+a405a+8×729a+t221+2a31+3a5a+61+2a25a.Proof.
LetG1 be the rhombus-type silicate network. Table 1 shows such an edge partition of RHSLt. Thus, from [9], it follows that(20)M1aG5G=∑rs∈EGSGr+SGsa.
By using edge partitions in Table1, we obtain(21)M1aG5G1=12+12a|E1G1t|+12+24a|E2G1t|+15+15a|E3G1t|+15+24a|E4G1t|+15+27a|E5G1t|+18+24a|E6G1t|+18+27a|E7G1t|+18+30a|E8G1t|+24+27a|E9G1t|+27+27a|E10G1t|+27+30a|E11G1t|+30+30a|E12G1t|,=12+12a6+12+24a6+15+15a4t−4+15+24a8+15+24a16t−24+18+24a2+18+27a8t−12+18+30a6t2−20t+16+24+27a8+27+27a8t−14+27+30a8t−16+30+30a6t2−24t+24.
By doing some calculations, we have(22)M1aG5G1=21+3a31+a+23+2a31+a5a−4×31+2a5a+61+2a−23+a31+a7a−22+a15a+3a161+a+21+a21a−7×21+a27a+8×39a+8×51a−16×57a+t−5×3a41+2a−23+2a31+a5a+22+a15a+24+a21a+23+a27a+8×45a+8×57a+t221+4a31+a+21+2a31+a5a.
From [12], we have(23)M2aG5G=∑rs∈EGSGr+SGsa.
By using edge partitions in Table1, we obtain(24)M2aG5G1=12×12a|E1G1t|+12×24a|E2G1t|+15×15a|E3G1t|+15×24a|E4G1t|+15×27a|E5G1t|+18×24a|E6G1t|+18×27a|E7G1t|+18×30a|E8G1t|+24×27a|E9G1t|+27×27a|E10G1t|+27×30a|E11G1t|+30×30a|E12G1t|,=12×12a6+12×24a6+15×15a4t−4+15×24a8+15×24a16t−24+18×24a2+18×27a8t−12+18×30a6t2−20t+16+24×27a8+27×27a8t−14+27×30a8t−16+30×30a6t2−24t+24.
By doing some calculations, we have(25)M2aG5G1=21+4a31+2a+21+5a31+2a−22+a31+5a−8×31+4a5a+23+2a31+2a25a+21+4a27a+81+a45a+81+a81a+42+a135a−4×225a−24+a405a−14×729a+t−23+2a31+2a25a−201+a27a+4×225a+23+a243a+16×405a+23+a405a+8×729a+t221+2a31+3a5a+61+2a25a.Theorem 2.1.3.
Consider the rhombus-type silicate networkG1≅RHSLt for t∈ℕ. Then, the first and second hyper-fifth M-Zagreb indices are equal to(26)HM1G5G1=36221−976t+984t2,HM2G5G1=16228307−68242t+40800t2.Proof.
LetG1 be the rhombus type of silicate network. Table 1 shows such an edge partition of RHSLt. Thus, from [13], it follows that(27)HM1G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table1, we obtain(28)HM1G5G1=12×122|E1G1t|+12×242|E2G1t|+15×152|E3G1t|+15×242|E4G1t|+15×272|E5G1t|+18×242|E6G1t|+18×272|E7G1t|+18×302|E8G1t|+24×272|E9G1t|+27×272|E10G1t|+27×302|E11G1t|+30×302|E12G1t|,=12×1226+12×2426+15×1524t−4+15×2428+15×24216t−24+18×2422+18×2728t−12+18×3026t2−20t+16+24×2728+27×2728t−14+27×3028t−16+30×3026t2−24t+24.
By doing some calculations, we have(29)HM1G5G1=36221−976t+984t2.
From [14], we have(30)HM22G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table1, we obtain(31)HM2G5G1=12×122|E1G1t|+12×242|E2G1t|+15×152|E3G1t|+15×242|E4G1t|+15×272|E5G1t|+18×242|E6G1t|+18×272|E7G1t|+18×302|E8G1t|+24×272|E9G1t|+27×272|E10G1t|+27×302|E11G1t|+30×302|E12G1t|,=12×1226+12×2426+15×1524t−4+15×2428+15×24216t−24+18×2422+18×2728t−12+18×3026t2−20t+16+24×2728+27×2728t−14+27×3028t−16+30×3026t2−24t+24.
By doing some calculations, we have(32)HM2aG5G1=16228307−68242t+40800t2.Theorem 2.1.4.
Consider the rhombus-type silicate networkG1≅RHSLt for t∈ℕ. Then, the third M-Zagreb index is equal to(33)M3G5G1=−232+248t+12t2.Proof.
LetG1 be the rhombus silicate network. Table 1 shows such an edge partition of RHSLt. Thus, from [15], it follows that(34)M3G5G=∑rs∈EGSGr+SGs.
By using edge partitions in Table1, we obtain(35)M3G5G1=|12−12E1G1t|+|12−24E2G1t|+|15−15E3G1t|+|15−24E4G1t|+|15−27E5G1t|+|18−24E6G1t|+|18−27E7G1t|+|18−30E8G1t|+|24−27E9G1t|+|27−27E10G1t|+|27−30E11G1t|+|30−30E12G1t|,=|12−12|6+|12−24|6+|15−15|4t−4+|15−24|8+|15−27|16t−24+|18−24|2+|18−27|8t−12+|18−30|6t2−20t+16+|24−27|8+|27−27|8t−14+|27−30|8t−16+|30−30|6t2−24t+24.
By doing some calculations, we have(36)M3G5G1=−232+248t+12t2.Corresponding to the above indices, we are going to compute general fifthM-Zagreb polynomials for rhombus-type silicate network RHSLt.Theorem 2.1.5.
LetG1≅RHSLt be the first type of rhombus-type silicate network; then, general fifth M-Zagreb polynomials of first and second type are equal to(37)M1aG5G1,x=6x24a+4t−4x30a+6x36a+8x39a+16t−22x42a+8t−12x45a+6t2−20t+16x48a+8x51a+8t−14x54a+8t−16x57a+6t2−24t+24x60a,M2aG5G1,x=6x144a+4t−4x225a+6x288a+8x360a+82t−3x405a+2x432a+42t−3x486a+2t−23t−4x540a+8x648a+24t−7x729a+8t−2x810a+6t−22x900a.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [1] that(38)M1aG5G1,x=∑rs∈EG1xSGr+SGsa,M1aG5G1,x=x12+12a|E1G1t|+x12+24a|E2G1t|+x15+15a|E3G1t|+x15+24a|E4G1t|+x15+27a|E5G1t|+x18+24a|E6G1t|+x18+27a|E7G1t|+x18+30a|E8G1t|+x24+27a|E9G1t|+x27+27a|E10G1t|+x27+30a|E11G1t|+x30+30a|E12G1t|,M1aG5G1,x=x12+12a6+x12+24a6+x15+15a4t−4+x15+24a8+x15+27a16t−24+x18+24a2+x18+27a8t−12+x18+30a6t2−20t+16+x24+27a8+x27+27a8t−14+x27+30a8t−16+x30+30a6t2−24t+24.
