Applied Mathematics and Computation 240 (2014) 213–228
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Computation of topological indices of certain networks q Sakander Hayat, Muhammad Imran ⇑ Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan
a r t i c l e
i n f o
Keywords: Topological index ABC index Silicate network Hexagonal network Oxide network Honeycomb network
a b s t r a c t There are certain types of topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. Among degree based topological indices, the so-called atom-bond connectivity ðABCÞ, geometric– arithmetic ðGAÞ are of vital importance. These topological indices correlate certain physico-chemical properties such as boiling point, stability and strain energy etc. of chemical compounds. In this paper, we compute ABC 4 and GA5 indices for certain networks like silicate, chain silicate, hexagonal, oxide and honeycomb networks. The atom-bond connectivity index (ABC index) and geometric–arithmetic index (GA index) for oxide and chain silicate networks are also computed in this paper. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Cheminformatics is new subject which is a combination of chemistry, mathematics and information science. There is a considerable usage of graph theory in chemistry. Chemical graph theory is the topology branch of mathematical chemistry which implements graph theory to mathematical modeling of chemical occurrence. There is lot of research which is done in this area in the last few decades. This theory contributes a major role in the field of chemical sciences. A topological index is actually a numeric quantity associated with chemical constitution purporting for correlation of chemical structure with many physico–chemical properties, chemical reactivity or you can say that biological activity. Actually topological indices are designed on the ground of transformation of a molecular graph into a number which characterize the topology of that graph. In molecular modeling, we study the relationship between structure, properties and activity of chemical compounds. The major subjects about chemical structures like chemistry, pharmacology etc. found a significant role of molecular descriptors. Among all molecular descriptors topological indices found a vital role in (QSAR)/(QSPR) study. A fixed interconnection parallel architecture is characterized by a graph, with vertices corresponding to processing nodes and edges representing communication links. Interconnection networks are notoriously hard to compare in abstract terms. Researchers in parallel processing are thus motivated to propose new or improved interconnection networks, arguing the benefits and offering performance evaluations in different contexts. The silicates are the largest, the most interesting and the most complicated class of minerals by far. Silicates are obtained by fusing metal oxides or metal carbonates with sand. A few networks such as hexagonal, honeycomb, and grid networks, for instance, bear resemblance to atomic or molecular lattice structures. These networks have very interesting topological properties which have been studied in different aspects in [7–9,12]. q
This research is partially supported by National University of Sciences and Technology, Islamabad, Pakistan.
⇑ Corresponding author.
E-mail addresses:
[email protected] (S. Hayat),
[email protected] (M. Imran). http://dx.doi.org/10.1016/j.amc.2014.04.091 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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The hexagonal and honeycomb networks have also been recognized as crucial evolutionary biology, in particular for the evolution of cooperation, where the overlapping triangles are vital for the propagation of cooperation in social dilemmas. Relevant research that applies this theory and which could benefit further from the insights of the new research is found in [10,11,17,21]. A graph can be recognized by a numeric number, a polynomial, a sequence of numbers or a matrix which represents the whole graph, and these representations are aimed to be uniquely defined for that graph. A topological index is a numeric quantity associated with a graph which characterize the topology of graph and is invariant under graph automorphism. There are some major classes of topological indices such as distance based topological indices, degree based topological indices and counting related polynomials and indices of graphs. Among these classes degree based topological indices are of great importance and play a vital role in chemical graph theory and particularly in chemistry. In more precise way, a topoP P logical index is a function ‘‘Top’’ from to the set of real numbers, where ‘ ’ is the set of finite simple graphs with property that TopðGÞ ¼ TopðHÞ if both G and H are isomorphic. Obviously, the number of edges and vertices of a graph are topological indices. Throughout in this article, G is considered to be simple and connected graph with vertex set VðGÞ and edge set EðGÞ; du is P the degree of vertex u 2 VðGÞ and Su ¼ v 2NG ðuÞ dðv Þ where N G ðuÞ ¼ fv 2 VðGÞjuv 2 EðGÞg. The notions used in this article are mainly taken from books [18,2,6]. The first degree based topological index is Randic´ index [13] vðGÞ introduced by Milan Randic´ in 1975. Randic´ index is defined as
vðGÞ ¼
X
1 pffiffiffiffiffiffiffiffiffiffi : d u dv uv 2EðGÞ
The widely used connectivity topological index is atom-bond Connectivity ðABCÞ index introduced by Estrada et al. [3] and is defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 : ABCðGÞ ¼ du dv uv 2EðGÞ X
The fourth version of ABC index (ABC 4 ) is introduced by Ghorbani et al. [4] in 2010 and is defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
Geometric–arithmetic ðGAÞ index which is introduced by Vukicˇevic´ et al. [20] and defined as
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv : ðdu þ dv Þ uv 2EðGÞ
Recently fifth version of GA index (GA5 ) is proposed by Graovac et al. [5] in 2011 and defined as
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv : ðSu þ Sv Þ uv 2EðGÞ
In this paper we discuss ABC 4 and GA5 topological index for silicate, chain silicate, oxide, honeycomb and hexagonal networks. We also computed close formulae of ABC and GA index for oxide and chain silicate networks. 2. Main results and discussion In this section, we discuss the silicate, chain silicate, hexagonal, oxide and honeycomb networks and give close formulae of certain topological indices for these networks. For further study of topological indices and related concepts please consult [1,14–16,19]. 2.1. Silicate networks Silicates are building blocks of the common rock-forming minerals and the largest, very interesting and most complicated minerals by far. The tetrahedron ðSiO4 Þ is basic unit of silicates. Silicates are obtained by fusing metal oxides or metal carbonates with sand. Almost all silicates contain ðSiO4 Þ tetrahedra. A silicate sheet is a ring of tetrahedrons which are linked by shared oxygen vertices to other rings in a two dimensional plane that produces a sheet-like structure. Cyclic silicates are structures which give cycles of different length after being linked by shared oxygen vertices. Some sheet and cyclic silicates are shown in Fig. 3. There are also other types of silicates which are shown in Fig. 2. From chemical point of view, the corner vertices of tetrahedron ðSiO4 Þ are actually the oxygen atoms and central vertex represents silicon atom. Usually, we call these corner atoms as oxygen vertices, central atom as
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Fig. 1. A ðSiO4 Þ tetrahedron in which corner vertices are oxygen vertices and central vertex is silicon vertex.
