THE AVERAGE ECCENTRICITY AND ECCENTRICITY BASED GEOMETRIC-ARITHMETIC INDEX OF TETRA SHEETS

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1 International Journal of Pure and Applied Mathematics Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v117i3.11 PAijpam.eu THE AVERAGE ECCENTRICITY AND ECCENTRICITY BASED GEOMETRIC-ARITHMETIC INDEX OF TETRA SHEETS Xiujun Zhang 1, Abdul Qudair Baig, Muhammad Razwan Azhar, Mohammad Reza Farahani 3, Muhammad Imran 4 1 School of Information Science and Technology Chengdu University Chengdu, , P.R. CHINA Department of Mathematics COMSATS Institute of Information Technology Attock Campus, PAKISTAN 3 Department of Applied Mathematics Iran University of Science and Technology Narmak, 16844, Tehran, IRAN 4 Department of Mathematical Sciences United Arab Emirates University Al Ain, P.O. Box 15551, UNITED ARAB EMIRATES Abstract: Among topological descriptor, connectivity indices are very important and they have a prominent role in chemistry. The average eccentricity is the mean of all the eccen- tricities of a graph, i.e; avec(g) = 1 n εi. The eccentricity based geometric-arithmetic index is GA 4(G) = ε(u)ε(v) uvǫe(g). In this present paper we compute the average eccentricity and eccentricity based geometric-arithmetic index for an infinite families of tetra ε(u)+ε(v) sheets equilateral triangular and rectangular. AMS Subject Classification: 05C1, 05C90 Key Words: molecular graph, average eccentricity, eccentric polynomial, geometric-arithmetic index, equilateral triangular tetra sheets, rectangular tetra sheets Received: September 8, 017 Revised: October 7, 017 Published: January 15, 018 c 017 Academic Publications, Ltd. url: Correspondence author

2 468 X. Zhang, A.Q. Baig, M.R. Azhar, M.R. Farahani, M. Imran 1. Introduction Let G = (V,E) be a graph,where V is a non-empty set of vertices and E is a set of edges. In chemical graph theory, there are many topological indices for a connected graph, which are helpful in study of chemical molecules. Development of chemical science had an important effect by this theory. A topological index of any graph is a number related to a graph which is constant under graph auto-morphisms. Among topological descriptors, connectivity indices play an important and prominent role in the development in chemistry. If p,q V(G), then the distance d(p,q) between p and q is defined as the length of any shortest path in G connecting p and q. In mathematical form, eccentricity is the maximum distance with first point v in G [34]. ε(u) = max{d(u,v) u V(G)}. (1) The average eccentricity avec(g) of a graph G is the mean value of eccentricities of all vertices of a graph, that is [3, 3, 33, 35, 36], avec(g) = 1 n εi. () The eccentricity based geometric-arithmetic index of a graph G is defined as [9, 30, 4, 5], GA 4 (G) = ε(u)ε(v) ε(u)+ε(v). (3) uvǫe(g) Form more information and history about eccentric connectivity indices, reader can see [6]-[8].. Main Results and Discussion The purpose of this paper is to apply topological indices avec(g) and GA 4 for an infinite families of tetra sheets. In this section, we discussed the average eccentricity index avec(g) and eccentricity based geometric-arithmetic index GA 4 of tetra sheets. The triangular tetra sheet graph has 1 (3n + 3n + ) vertices and 3n (3n+1) edges, similarly the rectangular tetra sheet graph has 3mn n+m vertices and 9mn 8n+ edges [6].

3 THE AVERAGE ECCENTRICITY AND ECCENTRICITY Figure 1: Equilateral Triangular Tetra Sheet ETTS() (ε(u), ε(v)) frequency Range of i 3i (i, i) (3i+1) i = n Table 1: Edge partition of Equilateral Triangular Tetra Sheet for (nlevel) based on eccentricity of end vertices of each edge with existence of their frequencies. Figure : Rectangular Tetra Sheet RTS(,7) (ε(u), ε(v)) frequency Range of m and i Range of k (k,k) m k =, n () (k,k +1) 10m 8 m k n 1, n m+1 (m+n+1) (k,k) 8m 8 m k n, n m+1 (k,k +1) 10(m i) m, 1 i n k m + n, n (k,k) 8(m i) 6 m, 1 i n + 1 k m + n 1, n Table : Edge partition of Rectangular Tetra Sheet for ((m, n)- levels)where mandn have opposite pairity, based on eccentricity of end vertices of each edge with existence of their frequencies.