By doing some calculations, we obtain(39)M1aG5G1,x=6x24a+4t−4x30a+6x36a+8x39a+16t−22x42a+8t−12x45a+6t2−20t+16x48a+8x51a+8t−14x54a+8t−16x57a+6t2−24t+24x60a.
Also, from [3],(40)M2aG5G1,x=∑rs∈EG1xSGr+SGsa,M2aG5G1,x=x12+12a|E1G1t|+x12+24a|E2G1t|+x15+15a|E3G1t|+x15+24a|E4G1t|+x15+27a|E5G1t|+x18+24a|E6G1t|+x18+27a|E7G1t|+x18+30a|E8G1t|+x24+27a|E9G1t|+x27+27a|E10G1t|+x27+30a|E11G1t|+x30+30a|E12G1t|,M2aG5G1,x=x12+12a6+x12+24a6+x15+15a4t−4+x15+24a8+x15+27a16t−24+x18+24a2+x18+27a8t−12+x18+30a6t2−20t+16+x24+27a8+x27+27a8t−14+x27+30a8t−16+x30+30a6t2−24t+24.
By making some calculations, we obtain(41)M2aG5G1,x=6x144a+4t−4x225a+6x288a+8x360a+82t−3x405a+2x432a+42t−3x486a+2t−23t−4x540a+8x648a+24t−7x729a+8t−2x810a+6t−22x900a.Corresponding to the above indices, we are going to compute fifthM-Zagreb polynomials for rhombus-type silicate network RHSLt.Theorem 2.1.6.
LetG1≅RHSLt be the rhombus type of silicate network; then, fifth M-Zagreb polynomials of first and second type are equal to(42)M1G5G1,x=6x24+4t−4x30+6x36+8x39+16t−22x42+8t−12x45+6t2−20t+16x48+8x51+8t−14x54+8t−16x57+6t2−24t+24x60,M2G5G1,x=6x144+4t−4x225+6x288+8x360+82t−3x405+2x432+42t−3x486+2t−23t−4x540+8x648+24t−7x729+8t−2x810+6t−22x900.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [16] that(43)M1G5G1,x=∑rs∈EG1xSGr+SGs,M1G5G1,x=x12+12|E1G1t|+x12+24|E2G1t|+x15+15|E3G1t|+x15+24|E4G1t|+x15+27|E5G1t|+x18+24|E6G1t|+x18+27|E7G1t|+x18+30|E8G1t|+x24+27|E9G1t|+x27+27|E10G1t|+x27+30|E11G1t|+x30+30|E12G1t|,=x12+126+x12+246+x15+154t−4+x15+248+x15+2716t−24+x18+242+x18+278t−12+x18+306t2−20t+16+x24+278+x27+278t−14+x27+308t−16+x30+306t2−24t+24.
By doing some calculations, we obtain(44)M1G5G1,x=6x24+4t−4x30+6x36+8x39+16t−22x42+8t−12x45+6t2−20t+16x48+8x51+8t−14x54+8t−16x57+6t2−24t+24x60.
Also, from [4],(45)M2G5G1,x=∑rs∈EG1xSGr+SGs,M2G5G1,x=x6×6|E1G1t|+x6×11|E2G1t|+x6×12|E3G1t|+x6×14|E4G1t|+x7×9|E5G1t|+x7×12|E6G1t|+x8×11|E7G1t|+x8×13|E8G1t|+x9×9|E9G1t|+x9×14|E10G1t|+x11×11|E11G1t|+x11×12|E12G1t|+x11×13|E13G1t|+x11×14|E14G1t|+x11×16|E15G1t|+x12×14|E16G1t|+x13×14|E17G1t|+x13×16|E18G1t|+x14×14|E19G1t|+x14×16|E20G1t|,=x6×64t+x6×114t+x6×124+x6×144t−4+x7×94t−4+x7×124t−4+x8×1112t−8+x8×134t−4+x9×92t−2+x9×144t−4+x11×119t2−7t+3+x11×124+x11×134t−4+x11×1436t2−68t+32+x11×164t−4+x12×144t−4+x13×144t−4+x13×164t−4+x14×144t−4+x14×1636t2−76t+40.
By making some calculations, we obtain(46)M2G5G1,x=6x144+4t−4x225+6x288+8x360+82t−3x405+2x432+42t−3x486+2t−23t−4x540+8x648+24t−7x729+8t−2x810+6t−22x900.Theorem 2.1.7.
LetG1≅RHSLt be the rhombus-type silicate network; then, hyper-fifth M-Zagreb polynomials of first and second type are equal to(47)HM1G5G1,x=6x576+4t−4x900+6x1296+8x1521+16t−22x1764+8t−12x2025+6t2−20t+16x2304+8x2601+8t−14x2916+8t−16x3249+6t2−24t+24x3600,HM2G5G1,x=6x20736+4t−4x50625+6x82944+8x129600+16t−24x164025+2x186624+8t−12x236196+6t2−20t+16x291600+8x419904+8t−14x531441+8t−16x656100+6t2−24t+24x810000.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [2] that(48)HM1G5G1,x=∑rs∈EG1xSGr+SGs2,HM1G5G1,x=x12+122|E1G1t|+x12+242|E2G1t|+x15+152|E3G1t|+x15+242|E4G1t|+x15+272|E5G1t|+x18+242|E6G1t|+x18+272|E7G1t|+x18+302|E8G1t|+x24+272|E9G1t|+x27+272|E10G1t|+x27+302|E11G1t|+x30+302|E12G1t|,=x12+1226+x12+2426+x15+1524t−4+x15+2428+x15+27216t−24+x18+2422+x18+2728t−12+x18+3026t2−20t+16+x24+2728+x27+2728t−14+x27+3028t−16+x30+3026t2−24t+24.
By doing some calculations, we obtain(49)HM1G5G1,x=6x576+4t−4x900+6x1296+8x1521+16t−22x1764+8t−12x2025+6t2−20t+16x2304+8x2601+8t−14x2916+8t−16x3249+6t2−24t+24x3600.
Also, from [10],(50)HM2aG5G1,x=∑rs∈EG1xSGr×SGs2,HM2aG5G1,x=x12×122|E1G1t|+x12×242|E2G1t|+x15×152|E3G1t|+x15×242|E4G1t|+x15×272|E5G1t|+x18×242|E6G1t|+x18×272|E7G1t|+x18×302|E8G1t|+x24×272|E9G1t|+x27×272|E10G1t|+x27×302|E11G1t|+x30×302|E12G1t|,=x12×1226+x12×2426+x15×1524t−4+x15×2428+x15×27216t−24+x18×2422+x18×2728t−12+x18×3026t2−20t+16+x24×2728+x27×2728t−14+x27×3028t−16+x30×3026t2−24t+24.
By making some calculations, we obtain(51)HM2G5G1,x=6x20736+4t−4x50625+6x82944+8x129600+16t−24x164025+2x186624+8t−12x236196+6t2−20t+16x291600+8x419904+8t−14x531441+8t−16x656100+6t2−24t+24x810000.


### 2.2. Results for the Rhombus Type of Oxide Networks

Now, we are calculating fifth M-Zagreb topological indices of the rhombus-type oxide networkG2≅RHOXt, where t∈ℕ.Theorem 2.2.1.