Orthosilicates
Pyrosilicates
Chain Silicates Fig. 2. Ortho, Pyro and chain silicates.
Cyclic Silicates
Sheet Silicates Fig. 3. Cyclic and sheet silicates.
silicon vertex and bonds between them as edges from graphical point of view. A ðSiO4 Þ tetrahedron is shown in Fig. 1. A silicate network of dimension n symbolizes as ðSLn Þ, where n is the number of hexagons between the center and boundary of SLn . A silicate network of dimension three is shown in Fig. 4. The number of vertices in SLn are 15n2 þ 3n and number of edges
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Fig. 4. A silicate network SLn with n ¼ 3.
Table 1 Edge partition of silicate networks based on degree sum of neighbors of end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
Number of edges
ð15; 15Þ ð15; 24Þ ð15; 27Þ ð18; 27Þ ð18; 30Þ ð24; 27Þ ð27; 27Þ ð27; 30Þ ð30; 30Þ
6n 24 24ðn 1Þ 12ðn 1Þ 18n2 30n þ 12 12 6ð2n 3Þ 12ðn 1Þ 18n2 36n þ 18
are 36n2 . Table 1 shows the partition of edge set of ðSLn Þ based on the degree sum of vertices lying at unit distance from end vertices of each edge, and by using this partition we compute the ABC 4 and GA5 indices of silicate networks. In the following theorem, the exact formula of ABC 4 index for silicate networks is computed. Theorem 2.1.1. Consider the silicate networks SLn , then its ABC 4 index is equal to
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 4 7 16 2 2 258 8 13 2 22 2 370 3 58 690 þ 3 58 2 690 6 58 n þ nþ ABC 4 ðSLn Þ ¼ þ þ þ þ þ 5 3 9 9 3 3 5 5 5 15 pffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 690 7 2 16 2 2 258 4 13 2 22 þ : þ 15 3 3 9 3 3
Proof. Let G be the graph of silicate networks SLn . We have jVðSLn Þj ¼ 15n2 þ 3n and jEðSLn Þj ¼ 36n2 . We find the edge partition of silicate networks SLn based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 1 explains such partition for SLn . Now by using the partition given in Table 1 we can apply the formula of ABC 4 index to compute this index for G. Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
This implies that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 þ 15 2 15 þ 24 2 15 þ 24 2 18 þ 27 2 þ ð24Þ þ 24ðn 1Þ þ 12ðn 1Þ ABC 4 ðGÞ ¼ ð6nÞ 15 15 15 24 15 24 18 27 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18 þ 30 2 24 þ 27 2 27 þ 27 2 27 þ 30 2 2 þ ð12Þ þ 6ð2n 3Þ þ 12ðn 1Þ þ ð18n 30n þ 12Þ 18 30 24 27 27 27 27 30 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 þ 30 2 : þ ð18n2 36n þ 18Þ 30 30
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After an easy simplification, we get
ABC 4 ðSLn Þ ¼
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 4 7 16 2 2 258 8 13 2 22 2 370 3 58 690 þ 3 58 2 690 6 58 n þ nþ þ þ þ þ þ 5 3 9 9 3 3 5 5 5 15 pffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 690 7 2 16 2 2 258 4 13 2 22 þ : þ 15 3 3 9 3 3
Following theorem computes the GA5 index of silicate networks SLn . Theorem 2.1.2. Consider the silicate networks SLn , then its GA5 index is equal to
GA5 ðSLn Þ ¼
! pffiffiffiffiffiffi! pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffi 72 5 24 6 36 þ 9 15 2 72 5 24 6 72 10 15 15 96 10 144 2 n þ þ þ 18 n þ þ 3 15 2 7 5 19 2 13 17 7 5 pffiffiffiffiffiffi 72 10 : 19
Proof. Let G be the graph of silicate networks SLn . The edge partition of silicate networks SLn based on the degree sum of vertices lying at unit distance from end vertices of each edge is given in Table 1. Now we apply the formula of GA5 index to compute this index for G. Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv : S þ Sv uv 2EðGÞ u
This gives that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 15 15 2 15 24 2 15 27 2 18 27 GA5 ðGÞ ¼ ð6nÞ þ ð24Þ þ 24ðn 1Þ þ 12ðn 1Þ 15 þ 15 15 þ 24 15 þ 27 18 þ 27 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 27 30 18 30 24 27 27 27 2 þ ð12Þ þ 6ð2n 3Þ þ 12ðn 1Þ þ ð18n 30n þ 12Þ 18 þ 30 24 þ 27 27 þ 27 27 þ 30 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30 30 2 : þ ð18n2 36n þ 18Þ 30 þ 30 After an easy simplification, we get
GA5 ðSLn Þ ¼
! pffiffiffiffiffiffi! pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffi 72 5 24 6 36 þ 9 15 2 72 5 24 6 72 10 15 15 96 10 144 2 n þ þ þ 18 n þ þ 3 15 2 7 5 19 2 13 17 7 5 pffiffiffiffiffiffi 72 10 : 19
Now, we define a new family of silicate networks named as chain silicate networks and then compute its certain degree based topological indices. 2.2. Chain silicate networks When tetrahedra are arranged linearly, chain silicate are obtained. We define chain silicate networks of dimension n as follows: A chain silicate network of dimension n symbolizes as ðCSn Þ is obtained by arranging n tetrahedra linearly. The number of vertices in ðCSn Þ with n > 1 are 3n þ 1 and number of edges are 6n. A chain silicate network of dimension n is shown in Fig. 5. Now we find the partition of edge set of ðCSn Þ based on the degrees of end vertices of each edge, and by using this partition we compute certain topological indices which are based on this partition. Table 2 shows such a partition. Now in the following theorem, we computed the ABC index of chain silicate networks CSn . 3
1
2
5
4
7
6
n
8
Fig. 5. Chain silicate networks of dimension n.