4 470 X. Zhang, A.Q. Baig, M.R. Azhar, M.R. Farahani, M. Imran (ε(u), ε(v)) frequency Range of m and i Range of k (k,k) 10m 9 m k = m+n, n m (m+n) (k,k +1) 10m 8 m k n 1, n m+1 (m+n+) (k,k) 8m 8 m k n, n m+1 (k,k +1) 10(m i) m, 1 i n k m+n, n (k,k) 8(m i) 6 m, 1 i n+1 k, n Table 3: Edge partition of rectangular tetra sheet for ((m, n)-levels) where mandn have same pairity, based on eccentricity of end vertices of each edge with existence of their frequencies..1. Average Eccentricity of Equilateral Triangular Tetra Sheet (ET T S) In this section we find the average eccentricity of equilateral triangular tetra sheets avec(etts(n)). Theorem 1. Let ETTS(n) for all n N be the equilateral triangular tetra sheet, then the average eccentricity avec(et T S(n)) is equal to avec(etts(n)) = 1 (3i 3 +3i +i). (4) Proof. For n 1, the ETTS(n) equilateral triangular tetra sheet graph contains 1 (3n +3n+) vertices, all the vertices have eccentricity n. With the help of Table 4 and () we have the following computations i=n ε(v) No. of vertices Rang of i i 1 (3i +3i+) i = n Table 4: Vertices partition of Equilateral Triangular Tetra Sheet for (n-level) based on eccentricity of each vertex with existence of their frequencies. avec(g) = 1 n n ε i f i,

5 THE AVERAGE ECCENTRICITY AND ECCENTRICITY avec(etts(n)) = 1 i 1 i (3i +3i+) i=n = 1 i(3i +3i+). i Finally, for all n N, the average eccentricity of the equilateral triangular tetra sheet ETTS(n) is equal to i=n avec(etts(n)) = 1 (3i 3 +3i +i). i.. Average Eccentricity of Rectangular Tetra Sheet (RT S) In this section we find the average eccentricity of the rectangular tetra sheet RTS(m,n) which is denoted by avec(rts(m,n)). Theorem. Let RTS(m,n) for all m,n N, where m and n have opposite parity, be the rectangular tetra sheet, then the average eccentricity avec of RTS(m,n) is equal to i=n avec(rts(m,n)) = {m m+n+1 +(6m 4) m n i= m+n+1 i= (i) (i)+ (6(m p) ) p=1 i=n+1 (i)}. Proof. Let RTS(m,n), where m and n have opposite parity, be rectangular tetra sheet graph contains 3mn n + m vertices and m+n+1 eccentricities. With the help of Table 5 and () we have the following computations ε(v) No. of vertices Rang of i Rang of m, nandp i m i = m,n m+n+1 i 6m 4 i n m,n m+1 i 6(m p) n+1 i m,n,1 p Table 5: Vertices partition of Rectangular Tetra Sheet for ((m, n)-level) based on eccentricity of each vertex with existence of their frequencies. avec(g) = 1 n n ε i f i,

6 47 X. Zhang, A.Q. Baig, M.R. Azhar, M.R. Farahani, M. Imran 1 avec(rts(m,n)) = (m+n+1)/ { m m + m (6m 4) n i= m+n+1 (i)+ m i= (i) (6(m p) ) p=1 n+1 (i)}. Finally, for all m,n N, where m and n have opposite parity, the average eccentricity of rectangular tetra sheet RT S(m, n) is equal to avec(rts(m,n)) = m+n+1 +(6m 4) {m m n i= m+n+1 i= (i) (i)+ (6(m p) ) p=1 n+1 (i)}. Theorem 3. Let RTS(m,n) for all m,n N, where m and n have same parity, be the rectangular tetra sheet, then the average eccentricity avec of RTS(m,n) is equal to avec(rts(m,n)) = m+n {(4m ) (i)+(6m 4) m i= m+n (6(m p) ) + p=1 n i= m+n+ i=n+1 (i) (i)}. Proof. Let RTS(m,n), where m and n have same parity, be rectangular tetra sheet graph contains 3mn n+m vertices and m+n eccentricities. With ε(v) No. of vertices Rang of i Rang of m, nandp i (4m ) i = m+n m,n m m+n+ i 6m 4 i n m,n m+ i 6(m p) n+1 i m,n,1 p Table 6: Vertices partition of Rectangular Tetra Sheet for ((m, n)-level) based on eccentricity of each vertex with existence of their frequencies. the help of Table 6 and () we have the following computations avec(g) = 1 n n ε i f i,