LetG2≅RHOXt be the rhombus-type silicate network; then, the first and second fifth M-Zagreb indices are equal to(52)M1G5G2=161−8t+12t2,M2G5G2=162t−148t−35.Proof.
The outcome can be obtained by using the edge partition in Table2.
By using equation [5],(53)M1G5G2=∑rs∈EG2SGr+SGs,M1G5G2=6+6|E1G2t|+6+12|E2G2t|+8+12|E3G2t|+8+14|E4G2t|+12+14|E5G2t|+14+14|E6G2t|+14+16|E7G2t|+16+16|E8G2t|,=6+62+6+124+8+124+8+148t−12+12+148+14+148t−14+14+168t−16+16+166t−22.
By doing some calculations, we obtain(54)M1G5G2=161−8t+12t2.
Thus, from [6],(55)M2G5G=∑rs∈EGSGr+SGs,M2G5G2=6+6|E1G2t|+6+12|E2G2t|+8+12|E3G2t|+8+14|E4G2t|+12+14|E5G2t|+14+14|E6G2t|+14+16|E7G2t|+16+16|E8G2t|,=6+62+6+124+8+124+8+148t−12+12+148+14+148t−14+14+168t−16+16+166t−22,
By doing some calculations, we obtain(56)M2G5G2=162t−148t−35.Table 2
Edge partition of rhombus-type oxide network (RHOXt) based on sum of degrees of end vertices of each edge.
Sr,SsNumber of edgesWherers∈EG26,626,1248,1248,1442t−3Wherers∈EG212,14814,1424t−714,168t−216,166t−22Theorem 2.2.2.
Consider the rhombus-type oxide networkG2≅RHOXt for t∈ℕ. Then, the first and second general fifth M-Zagreb indices are equal to(57)M1aG5G2=2a3×23+4a+21+a3a+22+a5a+4×9a−12×11a+8×13a−141+a−16×15a+2at−3×23+4a+23+a7a+8×11a+8×15a+3×21+5at2,M2aG5G2=3×23+8a+22+5a3a−24+5a7a−3×41+2a7a+21+2a9a+22+3a9a−141+2a+81+a21a+t−3×23+8a+23+4a7a+23+5a7a+23+2a49a+3×21+8at2.Proof.
LetG2 be the rhombus-type oxide network. Table 2 shows such an edge partition of RHOXt. Thus, from [9], it follows that(58)M1aG5G=∑rs∈EGSGr+SGsa.By using edge partitions in Table 2, we obtain(59)M1aG5G2=6+6a|E1G2t|+6+12a|E2G2t|+8+12a|E3G2t|+8+14a|E4G2t|+12+14a|E5G2t|+14+14a|E6G2t|+14+16a|E7G2t|+16+16a|E8G2t|,=6+6a2+6+12a4+8+12a4+8+14a8t−12+12+14a8+14+14a8t−14+14+16a8t−16+16+16a6t−22.
By doing some calculations, we have(60)M1aG5G2=2a3×23+4a+21+a3a+22+a5a+4×9a−12×11a+8×13a−141+a−16×15a+2at−3×23+4a+23+a7a+8×11a+8×15a+3×21+5at2.
From [12], we have(61)M2aG5G=∑rs∈EGSGr+SGsa.
By using edge partitions in Table2, we obtain(62)M2aG5G2=6+6a|E1G2t|+6+12a|E2G2t|+8+12a|E3G2t|+8+14a|E4G2t|+12+14a|E5G2t|+14+14a|E6G2t|+14+16a|E7G2t|+16+16a|E8G2t|,=6+6a2+6+12a4+8+12a4+8+14a8t−12+12+14a8+14+14a8t−14+14+16a8t−16+16+16a6t−22.
By doing some calculations, we have(63)M2aG5G2=3×23+8a+22+5a3a−24+5a7a−3×41+2a7a+21+2a9a+22+3a9a−141+2a+81+a21a+t−3×23+8a+23+4a7a+23+5a7a+23+2a49a+3×21+8at2.Theorem 2.2.3.
Consider the rhombus-type oxide networkG2≅RHOXt for t∈ℕ. Then, the first and second hyper fifth M-Zagreb indices are equal to(64)HM1aG5G2=6431−113t+96t2,HM2aG5G2=1921915−3978t+2048t2.Proof.
LetG2 be the rhombus-type oxide network. Table 2 shows such an edge partition of RHOXt. Thus, from [13], it follows that(65)HM1G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table2, we obtain(66)HM1G5G2=6+62|E1G2t|+6+122|E2G2t|+8+122|E3G2t|+8+142|E4G2t|+12+142|E5G2t|+14+142|E6G2t|+14+162|E7G2t|+16+162|E8G2t|,=6+622+6+1224+8+1224+8+1428t−12+12+1428+14+1428t−14+14+1628t−16+16+1626t−22.
By doing some calculations, we have(67)HM1G5G2=6431−113t+96t2.
From [14], we have(68)HM2G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table2, we obtain(69)HM2G5G2=6+62|E1G2t|+6+122|E2G2t|+8+122|E3G2t|+8+142|E4G2t|+12+142|E5G2t|+14+142|E6G2t|+14+162|E7G2t|+16+162|E8G2t|,=6+622+6+1224+8+1224+8+1428t−12+12+1428+14+1428t−14+14+1628t−16+16+1626t−22.
By doing some calculations, we have(70)HM2G5G2=1921915−3978t+2048t2.Theorem 2.2.4.
Consider the rhombus-type silicate networkG2≅RHOXt for t∈ℕ. Then, the third M-Zagreb index is equal to(71)M3G5G2=−48+64t.Proof.
LetG2 be the rhombus-type oxide network. Table 2 shows such an edge partition of RHOXt. Thus, from [15], it follows that(72)M3G5G=∑rs∈EGSGr−SGs.
By using edge partitions in Table2, we obtain(73)M3G5G2=|6−6E1G2t|+|6−12E2G2t|+|8−12E3G2t|+|8−14E4G2t|+|12−14E5G2t|+|14−14E6G2t|+|14−16E7G2t|+|16−16E8G2t|,=|6−6|2+|6−12|4+|8−12|4+|8−14|8t−12+|12−14|8+|14−14|8t−14+|14−16|8t−16+|16−16|6t−22.
By doing some calculations, we have(74)M3G5G2=−48+64t.Corresponding to the above indices, we are going to compute general fifth M-Zagreb polynomials for rhombus type of oxide networkRHOXt.Theorem 2.2.5.
LetG2≅RHOXt be the rhombus-type oxide network; then, general fifth M-Zagreb polynomials of first and second type are equal to(75)M1aG5G2,x=2x12a+4x18a+4x20a+42t−3x22a+8x26a+24t−7x28a+8t−2x30a+6t−22x32a,M2aG5G2,x=2x36a+4x72a+4x96a+42t−3x112a+8x168a+24t−7x196a+8t−2x224a+6t−22x256a.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [1] that(76)M1aG5G2,x=∑rs∈EG2xSGr+SGsa,M1aG5G2,x=x6+6a|E1G2t|+x6+12a|E2G2t|+x8+12a|E3G2t|+x8+14a|E4G2t|+x12+14a|E5G2t|+x14+14a|E6G2t|+x14+16a|E7G2t|+x16+16a|E8G2t|,=x6+6a2+x6+12a4+x8+12a4+x8+14a8t−12+x12+14a8+x14+14a8t−14+x14+16a8t−16+x16+16a6t−22.