n-1
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Table 2 Edge partition of CSn based on degrees of end vertices of each edge. ðdu; dv Þ where uv 2 EðGÞ
ð3; 3Þ
ð3; 6Þ
ð6; 6Þ
Number of edges
nþ4
2ð2n 1Þ
n2
Theorem 2.2.1. Consider the chain silicate networks ðCSn Þ; n > 1, then its ABC index is equal to
ABCðCSn Þ ¼
pffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi! 10 14 10 2 2 14 8 nþ : þ þ 3 3 3 6 3 3
Proof. Let G be the graph of chain silicate networks ðCSn Þ. The number of vertices in CSn are 3n þ 1 and number of edges are 6n. Now by using the edge partition based on the degrees of end vertices of each edge of chain silicate networks ðCSn Þ given in Table 2 we compute the ABC index of chain silicate networks ðCSn Þ. Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ : du dv uv 2EðGÞ X
This implies that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ32 3þ62 6þ62 þ 2ð2n 1Þ þ ðn 2Þ : 33 36 66
ABCðCSn Þ ¼ ðn þ 4Þ
After an easy simplification, we get
ABCðCSn Þ ¼
pffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi! 2 2 14 8 10 14 10 þ þ nþ : 3 3 3 6 3 3
Following theorem gives GA index of chain silicate networks CSn . Theorem 2.2.2. Consider the chain silicate networks ðCSn Þ; n > 1, then its GA index is equal to
GAðCSn Þ ¼
pffiffiffi! pffiffiffi! 6þ8 2 64 2 nþ : 3 3
Proof. By using the edge partition based on the degrees of end vertices of each edge of chain silicate networks ðCSn Þ given in Table 2, we compute the GA index of chain silicate networks ðCSn Þ. Since
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv : d þ dv uv 2EðGÞ u
This gives that
GAðCSn Þ ¼ ðn þ 4Þ
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 33 2 36 2 66 þ 2ð2n 1Þ þ ðn 2Þ : 3þ3 3þ6 6þ6
After simplification, we get
GAðCSn Þ ¼
pffiffiffi! pffiffiffi! 6þ8 2 64 2 nþ : 3 3
There are seven types of edges in CSn based on degree sum of vertices of neighbors of end vertices for each edge. In Table 3, such partition of CSn with n > 3 is presented. Now by using above Table, we compute ABC 4 and GA5 index of chain silicate networks CSn . Following theorem presents the ABC 4 index of chain silicate networks CSn . Theorem 2.2.3. Consider the chain silicate networks ðCSn Þ, then its ABC 4 index is equal to
8 pffiffiffiffi pffiffiffiffi 3 22þ2 42 > ; n ¼ 2; > 6 > > qffiffiffiffi < pffiffiffiffi pffiffi pffiffiffiffiffiffi pffiffiffiffi 22 2 7 217 2 10 4 34 þ 15 þ 7 þ 21 þ 3 35; n ¼ 3; ABC 4 ðCSn Þ ¼ 2 > qffiffiffiffi pffiffiffiffi pffiffi pffiffiffiffiffiffi pffiffiffiffi > pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffi > > : 2 7 þ 370 þ 46 n þ 22 þ 217 þ 602 þ 4 34 46 4 7 370 ; n > 3: 3 35 15 24 2 7 42 6 5 15 15
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S. Hayat, M. Imran / Applied Mathematics and Computation 240 (2014) 213–228 Table 3 Edge partition of CSn based on degree sum of neighbors of end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
Number of edges
ð12; 12Þ ð12; 21Þ ð15; 15Þ ð15; 21Þ ð15; 24Þ ð21; 24Þ ð24; 24Þ
6 6 n2 4 4ðn 3Þ 2 n4
Proof. There are three cases to discuss while proving this result. Firstly, we prove this result for n ¼ 2. Let G be the graph of CS2 , there are two type of edges in CS2 based on the degree sum of vertices lying at unit distance from end vertices of each edge as follows: first type is, for e ¼ uv 2 EðGÞ such that Su ¼ Sv ¼ 12 and other type is, for e ¼ uv 2 EðGÞ such that Su ¼ 12 and Su ¼ 18. There are six edges in each partite set of CS2 . Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
This implies that
ABC 4 ðCS2 Þ ¼ ð6Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 þ 12 2 12 þ 18 2 þ ð6Þ : 12 12 12 18
After a bit calculation we get,
pffiffiffiffiffiffi pffiffiffiffiffiffi 3 22 þ 2 42 ABC 4 ðCS2 Þ ¼ : 6 Now we prove this result for n ¼ 3. For this, we need edge partition of CS3 as we have in first case. Following table shows such partition of CS3 . Since
ABC 4 ðGÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : Su Sv uv 2EðGÞ X
This gives that
ABC 4 ðCS3 Þ ¼ ð6Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 þ 12 2 15 þ 15 2 12 þ 21 2 21 þ 21 2 15 þ 21 2 þ ð1Þ þ ð6Þ þ ð1Þ þ ð4Þ : 12 12 15 15 12 21 21 21 15 21
After an easy calculation,
ABC 4 ðCS3 Þ ¼
rffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi 22 2 7 217 2 10 4 34 : þ þ þ þ 3 35 2 7 21 15
Now we have third case to prove this result for n > 3. We find the edge partition of chain silicate networks CSn for n > 3 based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 3 explains such partition for CSn . Since now by using the partition given in Table 3, we can apply the formula of ABC 4 index to compute this index for G.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 ABC 4 ðGÞ ¼ : Su Sv uv 2EðGÞ X
This implies that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 þ 12 2 12 þ 21 2 15 þ 15 2 15 þ 21 2 15 þ 24 2 þ ð6Þ þ ðn 2Þ þ ð4Þ þ 4ðn 3Þ 12 12 12 21 15 15 15 21 15 24 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 21 þ 24 2 24 þ 24 2 þ ð2Þ þ ðn 4Þ : 21 24 24 24
ABC 4 ðGÞ ¼ ð6Þ
After an easy simplification, we get
ABC 4 ðCSn Þ ¼
rffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffi pffiffiffiffiffiffiffiffiffi 2 7 370 46 22 217 602 4 34 46 4 7 370 nþ þ þ þ þ þ : 3 35 15 24 2 7 42 6 5 15 15