7 THE AVERAGE ECCENTRICITY AND ECCENTRICITY avec(rts(m,n)) = (m+n)/ { (4m ) (i) m + m (6m 4) n i= m+n+ (i)+ m i= m+n (6(m p) ) p=1 n+1 (i)}. Finally, for all m,n N, where m and n have same parity, the average eccentricity of rectangular tetra sheet RTS(m,n) is equal to avec(rts(m,n)) = m+n +(6m 4) {(4m ) (i) m n i= m+n+ i= m+n (i)+ (6(m p) ) p=1.3. Geometric-Arithmetic Index of Equilateral Triangular Tetra Sheet (ET T S) n+1 (i)}. In this section we find the eccentricity based geometric-arithmetic index of equilateral triangular tetra sheets GA 4 (ETTS(n)). Theorem 4. Let ETTS(n) for all n N be the equilateral triangular tetra sheet, then the geometric-arithmetic index GA 4 (ETTS(n)) is equal to GA 4 (ETTS(n)) = 3 (3i +i). (5) Proof. Let ETTS(n) for all n N be the equilateral triangular tetra sheet. With the help of Table 1 and (3) we have the following computations GA 4 (G) = i=n uvǫe(g) ε(u)ε(v) ε(u)+ε(v), GA 4 (ETTS(n)) = 3i(3i+1) i.i i+i, i=n GA 4 (ETTS(n)) = i i i=n3i(3i+1)

8 474 X. Zhang, A.Q. Baig, M.R. Azhar, M.R. Farahani, M. Imran Finally, for all n N, the geometric-arithmetic index of the equilateral triangular tetra sheet ETTS(n) is equal to GA 4 (ETTS(n)) = 3 i(3i+1). i=n.4. Geometric-Arithmetic Index of Rectangular Tetra Sheet (RT S) In this section we find the eccentricity based geometric-arithmetic index of rectangular tetra sheets, which is denoted as GA 4 (RTS(m,n)). Theorem 5. Let RTS(m,n), for all m,n N, where m and n have opposite parity, be the rectangular tetra sheet, then the eccentricity based geometricarithmetic index GA 4 of RTS(m,n) is equal to GA 4 (RTS(m,n)) = m {9()+ (8(m i) 6)} + m {(10m 8) n 1 k= 1 () k +k k +1 + m+n (10(m i) ) k +k k +1 }. Proof. Let RTS(m,n), where m and n have opposite parity, be rectangular tetra sheet graph contains 3mn n + m vertices. With the help of Table and (3), we have the computations GA 4 (G) = uvǫe(g) ε(u)ε(v) ε(u)+ε(v), GA 4 (RTS(m,n)) = m {() +(10m 8) n 1 k= 1 () (10(m i) ) + = m {() k= 1 () k.k (k+k) k(k +1) (k +k+1) +(8m 8) n m+n k= 1 () k k(k +1) (k +k+1) + (k) +(10m 8) k= 1 (m+n+1) k.k (k+k) (8(m i) 6) n 1 k= 1 () k +k (k +1) +1 k.k (k +k) }

9 THE AVERAGE ECCENTRICITY AND ECCENTRICITY (8m 8) + n k= 1 (m+n+1) (8(m i) 6) k (k) + +1 k (k) } = {( + 8m m + m {(10m 8) (10(m i) ) + n 1 k= 1 () m+n m+n (10(m i) ) (8(m i) 6)} k +k (k +1) k +k (k +1) }. k +k (k +1) Finally, for all m,n N, where m and n have opposite parity, the eccentricity based geometric-arithmetic index of rectangular tetra sheet RT S(m, n) is equal to GA 4 (RTS(m,n)) = m {9()+ (8(m i) 6)} + m {(10m 8) n 1 k= 1 () k +k k +1 + m+n (10(m i) ) k +k k +1 }. Theorem 6. Let RTS(m,n), for all m,n N, where m and n have same parity, be the rectangular tetra sheet, then the eccentricity based geometricarithmetic index GA 4 of RTS(m,n) is equal to GA 4 (RTS(m,n)) = m {(187)+ (8(m i) 6)} + m {(10m 8) n 1 k= 1 (m+n) k +k k+1 + m+n (10(m i) ) k +k k +1 }. Proof. Let RTS(m,n), where m and n have same parity, be rectangular tetra sheet graph contains 3mn n + m vertices. With the help of Table 3

