NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n

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1 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n M. NAEEM, M. K. SIDDIQUI, J. L. G. GUIRAO AND W. GAO 3 Abstract. The eccentricity ε u of vertex u in a connected graph G, is the distance between u and a vertex farthermost from u. The aim of the present paper is to introduce new eccentricity based index and eccentricity based polynomial, namely modified augmented eccentric connectivity index and modified augmented eccentric connectivity polynomial respectively. As an application we compute these new indices for octagonal grid On m and we compare the results obtained with the ones obtained by other indices like Ediz eccentric connectivity index, modified eccentric connectivity index and modified eccentric connectivity polynomial ECP (G, x).. Introduction In recent years graph theory is extensively used in the branch of mathematical chemistry and some people call it as chemical graph theory because this theory is related with the practical applications of graph theory for solving the molecular problems. In mathematics a model of chemical system portrays a chemical graph that deals to explain the relations between its segments such as its atoms, bonds between atoms, cluster of atoms or molecules. A connected simple graph G = (V (G) E(G)) is a graph consisting of n vertices (V (G)) and m edges (E(G)) in which there is path between any of two its vertices. A network is merely a connected graph consisting of no multiple edges and loops. The degree of a vertex v in G is the number of edges which are incident to the vertex v and will be represented by d v. In a graph G, if there is no repetition of vertices in (u v) walk then such kind of walk is called (u v) path. The number of edges in (u v) path is called its length. The distance d(u, v) from vertex u to vertex v is the length of a shortest (u v) path in a graph G where u, v G. In a connected graph G, the eccentricity ε v of a vertex v is the distance between v and a vertex furthest from v in G. Thus, ε v = max v V (G) d(v, u). Therefore the maximum eccentricity over all vertices of G is the diameter of G which is denoted by D(G). A graph can be recognized by a different type of numeric number, a polynomial, a sequence of numbers or a matrix. A topological index is a numeric quantity that is 00 Mathematics Subject Classification. Primary: 05C, 05C90. Key words and phrases. Degree, Eccentricity, Ediz eccentric connectivity index, modified eccentric connectivity index, modified eccentric connectivity polynomial ECP (G, x), octagonal grid O m n.

2 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO associated with a graph which characterize the topology of graph and is invariant under graph automorphism. Over the years topological indices like Wiener index Balabans index [4, 5, 6, Hosoya index [6, 7, Randić index [9 and so on, have been studied extensively and recently the research and interest in this area has been increased exponentially. See too for more information [3, 3, 4, 8,, 3. There are some major classes of topological indices such as distance based topological indices, eccentricity based topological indices, degree based topological indices and counting related polynomials and indices of graphs. In this article we shall consider the eccentricity based indices. We note that in [5 is introduced the total eccentricity of a graph G and is defined as the sum of eccentricities of all vertices of a given graph G and denote by ζ(g). It is easy to see that for a k regular graph G is held ζ(g) = kζ(g). The Eccentric-connectivity index ξ(g) which was proposed by Sharma, Goswami and Madan defined as [0: () ξ(g) = u V (G) d u ɛ u, Another very relevant and special eccentricity based topological index is connective Eccentric index C ξ (G) that was proposed by Gupta et al. in [. The connective eccentric index is defined as. () C ξ (G) = d u, ɛ u u V (G) In 00, A. R. Ashrafi and M. Ghorbani [ introduces the so called modified eccentric connectivity index ξ c (G) and it is defined as (3) ξ c (G) = (S v ɛ v ), where S v = v. u N(v) v V (G) d u that is S v is the sum of degrees of all vertices adjacent to vertex In 00, S. Ediz et al., [8, defined Ediz eccentric connectivity index of G as (4) E ξ c (G) = ( S v ), ɛ v v V (G) Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as, [6: (5) ξ c (G, x) = S u x ɛu, u V (G)