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Now we present GA5 index of chain silicate networks ðCSn Þ. Theorem 2.2.4. Consider the chain silicate networks ðCSn Þ, then its GA5 index is equal to pffiffi 8 30þ12 6 > ; n ¼ 2; > 5 > > > < pffiffiffiffi pffiffi 24 7 2 35 n ¼ 3; GA4 ðCSn Þ ¼ 8 þ 11 þ 3 ; > > > p ffiffiffiffi p ffiffiffiffi p ffiffiffiffi p ffiffiffiffi p ffiffi > > : 26þ16 10 n þ 24 7 þ 2 35 þ 8 14 48 10 ; n > 3: 13 3 15 13 11
Proof. There are three cases to discuss while proving this result. Firstly, we prove this result for n ¼ 2. Let G be the graph of CS2 ,there are two type of edges in CS2 based on the degree sum of vertices lying at unit distance from end vertices of each edge as follows: first type is, for e ¼ uv 2 EðGÞ such that Su ¼ Sv ¼ 12 and other type is, for e ¼ uv 2 EðGÞ such that Su ¼ 12 and Su ¼ 18. There are six edges in each partite set of CS2 . Now we apply the formula of GA5 index to compute this index for CS2 . Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv ; S þ Sv uv 2EðGÞ u
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12 12 2 12 18 GA5 ðCS2 Þ ¼ ð6Þ þ ð6Þ : 12 þ 12 12 þ 18 After an easy calculation, we get
ABC 4 ðCS2 Þ ¼
pffiffiffi 30 þ 12 6 : 5
To prove this result for n ¼ 3 we use the same partition given in Table 4.
GA5 ðCS3 Þ ¼ ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12 12 2 15 15 2 12 21 2 21 21 2 15 21 þ ð1Þ þ ð6Þ þ ð1Þ þ ð4Þ : 12 þ 12 15 þ 15 12 þ 21 21 þ 21 15 þ 21
After an easy simplification, we get
pffiffiffi pffiffiffiffiffiffi 24 7 2 35 GA5 ðCS3 Þ ¼ 8 þ þ : 11 3 Now we discuss the third case for n > 3. Table 3 shows required partition for chain silicate networks CSn ; n > 3. Since
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv GA5 ðGÞ ¼ : S þ Sv uv 2EðGÞ u This implies that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12 12 2 12 21 2 15 15 2 15 21 2 15 25 þ ð6Þ þ ðn 2Þ þ ð4Þ þ 4ðn 3Þ 12 þ 12 12 þ 21 15 þ 15 15 þ 21 15 þ 24 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 24 24 : þ ðn 4Þ 24 þ 24
GA5 ðCSn Þ ¼ ð6Þ
After an easy calculation, we get
GA5 ðCSn Þ ¼
pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 26 þ 16 10 24 7 2 35 8 14 48 10 nþ þ þ : 13 3 15 13 11
In the following section, we study hexagonal networks and compute the ABC 4 and GA5 indices for these networks.
Table 4 Edge partition of CS3 based on degree sum of neighbors of end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
ð12; 12Þ
ð15; 15Þ
ð12; 21Þ
ð21; 21Þ
ð15; 21Þ
Number of edges
6
1
6
1
4
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2.3. Hexagonal networks It is well known fact, that there exist three regular plane tilings with composition of same kind of regular polygons such as triangular, hexagonal and square. In the construction of hexagonal networks, triangular tiling is being used. A hexagonal network of dimension n is usually denoted as HX n , where n is the number of vertices on each side of hexagon. The number of vertices in hexagonal networks HX n with n > 1 are 3n2 3n þ 1 and number of edges are 9n2 15n þ 6. A hexagonal network HX n with n ¼ 6 is depicted in Fig. 6. In any hexagonal network there are twelve types of edges based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 5 shows such edge partition of hexagonal networks HX n for n > 4. Now we calculate ABC 4 and GA5 indices of chain silicate networks HX n . Following theorem computed the ABC 4 index of hexagonal networks. Theorem 2.3.1. Consider the hexagonal networks HX n ; n > 1, then its ABC 4 index is equal to ffi pffiffi 8 pffiffiffiffiffi 130þ9 2 ; > > > pffiffiffiffiffi5ffi pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi > > 2 210 3 41 > þ 203 þ 6 29290 þ 6 29203 ; > 7 > > ffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffi > pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi < qffiffiffiffiffi 31 41 46 59 33 70 36 p21 ffiffiffiffi 3 203 ABC 4 ðHX n Þ ¼ 12 266 þ 6 406 þ 12 551 þ 3 58 þ 38 þ 29 þ 2 þ 3 þ 19 ; qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi > pffiffiffiffi pffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi > pffiffiffiffi > 70 2 3 38 3 5 33 3 62 17 70 31 37 49 59 41 > > > 4 n þ 10 þ 2 þ 2 þ 16 21 n þ 12 266 þ 12 380 þ 12 608 þ 12 928 þ 6 406 > > qffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi > p ffiffiffiffiffiffi p ffiffiffi > : þ12 46 þ 3 203 3 38 3 33 3 62 þ 2 70 6 5; 551
29
2
2
4
n ¼ 2; n ¼ 3; n ¼ 4;
n > 4:
Proof. Consider the graph of hexagonal networks HX n . We have jVðHX n Þj ¼ 3n2 3n þ 1 and jEðHX n Þj ¼ 9n2 15n þ 6. Now there are four cases to discuss while proving this result. Firstly, we prove this result for n ¼ 2. let G be the graph of HX 2 , there are two type of edges in HX 2 based on the degree sum of vertices lying at unit distance from end vertices of each edge as follows: first type is, for e ¼ uv 2 EðGÞ such that Su ¼ Sv ¼ 10 and other type is, for e ¼ uv 2 EðGÞ such that Su ¼ 10 and Su ¼ 18. There are six edges in each partite set of HX 2 . Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
This implies that
ABC 4 ðHX 2 Þ ¼ ð6Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 þ 10 2 10 þ 18 2 þ ð6Þ : 10 10 10 18
After a bit calculation we get,
ABC 4 ðHX 2 Þ ¼
pffiffiffiffiffiffiffiffiffi pffiffiffi 130 þ 9 2 : 5
Now we prove this result for n ¼ 3. For this, we need edge partition of HX 3 as we have in first case. Following Table shows such a partition of HX 3 . Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 ABC 4 ðGÞ ¼ : Su Sv uv 2EðGÞ X
Fig. 6. Hexagonal network HX n with n ¼ 6.
