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## Graph-Theoretic Techniques for the Study of Structures or Networks in Engineering

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Research Article | Open Access

Volume 2020 |Article ID 2797286 | https://doi.org/10.1155/2020/2797286

Jiangnan Liu, Lulu Cai, Abaid ur Rehman Virk, Waheed Akhtar, Shahzad Ahmed Maitla, Yang Wei, "Computation of Irregularity Indices of Certain Computer Networks", Mathematical Problems in Engineering, vol. 2020, Article ID 2797286, 17 pages, 2020. https://doi.org/10.1155/2020/2797286

# Computation of Irregularity Indices of Certain Computer Networks

Accepted04 Jul 2020
Published17 Aug 2020

#### Abstract

A graph is said to be a regular graph if all its vertices have the same degree; otherwise, it is irregular. In general, irregularity indices are used for computational analysis of nonregular graph topological composition. The creation of irregular indices is based on the conversion of a structural graph into a total count describing the irregularity of the molecular design on the map. It is important to be notified how unusual a molecular structure is in various situations and problems in structural science and chemistry. In this paper, we will compute irregularity indices of certain networks.

#### 1. Introduction

In mathematics, graph theory can be used to describe different types of graphs that are computational structures. It is also used to model sensible item connections [1, 2]. An irregularity index is a statistical value connected with a graph that defines a graph’s irregularity. The theory of networks is a part of computer science and network engineering graph theory.

A topological invariant is referred to as a graph irregularity index if the topological invariant and if G is a regular graph. However, if all its vertices have the same degree, it is said that the graph is regular. Topological invariant is a numeric value of a molecular structure of a chemical compound. Nonetheless, the creation of irregular indices is based on the conversion of a structural graph into a total count describing the irregularity of the molecular design on the map [3, 4]. Many networks including silicate, chain silicate, oxide, hexagonal, and honeycomb networks are identical to networks of atomic or chemical structure. There are very important unusual characteristics in such networks.

In this paper, we are concerned with simple connected graphs symbolized by , where and represent the set of vertices and edges of , respectively. The degree of a vertex of a graph is the count of first neighbors of . And represents an edge for , connecting vertices and [5, 6].

Graph theory was established in 1736 when Leonhard Euler presented “Solutio problematic as situspertinntis geometries” (the solution of a problem related to place theory). Wiener is the pioneer of topological indices; he discovered the first topological index and found out the boiling point of a compound (paraffin, a member of the alkane family) in 1947. It was named as path number, but latterly, it was renamed as the Wiener index .

Wiener also invented the Wiener polarity index. Milan Randic invented the first and oldest degree-based index named as the Randic index in 1975 :

The graph invariant denoted by is called the first Zagreb index, which is equal to the sum of square of the degrees of the vertices of a graph; it was introduced by Trinjastic and Gutman in 1972 . is linked with sum of quantities in the field of chemical graph theory. is known as the Gutman index; is bounded and attains lower and upper bound :

The second Zagreb index is a graph invariant denoted by which is defined as the aggregate of the product of degrees of connected pairs of vertices of the molecular compound, and it was introduced by Trinjastic and Gutman in 1972.

Bo-Zhou and Ivan Gutman presented the upper bound for these Zagreb indices w.r.t the min-max degree [11, 12].

Nevertheless, the emergence of new topological descriptors with extremely detecting capacity retains a concern without deformation for the scientific world [13, 14]. Therefore, there is a great willingness to change novel graph invariants with enormous detecting power combined with insignificant degeneration. In this paper, we compute irregularity indices for certain networks.

#### 2. Irregularity Indices

All these selected irregularity indices belong to the family of degree-based topological indices. Tamas Reti et al. selected these irregularity indices as a molecular descriptor in the QSPR study to predict physicochemical properties of octane isomers . The selected irregularity indices for certain networks are represented by

#### 3. Main Results and Discussion

In this section of the paper, we will discuss about the silicate network, chain silicate network, oxide network, hexagonal network, and honeycomb network briefly and will compute the irregularity indices for these networks.

##### 3.1. Silicate Network

Silicates are the most popular, significant, and most complex mineral class. Tetrahedron is the basic chemical unit of silicates. By fusing metal oxides or metal carbonates with sand, silicates are produced. The fact is that all silicates contain tetrahedron of . For chemistry, the corner vertices of tetrahedron reflect oxygen ions, and the silicon ion is represented by the middle vertex. We name the corner vertices as oxygen nodes in graph theory and the middle vertex as the silicon node. Although the tetrahedron is arranged linearly, chain silicates are produced (refer Figure 1).

The total number of vertices and edges in is and , respectively.

Theorem 1. The irregularity indices for the silicate network for are

Proof. By using the edge partition based on degrees of end vertices of each edge of the silicate network given in Table 1, we compute the irregularity indices of the silicate network , and the computations are given as follows:Specific values of irregularity indices of for different values of parameters are given in Table 2.