3 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 3 so that the modified eccentric connectivity index is the first derivative of this polynomial for x =. Motivated by these above eccentricity indices, in this article we introduce what we call modified augmented eccentric connectivity index MA ξ(g), as (6) MA ξ c (G) = (M v ɛ v ), where M v = v of G. u N(v) v V (G) d u that is denotes the product of degrees of all neighbors of vertex In the same way, we define the modified augmented eccentric connectivity polynomial MA ξ c (G, x), as (7) MA ξ c (G, x) = M v x ɛv v V (G) For more information and properties of eccentricity based topological index, see for instance [, 7, 9, 0,, 5, 7. The aim of this paper is is the introduction of the augmented eccentric connectivity index and modified augmented eccentric connectivity polynomial. As an application we shall compute these new indices for octagonal grid On m and we shall compare the results obtained with the ones obtained by other indices like Ediz eccentric connectivity index, modified eccentric connectivity index and modified eccentric connectivity polynomial ECP (G, x) via their computation too.. Octagonal Grid O m n In [4 and [ Diudea et al. constructed a C 4 C 8 net as a trivalent decoration made by alternating squares C 4 and octagons C 8 in two different ways. One is by alternating squares C 4 and octagons C 8 in different ways denoted by C 4 C 8 (S) and other is by alternating rhombus and octagons in different ways denoted by C 4 C 8 (R). We denote C 4 C 8 (R) by On m see Figure. In [ they also called it as the Octagonal grid. For n, m the Octagonal grid O m n, is the grid with m rows and n columns of octagons. The symbols V (O m n ) and E(O m n ) will denote the vertex set and the edge set of O m n, respectively. V (O m n ) = {u t s : s n, t m } {v t s : s n; t m } {w t s : s n, t m} {y t s : s n, t m}. E(O m n ) = {u t sv t s : s n, t m } {u t sw t s; s n, t m} {w t sy t s : s n, t m} {v t sw t s : s n, t m} {vsy t s t : s n, t m } {u t s ys t : s n, t m}.

4 4 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO Figure. The Octagonal grid O m n. In this paper, we consider O m n with n = m. 3. Statement of main results As we have said previously for On m with n = mwe shall compute modified eccentric connective index, Ediz eccentric connectivity index, modified eccentric connective polynomial, modified augmented eccentric connective index and modified augmented eccentric connectivepolynomial and we shall compare the results obtained. For this we have discussed two cases of n, when n 0( mod ) and when n ( mod ). Also to avoid any ambiguity related to Figure note that the vertices u t s = u t s. Theorem. For every n 4 and n 0( mod ) consider the graph of G = O m n, with n = m. Then the modified eccentric connectivity index ξ c (G) of G is equal to

5 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 5 ξ c (On m ) = 5n n 8 n [ l n l n [ s {4n 3(s ) t} {4(n ) s 3t} t=s n [ n s s=l n l n n [ n s {3(n t) s } n [ l n l n [ n {3(n s) t } t=l n t=n s [ s n [ n {4s t n 3} t=l n s=l n t=s {n 3s t } {n s 3t } {s 3t 4}. Proof. Let G be the graph of On m. Note that graph of On m is a symmetric about reflection and rotation at right angles. Thus the eccentricities ε u t s = ε v t n s and from the symmetry at right angles we can obtain that the eccentricities ε y t s = ε u s t, ε w t s = ε v s t. Therefore, from Table and formula (3), given below, the modified eccentric connectivity index ξ c (G) of On m is equal to ξ c (G) = v V (G) ( Sv ɛ v ) = 4 u t s V (G) ( Su t s ε u t s),

6 6 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO [ l n n ξ c (On m ) = 4 4 4n 5{4n s} 5{3n s } 4 [ l n l n 7(4n t) [ l n t=s l n [ s 9{4(n ) s 3t} n l n [ l n 9{4n 3(s ) t} n s=l n 9{n 3s t } [ n s n 9{3(n t) s } 7{3(n ) t } t=l n [ n s l n [ n 9{3(n s) t } 9{n s 3t } t=l n t=n s n [ s n [ n 9{4s t n 3} 9{s 3t 4}. t=l n s=l n t=s After some easy calculations we get ξ c (On m ) = 5n n 8 l n [ l n l n [ s {4n 3(s ) t} {4(n ) s 3t} t=s n [ n s s=l n {3(n t) s } n [ l n {n 3s t } l n n l n [ n {3(n s) t } t=l n t=n s [ s n [ n {4s t n 3} t=l n s=l n [ n s t=s {n s 3t } {s 3t 4}.