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S. Hayat, M. Imran / Applied Mathematics and Computation 240 (2014) 213–228 Table 5 Edge partition of HX n based on degree sum of vertices lying at unit distance from end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
Number of edges
ð14; 19Þ ð19; 20Þ ð20; 20Þ ð19; 29Þ ð19; 32Þ ð20; 32Þ ð14; 29Þ ð29; 32Þ ð29; 36Þ ð32; 36Þ ð32; 32Þ ð36; 36Þ
12 12 6ðn 5Þ 12 12 12ðn 4Þ 6 12 6 12ðn 3Þ 6ðn 4Þ 9n2 51n þ 72
This gives that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14 þ 18 2 14 þ 29 2 18 þ 29 2 29 þ 36 2 þ ð6Þ þ ð12Þ þ ð12Þ : ABC 4 ðHX 3 Þ ¼ ð12Þ 14 18 14 29 18 29 29 36 After an easy calculation,
ABC 4 ðHX 3 Þ ¼
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 210 3 41 6 290 6 203 þ þ þ : 7 203 29 29
Now we discuss the third case to prove this result for n ¼ 4. To prove this result for HX 4 , we need edge partition of HX 4 based on the degree sum of neighbors of end vertices of each edge. Following Table shows such a partition for hexagonal network HX 4 . Now we apply the formula of ABC 4 index to compute this index for hexagonal index HX 4 . Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
This implies that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14 þ 19 2 19 þ 19 2 14 þ 29 2 19 þ 29 2 19 þ 32 2 þ ð6Þ þ ð6Þ þ ð12Þ þ ð12Þ ABC 4 ðHX 4 Þ ¼ ð12Þ 14 19 19 19 14 29 19 29 19 32 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 29 þ 32 2 29 þ 36 2 32 þ 36 2 36 þ 36 2 þ ð6Þ þ ð12Þ þ ð12Þ : þ ð12Þ 29 32 29 36 32 36 36 36 After an easy simplification, we get
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 33 70 36 21 3 203 p ffiffiffiffiffiffi þ ABC 4 ðHX 4 Þ ¼ þ þ þ : 19 29 2 3 38 Now we have fourth case to prove this result for n > 4. Let G be the graph of hexagonal networks HX n with n > 4. We find the edge partition of chain silicate networks HX n for n > 4 based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 5 explains such partition for HX n ; n > 4. Now by using the partition given in Table 5, we can apply the formula of ABC 4 index to compute this index for G. Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
This implies that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14 þ 19 2 19 þ 20 2 20 þ 20 2 19 þ 29 2 þ ð12Þ þ 6ðn 5Þ þ ð12Þ ABC 4 ðGÞ ¼ ð12Þ 14 19 19 20 20 20 19 29 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 þ 32 2 20 þ 32 2 14 þ 29 2 29 þ 32 2 þ ð12Þ þ 12ðn 4Þ þ ð6Þ þ ð12Þ 19 32 20 32 14 29 29 32 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 29 þ 36 2 32 þ 36 2 32 þ 32 2 36 þ 36 2 2 þ 12ðn 3Þ þ 6ðn 4Þ þ ð9n 51n þ 72Þ : þ ð6Þ 29 36 32 36 32 32 36 36
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After an easy simplification, we get
ABC 4 ðGÞ ¼
rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi! pffiffiffi pffiffiffiffiffiffi 3 38 3 5 31 37 49 59 70 2 33 3 62 17 70 n þ 12 þ 12 þ 12 þ 12 n þ þ þ þ 266 380 608 928 4 10 2 16 21 2 rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 41 46 3 203 3 38 3 33 3 62 þ 12 þ þ 2 70 6 5: þ6 406 551 29 2 2 4
We compute GA5 index of hexagonal networks HX n . In the following theorem, GA5 index of hexagonal networks HX n is being computed. Theorem 2.3.2. Consider the hexagonal networks HX n ; n > 1, then its GA5 index is equal to pffiffi 8 42þ18 5 > ; > 7 > > > pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffi > > > 9 7 þ 12 406 þ 72 58 þ 144 29 ; < 43 47 65 2 GA4 ðHX n Þ ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffi 8 266 > > þ 12 43406 þ 551 þ 321738 þ 966158 þ 726529 þ 14417 2 þ 18; > 11 2 > > > pffiffiffiffi pffiffi pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffi > > : 9n2 þ 48 10 þ 144 2 39 n þ 24 406 þ 16 95 þ 551 þ 32 38 þ 4 266 þ 96 58 þ 72 29 192 10 432 2 þ 18 13 17 43 13 2 17 11 61 65 13 17
n ¼ 2; n ¼ 3; n ¼ 4; n > 4:
Proof. Firstly, we prove this result for n ¼ 2. Let G be the graph of HX 2 , there are two type of edges in HX 2 based on the degree sum of vertices lying at unit distance from end vertices of each edge as follows: first type is, for e ¼ uv 2 EðGÞ such that Su ¼ Sv ¼ 10 and other type is, for e ¼ uv 2 EðGÞ such that Su ¼ 10 and Su ¼ 18. There are six edges in each partite set of HX 2 . Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv ; S þ Sv uv 2EðGÞ u