By doing some calculations, we obtain(77)M1aG5G2,x=2x12a+4x18a+4x20a+42t−3x22a+8x26a+24t−7x28a+8t−2x30a+6t−22x32a.
Also, from [3],(78)M2aG5G2,x=∑rs∈EG2xSGr+SGsa,M2aG5G2,x=x6+6a|E1G2t|+x6+12a|E2G2t|+x8+12a|E3G2t|+x8+14a|E4G2t|+x12+14a|E5G2t|+x14+14a|E6G2t|+x14+16a|E7G2t|+x16+16a|E8G2t|,=x6+6a2+x6+12a4+x8+12a4+x8+14a8t−12+x12+14a8+x14+14a8t−14+x14+16a8t−16+x16+16a6t−22.
By making some calculations, we obtain(79)M2aG5G2,x=2x36a+4x72a+4x96a+42t−3x112a+8x168a+24t−7x196a+8t−2x224a+6t−22x256a.Corresponding to the above indices, we are going to compute fifthM-Zagreb polynomials for rhombus-type oxide network RHOXt.Theorem 2.2.6.
LetG2≅RHOXt be the rhombus-type oxide network; then, fifth M-Zagreb polynomials of first and second type are equal to(80)M1aG5G2,x=2x12+4x18+4x20+42t−3x22+8x26+24t−7x28+8t−2x30+6t−22x32,M2G5G2,x=2x36+4x72+4x96+42t−3x112+8x168+24t−7x196+8t−2x224+6t−22x256.Proof.
We obtain the outcome with the edge partition in Table2. It follows from [16] that(81)M1G5G2,x=∑rs∈EG2xSGr+SGs,M1G5G2,x=x6+6|E1G2t|+x6+12|E2G2t|+x8+12|E3G2t|+x8+14|E4G2t|+x12+14|E5G2t|+x14+14|E6G2t|+x14+16|E7G2t|+x16+16|E8G2t|,=x6+62+x6+124+x8+124+x8+148t−12+x12+148+x14+148t−14+x14+168t−16+x16+166t−22.
By doing some calculations, we obtain(82)M1aG5G2,x=2x12+4x18+4x20+42t−3x22+8x26+24t−7x28+8t−2x30+6t−22x32.
Also, from [4],(83)M2G5G2,x=∑rs∈EG2xSGr+SGs,M2G5G2,x=x6×6|E1G2t|+x6×12|E2G2t|+x8×12|E3G2t|+x8×14|E4G2t|+x12×14|E5G2t|+x14×14|E6G2t|+x14×16|E7G2t|+x16×16|E8G2t|,=x6×62+x6×124+x8×124+x8×148t−12+x12×144+x14×148t−14+x14×168t−16+x16×166t−22.
By making some calculations, we obtain(84)M2G5G2,x=2x36+4x72+4x96+42t−3x112+8x168+24t−7x196+8t−2x224+6t−22x256.Theorem 2.2.7.
LetG2≅RHOXt be the rhombus-type oxide network; then, hyper-fifth M-Zagreb polynomials of first and second type are equal to(85)HM1G5G2,x=2x144+4x324+4x400+42t−3x484+8x676+24t−7x784+8t−2x900+6t−22x1024,HM2G5G2,x=2x1296+4x5184+4x9216+42t−3x12544+8x28224+24t−7x38416+8t−2x50176+6t−22x65536.Proof.
We obtain the outcome with the edge partition in Table2. It follows from [2] that(86)HM1G5G2,x=∑rs∈EG2xSGr+SGs2,HM1G5G2,x=x6+62|E1G2t|+x6+122|E2G2t|+x8+122|E3G2t|+x8+142|E4G2t|+x12+142|E5G2t|+x14+142|E6G2t|+x14+162|E7G2t|+x16+162|E8G2t|=x6+622+x6+1224+x8+1224+x8+1428t−12+x12+1428+x14+1428t−14+x14+1628t−16+x16+1626t−22.
By doing some calculations, we obtain(87)HM1G5G2,x=2x144+4x324+4x400+42t−3x484+8x676+24t−7x784+8t−2x900+6t−22x1024.
Also, from [10],(88)HM2aG5G2,x=∑rs∈EG2xSGr+SGs2,HM2G5G2,x=x6×62|E1G2t|+x6×122|E2G2t|+x8×122|E3G2t|+x8×142|E4G2t|+x12×142|E5G2t|+x14×142|E6G2t|+x14×162|E7G2t|+x16×162|E8G2t|,=x6×622+x6×1224+x8×1224+x8×1428t−12+x12×1424+x14×1428t−14+x14×1628t−16+x16×1626t−22.
By making some calculations, we obtain(89)HM2G5G2,x=2x1296+4x5184+4x9216+42t−3x12544+8x28224+24t−7x38416+8t−2x50176+6t−22x65536.


## 2.1. Results for the Rhombus Type of Silicate Networks

In this section, we calculate degree-based topological indices of the dimensiont for rhombus-type silicate networks. In the following theorems, we compute M-Zagreb indices and polynomials.Theorem 2.1.1.
LetG1≅RHSLt be the rhombus-type silicate network; then, the first and second fifth M-Zagreb indices are equal to(14)M1G5G1=361−10t+18t2,M2G5G1=18119−490t+480t2.Proof.
The outcome can be obtained by using the edge partition in Table1.
By using equation [5],(15)M1G5G1=∑rs∈EG1SGr+SGs,M1G5G1=12+12|E1G1t|+12+24|E2G1t|+15+15|E3G1t|+15+24|E4G1t|+15+27|E5G1t|+18+24|E6G1t|+18+27|E7G1t|+18+30|E8G1t|+24+27|E9G1t|+27+27|E10G1t|+27+30|E11G1t|+30+30|E12G1t|,=12+126+12+246+15+154t−4+15+248+15+2716t−24+18+242+18+278t−12+18+306t2−20t+16+24+278+27+278t−14+27+308t−16+30+306t2−24t+24.
By doing some calculations, we obtain(16)M1G5G1=361−10t+18t2.
Thus, from [6],(17)M2G5G=∑rs∈EG1SGr+SGs,M2G5G1=12×12|E1G1t|+12×24|E2G1t|+15×15|E3G1t|+15×24|E4G1t|+15×27|E5G1t|+18×24|E6G1t|+18×27|E7G1t|+18×30|E8G1t|+24×27|E9G1t|+27×27|E10G1t|+27×30|E11G1t|+30×30|E12G1t|,=12×126+12×246+15×154t−4+15×248+15×2716t−24+18×242+18×278t−12+18×306t2−20t+16+24×278+27×278t−14+27×308t−16+30×306t2−24t+24.
By doing some calculations, we obtain(18)M2G5G1=18119−490t+480t2.Table 1
Edge partition of rhombus-type silicate network (RHSLt) based on sum of degrees of end vertices of each edge.