7 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 7 Table. Partition of vertices of the type u t s of On m based on degree sum and eccentricity of each vertex when n 0( mod ). Representative S u t s eccentricity Range Frequency u t s 4 4n s t =, n ; s = u t s 5 4n s t =, n, n s l n u t s 5 3n s t =, n, n l n s n u t s 7 4n 3(s ) t s =, l n t l n u t s 9 4n 3(s ) t s l n, l n 8 l 3n 4 s t l n u t s 9 4(n ) s 3t s l n, l n 8 l n 4 t s u t s 9 3n s 3t l n n s n, l l n 8 4 t n s u t s 9 n 3s t l n n s n, l l n 8 4 n s t l n u t s 7 3(n s) t s =, l n l n t n u t s 9 3(n s) t s l n, l (n 4)(n ) 8 l n t n s u t s 9 n s 3t s l n, l n 8 l n 4 n s t n u t s 9 4s n t 3 l n n s n, l l n 8 4 l n t s u t s 9 s 3t 4 l n n s n, l l n 8 4 s t n Theorem. For every n 3 and n ( mod ) consider the graph of G = O m n, with n = m. Then the modified eccentric connectivity index ξ c (G) of G is equal to

8 8 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO ξ c (On m ) = 5n 3n 35 l n [ l n l n [ s {4n 3(s ) t} {4(n ) s 3t} n s=l n l n s= n s=l n t=s [ n s [ n s t=l n {3(n t) s } {3(n s) t } [ s {4s t n 3} t=l n n s=l n l n [ n n s=l n [ l n t=n s [ n t=s {n 3s t } {n s 3t } {s 3t 4}. Proof. Let G be the graph of On m and n 3 is odd. As above note that graph of On m is a symmetric about reflection and rotation at right angles. Thus the eccentricities ε u t s = ε v t n s and from the symmetry at right angles we can obtain that the eccentricities ε y t s = ε u s t, ε w t s = ε v s t. Therefore, by using Table and equation (3) the modified eccentric connectivity index ξ c (G), we get [ l n ξ c (On m ) = 4 4 4n 4 [ l n l n 7{4n t} n 5{4n s} 5{3n s } s=l n [ l n 9{4n 3(s ) t} l n t=s [ s 9{4(n ) s 3t} n s=l n l n n s=l n [ l n [ n s t=l n n s=l n 9{n 3s t } 9{3(n s) t } [ s 9{4s t n 3} t=l n [ n s n t=l n l n 9{3(n t) s } 7{3(n ) t } [ n n s=l n t=n s 9{n s 3t } [ n 9{s 3t 4}. t=s

9 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 9 After some easy calculations we get ξ c (On m ) = 5n 3n 35 l n [ l n l n [ s {4n 3(s ) t} {4(n ) s 3t} n s=l n t=s [ n s {3(n t) s } n s=l n [ l n {n 3s t } l n s= n s=l n [ n s {3(n s) t } [ s {4s t n 3} t=l n t=l n l n [ n n s=l n t=n s [ n t=s {n s 3t } {s 3t 4}. Theorem 3. For every n 4 and n 0( mod ) consider the graph of G = O m n, with n = m. Then the Ediz eccentric connectivity index of G is equal to E ξ c (On m ) = l 8 l n n 40 8 l n 4n s 40 4n t 8 n 3n s n 3(n ) t t=l n l n n s=l n l n n [ l n t=s [ n s 4n 3(s ) t 3(n t) s [ n s 3(n s) t t=l n [ s t=l n 4s t n 3 l n [ s n l n [ n n s=l n 4(n ) s 3t [ l n t=n s [ n t=s n 3s t n s 3t. s 3t 4