GA5 ðHX 2 Þ ¼ ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 10 10 2 10 18 þ ð6Þ : 10 þ 10 10 þ 18
After a bit calculation, we get
GA5 ðHX 2 Þ ¼
pffiffiffi 42 þ 18 5 : 7
Now we prove this result for n ¼ 3. For this, we need edge partition of HX 3 as we have in first case. Table 6 shows such partition of HX 3 . Now we compute the fifth GA index of HX 3 . Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv ; S þ Sv uv 2EðGÞ u
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 14 18 2 14 29 2 18 29 2 29 36 GA5 ðHX 3 Þ ¼ ð12Þ þ ð6Þ þ ð12Þ þ ð12Þ : 14 þ 18 14 þ 29 18 þ 29 29 þ 36 After an easy calculation, we get
GA5 ðHX 3 Þ ¼
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 9 7 12 406 72 58 144 29 þ þ þ : 43 47 65 2
Now we discuss the third case to prove this result for n ¼ 4. To prove this result for HX 4 , we need edge partition of HX 4 based on the degree sum of neighbors of end vertices of each edge. Table 7 shows such partition for hexagonal network HX 4 . Now we compute the result for n ¼ 4. Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv : S þ Sv uv 2EðGÞ u
Table 6 Edge partition of HX 3 based on degree sum of neighbors of end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
ð14; 18Þ
ð14; 29Þ
ð18; 29Þ
ð29; 36Þ
Number of edges
12
6
12
12
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This implies that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 14 19 2 19 19 2 14 29 2 19 29 2 19 32 þ ð6Þ þ ð6Þ þ ð12Þ þ ð12Þ 14 þ 19 19 þ 19 14 þ 29 19 þ 29 19 þ 32 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 29 32 2 29 36 2 32 36 2 36 36 þ ð6Þ þ ð12Þ þ ð12Þ : þ ð12Þ 29 þ 32 29 þ 36 32 þ 36 36 þ 36
GA5 ðHX 4 Þ ¼ ð12Þ
After an easy calculation, we get
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 551 32 38 96 58 72 29 144 2 8 266 12 406 GA5 ðHX 4 Þ ¼ þ þ þ þ þ þ þ 18: 11 43 2 17 61 65 17 Now we have fourth case to prove this result for n > 4. Let G be the graph of hexagonal networks HX n with n > 4. We find the edge partition of chain silicate networks HX n for n > 4 based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 5 explains such partition for HX n ; n > 4. Now by using the partition given in Table 5, we can apply the formula of ABC 4 index to compute this index for G. Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv : S þ Sv uv 2EðGÞ u
This gives that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 14 19 2 19 20 2 20 20 2 19 29 2 19 32 þ ð12Þ þ 6ðn 5Þ þ ð12Þ þ ð12Þ 14 þ 19 19 þ 20 20 þ 20 19 þ 29 19 þ 32 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 20 32 2 14 29 2 29 32 2 29 36 2 32 36 þ ð6Þ þ ð12Þ þ ð6Þ þ 12ðn 3Þ þ 12ðn 4Þ 20 þ 32 14 þ 29 29 þ 32 29 þ 36 32 þ 36 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 32 32 2 36 36 þ ð9n2 51n þ 72Þ : þ 6ðn 4Þ 32 þ 32 36 þ 36
GA5 ðHX n Þ ¼ ð12Þ
After an easy simplification, we get
! pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 48 10 144 2 24 406 16 95 551 32 38 4 266 96 58 72 29 þ 39 n þ þ þ þ þ þ þ 13 17 43 13 2 17 11 61 65 pffiffiffiffiffiffi pffiffiffi 192 10 432 2 þ 18: 13 17
GA5 ðHX n Þ ¼ 9n2 þ
2.4. Oxide networks Oxide networks play a vital role in the study of silicate networks. If we delete silicon vertices from a silicate network, we get an oxide networks (see Fig. 7). An n-dimensional oxide network is denoted as OX n . The number of vertices in 9n2 þ 3n and number of edges are 18n2 . There are two types of edges based on the degrees of end vertices of each edge in oxide networks OX n . Table 8 shows such types of edges for oxide networks OX n . Now we compute certain degree based topological indices of oxide networks OX n . In the following theorem, we present ABC index of oxide networks. Theorem 2.4.1. Consider the oxide networks ðOX n Þ; n > 1, then its ABC index is equal to
ABCðOX n Þ ¼
pffiffiffi pffiffiffi pffiffiffi 9 6 2 n þ 3nð2 2 6Þ: 2 Table 7 Edge partition of hexagonal network HX 4 based on degree sum of neighbors of end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
Number of edges
ð14; 19Þ ð19; 19Þ ð14; 29Þ ð19; 29Þ ð19; 32Þ ð29; 32Þ ð29; 36Þ ð32; 36Þ ð36; 36Þ
12 6 6 12 12 12 6 12 12
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Fig. 7. An oxide network OX 5 .