 Number of edges
 Irregularity indices n = 1 n = 2 n = 3 n = 4 n = 5 1.3333 1.8678 2.0000 2.0544 2.0828 126 468 1026 1800 2790 978 5344.3636 13324.5000 24929.1429 40160.7692 0.4159 0.4582 0.4497 0.4411 0.4344 216 756 1620 2808 4320 0.3077 0.2258 0.2041 0.1940 0.1882 0.6864 2.4024 5.1480 8.9232 13.7280 12.3532 43.2362 92.6490 160.5916 247.6640 0.5000 0.4470 0.4142 0.3945 0.3816 36 126 270 468 720 16.6344 58.2204 124.7580 216.2472 332.6880 24 84 180 312 480 16.9704 59.3964 127.2780 220.6152 339.4080 16 56 120 208 320 33.2712 116.4492 249.5340 432.5256 665.4240 1.4132 4.9462 10.5990 18.3716 28.2640
##### 3.2. Chain Silicate Network

Chain silicate network is obtained when tetrahedra are organized in a sequence. An n-dimensional chain silicate network is represented by , and it is generated by sequentially organizing n tetrahedra. An n-dimensional chain silicate network is shown in Figure 2.

The edge partition of the chain silicate network is given in Table 3.

 , where Number of edges

The total number of vertices and edges in is and , respectively.

Theorem 2. The irregularity indices for the chain silicate network for are

Proof. By using the edge partition based on degrees of end vertices of each edge of the chain silicate network given in Table 3, the computations for irregularities indices are given as follows:Specific values of irregularity indices for different values of involved parameters for the chain silicon network are given in Table 4.

 Irregularity indices n = 1 n = 2 n = 3 n = 4 n = 5 0.0000 1.1020 1.4400 1.5976 1.6875 6 18 30 42 54 0.0000 69.4286 129.6000 186.9231 243.0000 −0.2614 0.2457 0.3337 0.3669 0.3852 18 54 90 126 162 0.4000 0.3333 0.3226 0.3182 0.3158 0.0572 0.1716 0.2860 0.4003 0.5147 1.0294 3.0882 5.1470 7.2058 9.2646 −0.0858 0.1927 0.2570 0.2826 0.2956 3 9 15 21 27 1.3862 4.1586 6.9310 9.7034 12.4758 2 6 10 14 18 1.4142 4.2426 7.0711 9.8995 12.7279 1.3333 4.0000 6.6667 9.3333 12.0000 2.7726 8.3178 13.8629 19.4081 24.9533 0.7520 2.2561 3.7602 5.2643 6.7683
##### 3.3. Oxide Network

Oxide networks play a crucial role in the analysis of silicate networks. If we detach silicone vertices out of a silicate network, we get an oxide network that is referred to as an n-dimensional oxide network as shown in Figure 3.

There are two types of edge partition in the oxide network centered on degrees of end vertices. Table 5 shows the edge partition for .

 , where Number of edges

The total number of vertices and edges in is and , respectively.

Theorem 3. The irregularity indices of oxide networks for are

Proof. By using the edge partition based on degrees of end vertices of each edge of the oxide network given in Table 5, we have the following computations for irregularities of the oxide network :Specific values of irregularity indices for different values of involved parameters for the oxide network are given in Table 6.

 Irregularity indices n = 1 n = 2 n = 3 n = 4 n = 5 1.0000 0.8163 0.6400 0.5207 0.4375 72 144 216 288 360 72 205.7143 345.6000 487.3846 630.0000 0.2660 3.8744 7.7137 11.6265 15.5718 48 96 144 192 240 0.1458 0.0583 0.0365 0.0265 0.0208 0.5136 1.0272 1.5408 2.0544 2.5680 4.1177 8.2354 12.3531 16.4708 20.5885 0.2189 0.1809 0.1396 0.1124 0.0938 18 36 54 72 90 8.3177 16.6354 24.9531 33.2708 41.5885 12 24 36 48 60 8.4853 16.9706 25.4558 33.9411 42.4264 8 16 24 32 40 13.1833 26.3666 39.5499 52.7332 65.9165 0.7068 1.4136 2.1204 2.8272 3.5340
##### 3.4. Hexagonal Network

It is well known that there are three normal plane tilings with the same regular polygon type as triangle, hexagon-shaped, and square structures. Triangular tiling is used in the design of hexagonal networks. An n-dimensional hexagonal network is commonly referred to as , where is each side’s number of hexagonal vertices. A hexagonal network is shown in Figure 4.

Table 7 shows the edge partition of the hexagonal network . The total number of vertices and edges in hexagonal networks is and , respectively.

 , where Number of edges 12

Theorem 4. The irregularity indices of the hexagonal network for are