10 476 X. Zhang, A.Q. Baig, M.R. Azhar, M.R. Farahani, M. Imran and (3), we have the computations GA 4 (G) = uvǫe(g) ε(u)ε(v) ε(u)+ε(v), GA 4 (RTS(m,n)) = m {(10m 9) +(10m 8) + n 1 k= 1 (m+n) (10(m i) ) = m {(10m 9) +(8m 8) k= 1 (m+n) k.k (k +k) k(k +1) (k +k+1) +(8m 8) n m+n k= 1 (m+n) n k= 1 (m+n+) (8(m i) 6) + = {( 10m 9 m + m {(10m 8) k(k +1) (k +k+1) + k n 1 (k) +(10m 8) k (k) m 8 n 1 k= 1 (m+n) k (k) } k= 1 (m+n+) k.k (k +k) (8(m i) 6) k= 1 (m+n) m+n (10(m i) ) + 1 (8(m i) 6)} k +k (k +1) + k +k (k +1) +1 k +k (k +1) m+n (10(m i) ) k.k (k +k) } k +k (k +1) }. Finally, for all m,n N, where m and n have opposite parity, the eccentricity based geometric-arithmetic index of rectangular tetra sheet RT S(m, n) is equal to GA 4 (RTS(m,n)) = m {(187)+ (8(m i) 6)} + m {(10m 8) n 1 k= 1 (m+n) k +k k +1 + m+n (10(m i) ) k +k k +1 }.

11 THE AVERAGE ECCENTRICITY AND ECCENTRICITY Conclusions In this paper, we computed the average eccentricity avec(g) and geometricarithmetic index GA 4 of the families of tetra sheets, equilateral triangular and rectangular tetra sheets. Acknowledgments The authors are very grateful to the referees for their constructive suggestions and useful comments, which improved this work very much. This research was supported by Key Project of Sichuan Provincial Department of Education under grant 17ZA0079. References [1] M. Alaeiyan and J. Asadpour. Computing the MEC polynomial of an infinite family of the linear parallelogram P(n, n). Optoelectron. Adv. Mater.-Rapid Commun. 6(1-), 01, [] M. Alaeiyan and R. Mojarad, J. Asadpour. A new method for computing eccentric connectivity polynomial of an infinite family of linear polycene parallelogram benzenod. Optoelectron. Adv. Mater.-Rapid Commun. 5(7), 011, [3] P. Dankelmann, W. Goddard and C. S. Swart. The average eccentricity of a graph and its subgraphs. Util Math., 65, (004), [4] K.C. Das. On geometrical-arithmetic index of graphs. MATCH Commun Math Comput Chem., 64, (010), [5] K.C. Das and N. Trinajstic. Comparison between first geometric-arithmetic index and atom-bond connectivity index. Chem. Phys. Lett., 497(1-3), (010), , doi: doi.org/ /j.cplett [6] A.Q. Baig, M.R. Azhar, M.R. Farahani, S. Ediz. Some eccentricity based topological indices of tetra sheets. Communications in Applied Analysis. 1(4), 017, , doi: /caa.v1i4.9. [7] M.R. Farahani, Eccentricity version of atom bond connectivity index of benzenoid family ABC 5(Hk), World Appl. Sci. J. Chem., 1, (013), doi:10.589/idosi.wasj [8] M.R. Farahani. Computing Eccentricity Connectivity Polynomial of Circumcoronene Series of Benzenoid Hk by Ring-Cut Method. Annals of West University of Timisoara- Mathematics and Computer Science. 51(), (013), doi:10.478/awutm [9] M.R. Farahani. The Ediz Eccentric Connectivity index and the Total Eccentricity Index of a Benzenoid System. Journal of Chemica Acta., (013), -5.

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