Table 8 Edge partition of OX n based on degrees of end vertices of each edge. ðdu; dv Þ where uv 2 EðGÞ
ð2; 4Þ
ð4; 4Þ
Number of edges
12n
18n2 12n
Proof. By using the edge partition based on the degrees of end vertices of each edge of oxide networks ðOX n Þ given in 8 we compute the ABC index of oxide networks ðOX n Þ. Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 : ABCðGÞ ¼ du dv uv 2EðGÞ X
This gives that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 4þ42 þ ð18n2 12nÞ : 24 44
ABCðOX n Þ ¼ ð12nÞ
After an easy simplification, we get
ABCðOX n Þ ¼
pffiffiffi pffiffiffi pffiffiffi 9 6 2 n þ 3nð2 2 6Þ: 2
Following theorem gives exact formula GA index of oxide networks OX n . Theorem 2.4.2. Consider the oxide networks ðOX n Þ; n > 1, then its GA index is equal to
pffiffiffi GAðOX n Þ ¼ 18n2 þ 4nð2 2 3Þ: Proof. By using the edge partition based on the degrees of end vertices of each edge of oxide networks ðOX n Þ given in Table 8, we compute the GA index of oxide networks ðOX n Þ. Since
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv : d þ dv uv 2EðGÞ u
This implies that
GAðOX n Þ ¼ ð12nÞ
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 24 2 44 þ ð18n2 12nÞ : 2þ4 4þ4
After simplification, we get
pffiffiffi GAðOX n Þ ¼ 18n2 þ 4nð2 2 3Þ:
There are six types of edges based on degree sum of vertices lying at unit distance from end vertices of each edge in oxide networks OX n . We use this partition of edges of OX n to calculate ABC 4 and GA5 indices for oxide networks. Table 9 shows such types of edges of oxide networks OX n . Now we compute ABC 4 and GA5 indices of oxide networks OX n . Following theorem gives formula ABC 4 index of oxide networks OX n .
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S. Hayat, M. Imran / Applied Mathematics and Computation 240 (2014) 213–228 Table 9 Edge partition of OX n based on degree sum of vertices lying at unit distance from end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
Number of edges
ð8; 12Þ ð12; 14Þ ð8; 14Þ ð14; 14Þ ð14; 16Þ ð16; 16Þ
12 12 12ðn 1Þ 6ð2n 3Þ 12ðn 1Þ 18n2 36n þ 18
Theorem 2.4.3. Consider the oxide networks ðOX n Þ; n > 1, then its ABC 4 index is equal to
ABC 4 ðOX n Þ ¼
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 9 30 2 6 35 6 26 9 30 12 7 9 30 6 35 9 26 n þ þ þ3 2 nþ þ þ 3 3 3 2: 8 7 7 4 8 7 7 7
Proof. We find the edge partition of oxide networks OX n based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 9 explains such partition for OX n . Now by using the partition given in Table 9, we can apply the formula of ABC 4 index to compute this index for G. Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 : ABC 4 ðGÞ ¼ Su Sv uv 2EðGÞ X
This implies that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 þ 12 2 12 þ 14 2 8 þ 14 2 14 þ 14 2 þ ð12Þ þ 12ðn 1Þ þ 6ð2n 3Þ ABC 4 ðGÞ ¼ ð12Þ 8 12 12 14 8 14 14 14 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 14 þ 16 2 16 þ 16 2 þ ð18n2 36n þ 18Þ : þ 12ðn 1Þ 14 16 16 16 After an easy simplification, we get
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 9 30 2 6 35 6 26 9 30 12 7 9 30 6 35 9 26 ABC 4 ðOX n Þ ¼ n þ þ þ3 2 nþ þ þ 3 3 3 2: 7 8 7 7 4 8 7 7
In the following theorem, GA5 index of oxide networks OX n is computed. Theorem 2.4.4. Consider the oxide networks ðOX n Þ; n > 1, then its GA5 index is equal to
GA5 ðOX n Þ ¼ 18n2
! pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi 48 7 16 14 24 6 24 42 16 14 48 7 þ 24 n þ þ : 11 11 15 5 13 15
Proof. The edge partition of oxide networks OX n based on the degree sum of vertices lying at unit distance from end vertices of each edge is given in Table 9. Now we apply the formula of GA5 index to compute this index for G. Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv : S þ Sv uv 2EðGÞ u
This gives that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 8 12 2 12 14 2 8 14 2 14 14 2 14 16 þ ð12Þ þ 12ðn 1Þ þ 6ð2n 3Þ þ 12ðn 1Þ 8 þ 12 12 þ 14 8 þ 14 14 þ 14 14 þ 16 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 16 16 2 þ ð18n 36n þ 18Þ : 16 þ 16
GA5 ðGÞ ¼ ð12Þ
After an easy simplification, we get
GA5 ðOX n Þ ¼ 18n2
! pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi 48 7 16 14 24 6 24 42 16 14 48 7 þ 24 n þ þ : 15 5 13 15 11 11
227
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2.5. Honeycomb networks Honeycomb networks are widely used in computer graphics, cellular phone base stations, image processing and as a representation of benzenoid hydrocarbons in chemistry. If we recursively use hexagonal tiling in a particular pattern, honeycomb networks are formed. An n-dimensional honeycomb network is denoted as HC n , where n is the number of hexagons between central and boundary hexagon. Honeycomb network HC n is constructed from HC n1 by adding a layer of hexagons around boundary of HC n1 (see Fig. 8). The number of vertices in honeycomb network HC n are 6n2 and number of edges are 9n2 3n. There are four types of edges in HC n based on degree sum of vertices lying at unit distance from end vertices of each edge. Table 10 shows such types of edges. By using this partition of edges of HC n , we compute ABC 4 and GA5 indices of HC n . Now we compute ABC 4 index of honeycomb network HC n for n > 1. In the following theorem, we computed ABC 4 index of honeycomb network HC n . Theorem 2.5.1. Consider the honeycomb networks ðHC n Þ; n > 1, then its ABC 4 index is equal to
! pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi 28 pffiffiffi 16 12 14 12 2 12 14 nþ ABC 4 ðHC n Þ ¼ 4n þ þ2 2 2 2þ : 3 3 7 5 7 2
Proof. We find the edge partition of honeycomb networks HC n based on the degree sum of vertices lying at unit distance from end vertices of each edge. Table 10 explains such partition for HC n . Now by using the partition given in Table 10, we can apply the formula of ABC 4 index to compute this index for G. Since
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Su þ Sv 2 ABC 4 ðGÞ ¼ : Su Sv uv 2EðGÞ X
This implies that
ABC 4 ðGÞ ¼ ð6Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5þ52 5þ72 7þ92 9þ92 þ 12ðn 1Þ þ 6ðn 1Þ þ ð9n2 21n þ 12Þ : 55 57 79 99
After an easy simplification, we get
ABC 4 ðHC n Þ ¼ 4n2 þ
! pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi 28 pffiffiffi 16 12 14 12 2 12 14 þ2 2 2 2þ : nþ 3 3 7 5 7
In the following theorem, GA5 index of honeycomb network HC n is computed. Theorem 2.5.2. Consider the honeycomb networks ðHC n Þ; n > 1, then its GA5 index is equal to
! pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 9 7 9 7 GA5 ðHC n Þ ¼ 9n þ þ 2 35 21 n þ 18 2 35: 4 4 2
Fig. 8. A 4-dimensional honeycomb network.