Sr,SsNumber of edgesWherers∈EG112,12612,24615,154t−415,24815,2716t−2418,242Wherers∈EG118,278t−218,306t2−20t+1624,27827,278t−1427,308t−1630,306t2−24t+24Theorem 2.1.2.
Consider the rhombus-type silicate networkG1≅RHSLt for t∈ℕ. Then, the first and second general fifth M-Zagreb indices are equal to(19)M1aG5G1=21+3a31+a+23+2a31+a5a−4×31+2a5a+61+2a−23+a31+a7a−22+a15a+3a161+a+21+a21a−7×21+a27a+8×39a+8×51a−16×57a+t−5×3a41+2a−23+2a31+a5a+22+a15a+24+a21a+23+a27a+8×45a+8×57a+t221+4a3+21+2a31+a5a,M2aG5G1=21+4a31+2a+21+5a31+2a−22+a31+5a−8×31+4a5a+23+2a31+2a25a+21+4a27a+81+a45a+81+a81a+42+a135a−4×225a−24+a405a−14×729a+t−23+2a31+2a25a−201+a27a+4×225a+23+a243a+16×405a+23+a405a+8×729a+t221+2a31+3a5a+61+2a25a.Proof.
LetG1 be the rhombus-type silicate network. Table 1 shows such an edge partition of RHSLt. Thus, from [9], it follows that(20)M1aG5G=∑rs∈EGSGr+SGsa.
By using edge partitions in Table1, we obtain(21)M1aG5G1=12+12a|E1G1t|+12+24a|E2G1t|+15+15a|E3G1t|+15+24a|E4G1t|+15+27a|E5G1t|+18+24a|E6G1t|+18+27a|E7G1t|+18+30a|E8G1t|+24+27a|E9G1t|+27+27a|E10G1t|+27+30a|E11G1t|+30+30a|E12G1t|,=12+12a6+12+24a6+15+15a4t−4+15+24a8+15+24a16t−24+18+24a2+18+27a8t−12+18+30a6t2−20t+16+24+27a8+27+27a8t−14+27+30a8t−16+30+30a6t2−24t+24.
By doing some calculations, we have(22)M1aG5G1=21+3a31+a+23+2a31+a5a−4×31+2a5a+61+2a−23+a31+a7a−22+a15a+3a161+a+21+a21a−7×21+a27a+8×39a+8×51a−16×57a+t−5×3a41+2a−23+2a31+a5a+22+a15a+24+a21a+23+a27a+8×45a+8×57a+t221+4a31+a+21+2a31+a5a.
From [12], we have(23)M2aG5G=∑rs∈EGSGr+SGsa.
By using edge partitions in Table1, we obtain(24)M2aG5G1=12×12a|E1G1t|+12×24a|E2G1t|+15×15a|E3G1t|+15×24a|E4G1t|+15×27a|E5G1t|+18×24a|E6G1t|+18×27a|E7G1t|+18×30a|E8G1t|+24×27a|E9G1t|+27×27a|E10G1t|+27×30a|E11G1t|+30×30a|E12G1t|,=12×12a6+12×24a6+15×15a4t−4+15×24a8+15×24a16t−24+18×24a2+18×27a8t−12+18×30a6t2−20t+16+24×27a8+27×27a8t−14+27×30a8t−16+30×30a6t2−24t+24.
By doing some calculations, we have(25)M2aG5G1=21+4a31+2a+21+5a31+2a−22+a31+5a−8×31+4a5a+23+2a31+2a25a+21+4a27a+81+a45a+81+a81a+42+a135a−4×225a−24+a405a−14×729a+t−23+2a31+2a25a−201+a27a+4×225a+23+a243a+16×405a+23+a405a+8×729a+t221+2a31+3a5a+61+2a25a.Theorem 2.1.3.
Consider the rhombus-type silicate networkG1≅RHSLt for t∈ℕ. Then, the first and second hyper-fifth M-Zagreb indices are equal to(26)HM1G5G1=36221−976t+984t2,HM2G5G1=16228307−68242t+40800t2.Proof.
LetG1 be the rhombus type of silicate network. Table 1 shows such an edge partition of RHSLt. Thus, from [13], it follows that(27)HM1G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table1, we obtain(28)HM1G5G1=12×122|E1G1t|+12×242|E2G1t|+15×152|E3G1t|+15×242|E4G1t|+15×272|E5G1t|+18×242|E6G1t|+18×272|E7G1t|+18×302|E8G1t|+24×272|E9G1t|+27×272|E10G1t|+27×302|E11G1t|+30×302|E12G1t|,=12×1226+12×2426+15×1524t−4+15×2428+15×24216t−24+18×2422+18×2728t−12+18×3026t2−20t+16+24×2728+27×2728t−14+27×3028t−16+30×3026t2−24t+24.
By doing some calculations, we have(29)HM1G5G1=36221−976t+984t2.
From [14], we have(30)HM22G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table1, we obtain(31)HM2G5G1=12×122|E1G1t|+12×242|E2G1t|+15×152|E3G1t|+15×242|E4G1t|+15×272|E5G1t|+18×242|E6G1t|+18×272|E7G1t|+18×302|E8G1t|+24×272|E9G1t|+27×272|E10G1t|+27×302|E11G1t|+30×302|E12G1t|,=12×1226+12×2426+15×1524t−4+15×2428+15×24216t−24+18×2422+18×2728t−12+18×3026t2−20t+16+24×2728+27×2728t−14+27×3028t−16+30×3026t2−24t+24.
By doing some calculations, we have(32)HM2aG5G1=16228307−68242t+40800t2.Theorem 2.1.4.
Consider the rhombus-type silicate networkG1≅RHSLt for t∈ℕ. Then, the third M-Zagreb index is equal to(33)M3G5G1=−232+248t+12t2.Proof.
LetG1 be the rhombus silicate network. Table 1 shows such an edge partition of RHSLt. Thus, from [15], it follows that(34)M3G5G=∑rs∈EGSGr+SGs.
By using edge partitions in Table1, we obtain(35)M3G5G1=|12−12E1G1t|+|12−24E2G1t|+|15−15E3G1t|+|15−24E4G1t|+|15−27E5G1t|+|18−24E6G1t|+|18−27E7G1t|+|18−30E8G1t|+|24−27E9G1t|+|27−27E10G1t|+|27−30E11G1t|+|30−30E12G1t|,=|12−12|6+|12−24|6+|15−15|4t−4+|15−24|8+|15−27|16t−24+|18−24|2+|18−27|8t−12+|18−30|6t2−20t+16+|24−27|8+|27−27|8t−14+|27−30|8t−16+|30−30|6t2−24t+24.
By doing some calculations, we have(36)M3G5G1=−232+248t+12t2.Corresponding to the above indices, we are going to compute general fifthM-Zagreb polynomials for rhombus-type silicate network RHSLt.Theorem 2.1.5.