10 0 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO Table. Partition of vertices of the type u t s of On m based on degree sum and eccentricity of each vertex when n ( mod ). Representative S u t s eccentricity Range Frequency u t s 4 4n s t =, n ; s = u t s 5 4n s t =, n, n s l n u t s 5 3n s t =, n, n l n s n u t s 7 4n 3(s ) t = s, (l n ) t l n u t s 9 4n 3(s ) t s l n, l n 3 4 s t l n u t s 9 4(n ) s 3t s l n t s, l n 4 (l n ) (l n ) u t s 9 3n s 3t l n s n, l n n (l ) 4 t n s u t s 9 n 3s t l n s n, l n 4 n s t l n u t s 7 3(n s) t s =, (l n l n t n u t s 9 n s 3t s l n, l n 3 4 n s t n u t s 9 4s n t 3 l n s n, l n 4 l n t s u t s 9 s 3t 4 l n s n, l n 4 s t n (l n ) ) (l n ) (l n ) (l n ) Proof. Let G be the graph of On m and n 0( mod ). By using the arguments in proof of Theorem, Table and following formula the Ediz eccentric connectivity index E ξ c (G) of On m is equal to E ξ c (G) = v V (G) ( l S ) v ɛ v = 4 u t s V (G) ( l S ) u t s ɛ u t s

11 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n [ E ξ c (On m ) = 4 l 4 l n 4n 4 [ l n 7 4n t l n 5 4n s [ l n t=s l n [ s 9 4(n ) s 3t n l n n [ l n After an easy computation, we get E ξ c (On m ) = l 8 l n n 40 8 l n l n n s=l n n 5 3n s 9 4n 3(s ) t n s=l n 9 n 3s t [ n s 9 3(n s) t t=l n [ s t=l n 9 4s t n 3 4n s 40 4n t 8 [ l n t=s [ n s [ n s 9 3(n t) s n 7 3(n ) t t=l n l n [ n n s=l n t=n s [ n n 3n s n 3(n ) t t=l n 4n 3(s ) t 3(n t) s l n [ s n t=s 9 n s 3t 9. s 3t 4 4(n ) s 3t [ l n n 3s t l n n [ n s 3(n s) t t=l n [ s t=l n 4s t n 3 l n [ n n s=l n t=n s [ n t=s n s 3t. s 3t 4

12 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO Theorem 4. For every n 3 and n ( mod ) consider the graph of G = O m n, with n = m. Then the Ediz eccentric connectivity index of G is equal to E ξ c (On m ) = l 8 l n n 40 4n s 40 8 l n 4n t 8 l n n s=l n l n n s=l n [ l n t=s [ n s n t=l n n s=l n 4n 3(s ) t [ n t=l n [ s t=l n 3n s 3(n ) t 3(n t) s l n [ s n s=l n l n 3(n s) t 4s t n 3 [ n n s=l n 4(n ) s 3t [ l n t=n s [ n t=s n 3s t n s 3t. s 3t 4 Proof. Let G be the graph of On m and n ( mod ). By using the arguments in proof the of Theorem, Table and from formula (4) the result follows.

13 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 3 Theorem 5. For every n 4 and n 0( mod ) consider the graph of G = O m n, with n = m. Then the modified eccentric connectivity polynomial of G is equal to ( ξ c (On m, x) = l ( 8x 4n 40x 3n 40x 4n )(l n x x )(l ) 40x (3n) (l x )n 4x 4n ) 8x (7n/) 56x 4n 6x 4n l n n s=l n l n n [ l n t=s [ n s x (4n 3(s ) t) [ n s l n [ s x (4(n) s 3t) x (3(n t)s) t=l n x (3(n s)t) [ s t=l n x (4st n 3) n l n [ n n s=l n [ l n t=n s [ n t=s x (n3s t ) x (n s3t ) x (s3t 4). Proof. By using the arguments in the proof of Theorem, the values from Table and equation (5) given below we get ξ c (G, x) = u V (G) S u x ɛu = 4 S u t s x ɛut s u t s V (G)