Table 10 Edge partition of HC n based on degree sum of neighbors of end vertices of each edge. ðSu; Sv Þ where uv 2 EðGÞ
ð5; 5Þ
ð5; 7Þ
ð7; 9Þ
ð9; 9Þ
Number of edges
6
12ðn 1Þ
6ðn 1Þ
9n2 21n þ 12
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Proof. The edge partition of honeycomb networks HC n based on the degree sum of vertices lying at unit distance from end vertices of each edge is given in Table 10. Now we apply the formula of GA5 index to compute this index for G. Since
GA5 ðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 Su Sv : S þ Sv uv 2EðGÞ u
This implies that
GA5 ðGÞ ¼ ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 55 2 57 2 79 2 99 þ 12ðn 1Þ þ 6ðn 1Þ þ ð9n2 21n þ 12Þ : 7þ9 9þ9 5þ5 5þ7
After an easy simplification, we get
! pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 9 7 9 7 GA5 ðHC n Þ ¼ 9n þ þ 2 35 21 n þ 18 2 35: 4 4 2
3. Concluding remarks In this paper, we have computed certain topological indices of some networks like silicate networks, chain silicate networks, hexagonal networks, oxide networks and honeycomb networks. We determined the closed formulas of ABC 4 and GA5 indices for these networks. We also computed ABC and GA indices for chain silicate and oxide networks. These results are novel and significant contributions in the network science and provide a basis to understand deep topology of these important networks. The results are an eye-opener for the researchers working in network sciences that how these networks can be constructed and how good topological properties they have. In future, we are interested to study and compute topological indices of various inter-connecting networks like butterfly networks, Benes networks and hex derived networks. Acknowledgment The authors are very grateful to the referees for their careful reading with corrections and useful comments, which improved this work very much. References [1] K.C. Das, F.M. Bhatti, S.G. Lee, I. Gutman, Spectral properties of the He matrix of hexagonal systems, MATCH Commun. Math. Comput. Chem. 65 (2011) 753–774. [2] M.V. Diudea, I. Gutman, J. Lorentz, Molecular Topology, Nova, Huntington, 2001. [3] E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855. [4] A. Graovac, M.A. Hosseinzadeh, Computing ABC4 index of nanostar dendrimers, Optoelectron. Adv. Mater. Rapid Commun. 4 (2010) 1419–1422. [5] A. Graovac, M. Ghorbani, M.A. Hosseinzadeh, Computing fifth geometric–arithmetic index for nanostar dendrimers, J. Math. Nanosci. 1 (2011) 33–42. [6] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, New York, 1986. [7] P. Manuel, B. Rajan, I. Rajasingh, C. Monica, On minimum metric dimension of honeycomb networks, J. Discrete Algoritms 6 (2008) 20–27. [8] P. Manuel, I. Rajasingh, Minimum metric dimension of silicate networks, Ars Combinatoria 98 (2011) 501–510. [9] P. Manuel, I. Rajasingh, Topological properties of silicate networks, in: GCC Conference & Exhibition 5th IEEE, 2009, pp. 1–5. [10] M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L.M. Floría, Y. Moreno, Evolutionary dynamics of group interactions on structured populations: a review, J. R. Soc. Interface 10 (2013) 20120997. [11] M. Perc, A. Szolnoki, Coevolutionary games-a mini review, BioSystems 99 (2010) 109–125. [12] B. Rajan, A. William, C. Grigorious, S. Stephen, On certain topological indices of silicate, honeycomb and hexagonal networks, J. Comp. Math. Sci. 5 (2012) 530–535. [13] M. Randic´, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (23) (1975) 6609–6615. [14] S. Hayat, M. Imran, Computation of certain topological indices of nanotubes, J. Comput. Theor. Nanosci., Accepted, in press. [15] S. Hayat, M. Imran, Computation of certain topological indices of nanotubes covered by C 5 and C 7 , J. Comput. Theor. Nanosci., Accepted, in press. [16] S. Hayat, M. Imran, On some degree based topological indices of certain nanotubes, J. Comput. Theor. Nanosci., Accepted, in press. [17] A. Szolnoki, M. Perc, G. Szabó, Topology-independent impact of noise on cooperation in spatial public goods games, Phys. Rev. E 80 (2009) 056109. [18] N. Trinajstic´, Chemical Graph Theory, CRC Press, Boca Raton, FL, 1992. [19] I. Tomescu, S. Kanwal, Ordering trees having small general sum-connectivity index, MATCH Commun. Math. Comput. Chem. 69 (2013) 535–548. [20] D. Vukicˇevic´, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369–1376. [21] Z. Wang, A. Szolnoki, M. Perc, If players are sparse social dilemmas are too: importance of percolation for evolution of cooperation, Sci. Rep. 2 (2012) 369.