LetG1≅RHSLt be the first type of rhombus-type silicate network; then, general fifth M-Zagreb polynomials of first and second type are equal to(37)M1aG5G1,x=6x24a+4t−4x30a+6x36a+8x39a+16t−22x42a+8t−12x45a+6t2−20t+16x48a+8x51a+8t−14x54a+8t−16x57a+6t2−24t+24x60a,M2aG5G1,x=6x144a+4t−4x225a+6x288a+8x360a+82t−3x405a+2x432a+42t−3x486a+2t−23t−4x540a+8x648a+24t−7x729a+8t−2x810a+6t−22x900a.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [1] that(38)M1aG5G1,x=∑rs∈EG1xSGr+SGsa,M1aG5G1,x=x12+12a|E1G1t|+x12+24a|E2G1t|+x15+15a|E3G1t|+x15+24a|E4G1t|+x15+27a|E5G1t|+x18+24a|E6G1t|+x18+27a|E7G1t|+x18+30a|E8G1t|+x24+27a|E9G1t|+x27+27a|E10G1t|+x27+30a|E11G1t|+x30+30a|E12G1t|,M1aG5G1,x=x12+12a6+x12+24a6+x15+15a4t−4+x15+24a8+x15+27a16t−24+x18+24a2+x18+27a8t−12+x18+30a6t2−20t+16+x24+27a8+x27+27a8t−14+x27+30a8t−16+x30+30a6t2−24t+24.
By doing some calculations, we obtain(39)M1aG5G1,x=6x24a+4t−4x30a+6x36a+8x39a+16t−22x42a+8t−12x45a+6t2−20t+16x48a+8x51a+8t−14x54a+8t−16x57a+6t2−24t+24x60a.
Also, from [3],(40)M2aG5G1,x=∑rs∈EG1xSGr+SGsa,M2aG5G1,x=x12+12a|E1G1t|+x12+24a|E2G1t|+x15+15a|E3G1t|+x15+24a|E4G1t|+x15+27a|E5G1t|+x18+24a|E6G1t|+x18+27a|E7G1t|+x18+30a|E8G1t|+x24+27a|E9G1t|+x27+27a|E10G1t|+x27+30a|E11G1t|+x30+30a|E12G1t|,M2aG5G1,x=x12+12a6+x12+24a6+x15+15a4t−4+x15+24a8+x15+27a16t−24+x18+24a2+x18+27a8t−12+x18+30a6t2−20t+16+x24+27a8+x27+27a8t−14+x27+30a8t−16+x30+30a6t2−24t+24.
By making some calculations, we obtain(41)M2aG5G1,x=6x144a+4t−4x225a+6x288a+8x360a+82t−3x405a+2x432a+42t−3x486a+2t−23t−4x540a+8x648a+24t−7x729a+8t−2x810a+6t−22x900a.Corresponding to the above indices, we are going to compute fifthM-Zagreb polynomials for rhombus-type silicate network RHSLt.Theorem 2.1.6.
LetG1≅RHSLt be the rhombus type of silicate network; then, fifth M-Zagreb polynomials of first and second type are equal to(42)M1G5G1,x=6x24+4t−4x30+6x36+8x39+16t−22x42+8t−12x45+6t2−20t+16x48+8x51+8t−14x54+8t−16x57+6t2−24t+24x60,M2G5G1,x=6x144+4t−4x225+6x288+8x360+82t−3x405+2x432+42t−3x486+2t−23t−4x540+8x648+24t−7x729+8t−2x810+6t−22x900.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [16] that(43)M1G5G1,x=∑rs∈EG1xSGr+SGs,M1G5G1,x=x12+12|E1G1t|+x12+24|E2G1t|+x15+15|E3G1t|+x15+24|E4G1t|+x15+27|E5G1t|+x18+24|E6G1t|+x18+27|E7G1t|+x18+30|E8G1t|+x24+27|E9G1t|+x27+27|E10G1t|+x27+30|E11G1t|+x30+30|E12G1t|,=x12+126+x12+246+x15+154t−4+x15+248+x15+2716t−24+x18+242+x18+278t−12+x18+306t2−20t+16+x24+278+x27+278t−14+x27+308t−16+x30+306t2−24t+24.
By doing some calculations, we obtain(44)M1G5G1,x=6x24+4t−4x30+6x36+8x39+16t−22x42+8t−12x45+6t2−20t+16x48+8x51+8t−14x54+8t−16x57+6t2−24t+24x60.
Also, from [4],(45)M2G5G1,x=∑rs∈EG1xSGr+SGs,M2G5G1,x=x6×6|E1G1t|+x6×11|E2G1t|+x6×12|E3G1t|+x6×14|E4G1t|+x7×9|E5G1t|+x7×12|E6G1t|+x8×11|E7G1t|+x8×13|E8G1t|+x9×9|E9G1t|+x9×14|E10G1t|+x11×11|E11G1t|+x11×12|E12G1t|+x11×13|E13G1t|+x11×14|E14G1t|+x11×16|E15G1t|+x12×14|E16G1t|+x13×14|E17G1t|+x13×16|E18G1t|+x14×14|E19G1t|+x14×16|E20G1t|,=x6×64t+x6×114t+x6×124+x6×144t−4+x7×94t−4+x7×124t−4+x8×1112t−8+x8×134t−4+x9×92t−2+x9×144t−4+x11×119t2−7t+3+x11×124+x11×134t−4+x11×1436t2−68t+32+x11×164t−4+x12×144t−4+x13×144t−4+x13×164t−4+x14×144t−4+x14×1636t2−76t+40.
By making some calculations, we obtain(46)M2G5G1,x=6x144+4t−4x225+6x288+8x360+82t−3x405+2x432+42t−3x486+2t−23t−4x540+8x648+24t−7x729+8t−2x810+6t−22x900.Theorem 2.1.7.
LetG1≅RHSLt be the rhombus-type silicate network; then, hyper-fifth M-Zagreb polynomials of first and second type are equal to(47)HM1G5G1,x=6x576+4t−4x900+6x1296+8x1521+16t−22x1764+8t−12x2025+6t2−20t+16x2304+8x2601+8t−14x2916+8t−16x3249+6t2−24t+24x3600,HM2G5G1,x=6x20736+4t−4x50625+6x82944+8x129600+16t−24x164025+2x186624+8t−12x236196+6t2−20t+16x291600+8x419904+8t−14x531441+8t−16x656100+6t2−24t+24x810000.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [2] that(48)HM1G5G1,x=∑rs∈EG1xSGr+SGs2,HM1G5G1,x=x12+122|E1G1t|+x12+242|E2G1t|+x15+152|E3G1t|+x15+242|E4G1t|+x15+272|E5G1t|+x18+242|E6G1t|+x18+272|E7G1t|+x18+302|E8G1t|+x24+272|E9G1t|+x27+272|E10G1t|+x27+302|E11G1t|+x30+302|E12G1t|,=x12+1226+x12+2426+x15+1524t−4+x15+2428+x15+27216t−24+x18+2422+x18+2728t−12+x18+3026t2−20t+16+x24+2728+x27+2728t−14+x27+3028t−16+x30+3026t2−24t+24.
By doing some calculations, we obtain(49)HM1G5G1,x=6x576+4t−4x900+6x1296+8x1521+16t−22x1764+8t−12x2025+6t2−20t+16x2304+8x2601+8t−14x2916+8t−16x3249+6t2−24t+24x3600.
Also, from [10],(50)HM2aG5G1,x=∑rs∈EG1xSGr×SGs2,HM2aG5G1,x=x12×122|E1G1t|+x12×242|E2G1t|+x15×152|E3G1t|+x15×242|E4G1t|+x15×272|E5G1t|+x18×242|E6G1t|+x18×272|E7G1t|+x18×302|E8G1t|+x24×272|E9G1t|+x27×272|E10G1t|+x27×302|E11G1t|+x30×302|E12G1t|,=x12×1226+x12×2426+x15×1524t−4+x15×2428+x15×27216t−24+x18×2422+x18×2728t−12+x18×3026t2−20t+16+x24×2728+x27×2728t−14+x27×3028t−16+x30×3026t2−24t+24.
By making some calculations, we obtain(51)HM2G5G1,x=6x20736+4t−4x50625+6x82944+8x129600+16t−24x164025+2x186624+8t−12x236196+6t2−20t+16x291600+8x419904+8t−14x531441+8t−16x656100+6t2−24t+24x810000.


## 2.2. Results for the Rhombus Type of Oxide Networks

Now, we are calculating fifth M-Zagreb topological indices of the rhombus-type oxide networkG2≅RHOXt, where t∈ℕ.Theorem 2.2.1.
LetG2≅RHOXt be the rhombus-type silicate network; then, the first and second fifth M-Zagreb indices are equal to(52)M1G5G2=161−8t+12t2,M2G5G2=162t−148t−35.Proof.
The outcome can be obtained by using the edge partition in Table2.
By using equation [5],(53)M1G5G2=∑rs∈EG2SGr+SGs,M1G5G2=6+6|E1G2t|+6+12|E2G2t|+8+12|E3G2t|+8+14|E4G2t|+12+14|E5G2t|+14+14|E6G2t|+14+16|E7G2t|+16+16|E8G2t|,=6+62+6+124+8+124+8+148t−12+12+148+14+148t−14+14+168t−16+16+166t−22.
By doing some calculations, we obtain(54)M1G5G2=161−8t+12t2.
Thus, from [6],(55)M2G5G=∑rs∈EGSGr+SGs,M2G5G2=6+6|E1G2t|+6+12|E2G2t|+8+12|E3G2t|+8+14|E4G2t|+12+14|E5G2t|+14+14|E6G2t|+14+16|E7G2t|+16+16|E8G2t|,=6+62+6+124+8+124+8+148t−12+12+148+14+148t−14+14+168t−16+16+166t−22,
By doing some calculations, we obtain(56)M2G5G2=162t−148t−35.Table 2
Edge partition of rhombus-type oxide network (RHOXt) based on sum of degrees of end vertices of each edge.
Sr,SsNumber of edgesWherers∈EG26,626,1248,1248,1442t−3Wherers∈EG212,14814,1424t−714,168t−216,166t−22Theorem 2.2.2.
Consider the rhombus-type oxide networkG2≅RHOXt for t∈ℕ. Then, the first and second general fifth M-Zagreb indices are equal to(57)M1aG5G2=2a3×23+4a+21+a3a+22+a5a+4×9a−12×11a+8×13a−141+a−16×15a+2at−3×23+4a+23+a7a+8×11a+8×15a+3×21+5at2,M2aG5G2=3×23+8a+22+5a3a−24+5a7a−3×41+2a7a+21+2a9a+22+3a9a−141+2a+81+a21a+t−3×23+8a+23+4a7a+23+5a7a+23+2a49a+3×21+8at2.Proof.
LetG2 be the rhombus-type oxide network. Table 2 shows such an edge partition of RHOXt. Thus, from [9], it follows that(58)M1aG5G=∑rs∈EGSGr+SGsa.By using edge partitions in Table 2, we obtain(59)M1aG5G2=6+6a|E1G2t|+6+12a|E2G2t|+8+12a|E3G2t|+8+14a|E4G2t|+12+14a|E5G2t|+14+14a|E6G2t|+14+16a|E7G2t|+16+16a|E8G2t|,=6+6a2+6+12a4+8+12a4+8+14a8t−12+12+14a8+14+14a8t−14+14+16a8t−16+16+16a6t−22.
By doing some calculations, we have(60)M1aG5G2=2a3×23+4a+21+a3a+22+a5a+4×9a−12×11a+8×13a−141+a−16×15a+2at−3×23+4a+23+a7a+8×11a+8×15a+3×21+5at2.
From [12], we have(61)M2aG5G=∑rs∈EGSGr+SGsa.
By using edge partitions in Table2, we obtain(62)M2aG5G2=6+6a|E1G2t|+6+12a|E2G2t|+8+12a|E3G2t|+8+14a|E4G2t|+12+14a|E5G2t|+14+14a|E6G2t|+14+16a|E7G2t|+16+16a|E8G2t|,=6+6a2+6+12a4+8+12a4+8+14a8t−12+12+14a8+14+14a8t−14+14+16a8t−16+16+16a6t−22.
By doing some calculations, we have(63)M2aG5G2=3×23+8a+22+5a3a−24+5a7a−3×41+2a7a+21+2a9a+22+3a9a−141+2a+81+a21a+t−3×23+8a+23+4a7a+23+5a7a+23+2a49a+3×21+8at2.Theorem 2.2.3.
Consider the rhombus-type oxide networkG2≅RHOXt for t∈ℕ. Then, the first and second hyper fifth M-Zagreb indices are equal to(64)HM1aG5G2=6431−113t+96t2,HM2aG5G2=1921915−3978t+2048t2.Proof.
LetG2 be the rhombus-type oxide network. Table 2 shows such an edge partition of RHOXt. Thus, from [13], it follows that(65)HM1G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table2, we obtain(66)HM1G5G2=6+62|E1G2t|+6+122|E2G2t|+8+122|E3G2t|+8+142|E4G2t|+12+142|E5G2t|+14+142|E6G2t|+14+162|E7G2t|+16+162|E8G2t|,=6+622+6+1224+8+1224+8+1428t−12+12+1428+14+1428t−14+14+1628t−16+16+1626t−22.
By doing some calculations, we have(67)HM1G5G2=6431−113t+96t2.
From [14], we have(68)HM2G5G=∑rs∈EGSGr+SGs2.
By using edge partitions in Table2, we obtain(69)HM2G5G2=6+62|E1G2t|+6+122|E2G2t|+8+122|E3G2t|+8+142|E4G2t|+12+142|E5G2t|+14+142|E6G2t|+14+162|E7G2t|+16+162|E8G2t|,=6+622+6+1224+8+1224+8+1428t−12+12+1428+14+1428t−14+14+1628t−16+16+1626t−22.
By doing some calculations, we have(70)HM2G5G2=1921915−3978t+2048t2.Theorem 2.2.4.
Consider the rhombus-type silicate networkG2≅RHOXt for t∈ℕ. Then, the third M-Zagreb index is equal to(71)M3G5G2=−48+64t.Proof.
LetG2 be the rhombus-type oxide network. Table 2 shows such an edge partition of RHOXt. Thus, from [15], it follows that(72)M3G5G=∑rs∈EGSGr−SGs.
By using edge partitions in Table2, we obtain(73)M3G5G2=|6−6E1G2t|+|6−12E2G2t|+|8−12E3G2t|+|8−14E4G2t|+|12−14E5G2t|+|14−14E6G2t|+|14−16E7G2t|+|16−16E8G2t|,=|6−6|2+|6−12|4+|8−12|4+|8−14|8t−12+|12−14|8+|14−14|8t−14+|14−16|8t−16+|16−16|6t−22.
By doing some calculations, we have(74)M3G5G2=−48+64t.Corresponding to the above indices, we are going to compute general fifth M-Zagreb polynomials for rhombus type of oxide networkRHOXt.Theorem 2.2.5.
LetG2≅RHOXt be the rhombus-type oxide network; then, general fifth M-Zagreb polynomials of first and second type are equal to(75)M1aG5G2,x=2x12a+4x18a+4x20a+42t−3x22a+8x26a+24t−7x28a+8t−2x30a+6t−22x32a,M2aG5G2,x=2x36a+4x72a+4x96a+42t−3x112a+8x168a+24t−7x196a+8t−2x224a+6t−22x256a.Proof.
We obtain the outcome with the edge partition in Table1. It follows from [1] that(76)M1aG5G2,x=∑rs∈EG2xSGr+SGsa,M1aG5G2,x=x6+6a|E1G2t|+x6+12a|E2G2t|+x8+12a|E3G2t|+x8+14a|E4G2t|+x12+14a|E5G2t|+x14+14a|E6G2t|+x14+16a|E7G2t|+x16+16a|E8G2t|,=x6+6a2+x6+12a4+x8+12a4+x8+14a8t−12+x12+14a8+x14+14a8t−14+x14+16a8t−16+x16+16a6t−22.
By doing some calculations, we obtain(77)M1aG5G2,x=2x12a+4x18a+4x20a+42t−3x22a+8x26a+24t−7x28a+8t−2x30a+6t−22x32a.
Also, from [3],(78)M2aG5G2,x=∑rs∈EG2xSGr+SGsa,M2aG5G2,x=x6+6a|E1G2t|+x6+12a|E2G2t|+x8+12a|E3G2t|+x8+14a|E4G2t|+x12+14a|E5G2t|+x14+14a|E6G2t|+x14+16a|E7G2t|+x16+16a|E8G2t|,=x6+6a2+x6+12a4+x8+12a4+x8+14a8t−12+x12+14a8+x14+14a8t−14+x14+16a8t−16+x16+16a6t−22.
By making some calculations, we obtain(79)M2aG5G2,x=2x36a+4x72a+4x96a+42t−3x112a+8x168a+24t−7x196a+8t−2x224a+6t−22x256a.Corresponding to the above indices, we are going to compute fifthM-Zagreb polynomials for rhombus-type oxide network RHOXt.Theorem 2.2.6.
LetG2≅RHOXt be the rhombus-type oxide network; then, fifth M-Zagreb polynomials of first and second type are equal to(80)M1aG5G2,x=2x12+4x18+4x20+42t−3x22+8x26+24t−7x28+8t−2x30+6t−22x32,M2G5G2,x=2x36+4x72+4x96+42t−3x112+8x168+24t−7x196+8t−2x224+6t−22x256.Proof.
We obtain the outcome with the edge partition in Table2. It follows from [16] that(81)M1G5G2,x=∑rs∈EG2xSGr+SGs,M1G5G2,x=x6+6|E1G2t|+x6+12|E2G2t|+x8+12|E3G2t|+x8+14|E4G2t|+x12+14|E5G2t|+x14+14|E6G2t|+x14+16|E7G2t|+x16+16|E8G2t|,=x6+62+x6+124+x8+124+x8+148t−12+x12+148+x14+148t−14+x14+168t−16+x16+166t−22.
By doing some calculations, we obtain(82)M1aG5G2,x=2x12+4x18+4x20+42t−3x22+8x26+24t−7x28+8t−2x30+6t−22x32.
Also, from [4],(83)M2G5G2,x=∑rs∈EG2xSGr+SGs,M2G5G2,x=x6×6|E1G2t|+x6×12|E2G2t|+x8×12|E3G2t|+x8×14|E4G2t|+x12×14|E5G2t|+x14×14|E6G2t|+x14×16|E7G2t|+x16×16|E8G2t|,=x6×62+x6×124+x8×124+x8×148t−12+x12×144+x14×148t−14+x14×168t−16+x16×166t−22.
By making some calculations, we obtain(84)M2G5G2,x=2x36+4x72+4x96+42t−3x112+8x168+24t−7x196+8t−2x224+6t−22x256.Theorem 2.2.7.
LetG2≅RHOXt be the rhombus-type oxide network; then, hyper-fifth M-Zagreb polynomials of first and second type are equal to(85)HM1G5G2,x=2x144+4x324+4x400+42t−3x484+8x676+24t−7x784+8t−2x900+6t−22x1024,HM2G5G2,x=2x1296+4x5184+4x9216+42t−3x12544+8x28224+24t−7x38416+8t−2x50176+6t−22x65536.Proof.
We obtain the outcome with the edge partition in Table2. It follows from [2] that(86)HM1G5G2,x=∑rs∈EG2xSGr+SGs2,HM1G5G2,x=x6+62|E1G2t|+x6+122|E2G2t|+x8+122|E3G2t|+x8+142|E4G2t|+x12+142|E5G2t|+x14+142|E6G2t|+x14+162|E7G2t|+x16+162|E8G2t|=x6+622+x6+1224+x8+1224+x8+1428t−12+x12+1428+x14+1428t−14+x14+1628t−16+x16+1626t−22.
By doing some calculations, we obtain(87)HM1G5G2,x=2x144+4x324+4x400+42t−3x484+8x676+24t−7x784+8t−2x900+6t−22x1024.
Also, from [10],(88)HM2aG5G2,x=∑rs∈EG2xSGr+SGs2,HM2G5G2,x=x6×62|E1G2t|+x6×122|E2G2t|+x8×122|E3G2t|+x8×142|E4G2t|+x12×142|E5G2t|+x14×142|E6G2t|+x14×162|E7G2t|+x16×162|E8G2t|,=x6×622+x6×1224+x8×1224+x8×1428t−12+x12×1424+x14×1428t−14+x14×1628t−16+x16×1626t−22.
By making some calculations, we obtain(89)HM2G5G2,x=2x1296+4x5184+4x9216+42t−3x12544+8x28224+24t−7x38416+8t−2x50176+6t−22x65536.


## 3. Conclusion

In this study, we computed sum of degree-based indices forRHSLt and RHOXt graphs of rhombus oxide and silicate structures. We also computed certain sum of degree-based polynomials such as fifth M-Zagreb, fifth hyper M-Zagreb, and generalized fifth M-Zagreb indices for RHSLt and RHOXt graphs of rhombus oxide and silicate structures. These facts may be useful for people working in computer science and chemistry fields who encounter chemical networks. These results can also play a vital role in the determination of the significance of silicate and oxide networks. Like certain other topological indices, determining the representations of derived graphs like these is an open question.



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*Source: 1100024-2021-12-29.xml*