14 4 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO [ l n ξ c (On m, x) = 4 4x 4n 5x (4n s) 4 [ l n l n 7x (4n t) [ l n t=s l n [ s 9x (4(n) s 3t) n l n n [ l n [ n s 9x (4n t) n s=l n 9x (n3s t ) t=l n 9x (3(n s)t) [ s t=l n 9x (4st n 3) n 5x (3ns ) [ n s n 9x (3(n t)s) t=l n 7x (3(n )t) l n [ n n s=l n t=n s [ n t=s 9x (n s3t ) 9x (s3t 4). After some easy calculations we get ( ξ c (On m, x) = l ( 8x 4n 40x 3n 40x 4n )(l n x x )(l ) 40x (3n) (l x )n 4x 4n ) 8x (7n/) 56x 4n 6x 4n l n n s=l n l n n [ l n t=s [ n s x (4n 3(s ) t) [ n s l n [ s x (4(n) s 3t) x (3(n t)s) t=l n x (3(n s)t) [ s t=l n x (4st n 3) n l n [ n n s=l n [ l n t=n s [ n t=s x (n3s t ) x (n s3t ) x (s3t 4).

15 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 5 Theorem 6. For every n 3 and n ( mod ) consider the graph of G = O m n, with n = m. Then the modified eccentric connectivity polynomial of G is equal to ( ξ c (On m, x) = l (40x 3n 8x 4n 40x 4n )(l n x x )(l l 3 ) 40x (3n) (l x )n 4x 4n ) 8x (l 7n ) l 56x 4n 6x 4n l n n s=l n l n s= n s=l n [ l n t=s [ n s [ n s x (4n 3(s ) t) x (3(n t)s) t=l n x (3(n s)t) [ s t=l n x (4st n 3) l n [ s x (4(n) s 3t) n s=l n l n n s=l n [ n [ l n t=n s [ n t=s x (n3s t ) x (n s3t ) x (s3t 4). Proof. Let G = On m, n 3 and n ( mod ). By using the arguments in the proof of Theorem, as in Theorem 5, the values from Table and equation (5) the result follows. In Table 3 and Table 4 we have partitioned the vertices of the type u t s of On m based on degree product and eccentricity of each vertex. This will help us to develop the coming theorems. Theorem 7. For every n 4 and n 0( mod ) consider the graph of G = O m n, with n = m. Then the modified augmented eccentric connectivity index MA ξ c c(g) of G is equal to

16 6 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO MA ξc(o c n m ) = 34n 3n 48 l n [ l n 08 {4n 3(s ) t} t=s n [ n s s=l n l n n [ n s l n [ s {4(n ) s 3t} {3(n t) s } 08 n [ l n l n [ n {3(n s) t } 08 t=l n t=n s [ s n [ n {4s t n 3} 08 t=l n s=l n t=s {n 3s t } {n s 3t } {s 3t 4}. Proof. Let G be the graph of On m. Therefore, from Table 3 and formula (8), given below, the modified augmented eccentric connectivity index MA ξc(o c n m ) of On m can be calculated. Hence the result. (8) MA ξ c (G, x) = M v ɛ v v V (G) MA ξ c (O m n ) = 4 u t s V (G) ( Mu t ε ) s u t s [ l n n MA ξ c (On m ) = 4 4 4n 6{4n s} 6{3n s } 4 [ l n l n (4n t) [ l n t=s 7{4n 3(s ) t}

17 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 7 l n [ s 7{4(n ) s 3t} n l n [ l n n s=l n 7{n 3s t } [ n s n 7{3(n t) s } {3(n ) t } t=l n [ n s l n [ n 7{3(n s) t } 7{n s 3t } t=l n t=n s n [ s n [ n 7{4s t n 3} 7{s 3t 4}. t=l n s=l n t=s After some easy calculations we get MA ξc(o c n m ) = 34n 3n 48 l n [ l n 08 {4n 3(s ) t} t=s n [ n s s=l n l n n [ n s l n [ s {4(n ) s 3t} {3(n t) s } 08 n [ l n l n [ n {3(n s) t } 08 t=l n t=n s [ s n [ n {4s t n 3} 08 t=l n s=l n t=s {n 3s t } {n s 3t } {s 3t 4}. Theorem 8. For every n 3 and n ( mod ) consider the graph of G = O m n, with n = m. Then the modified augmented eccentric connectivity index ξ c (G) of G is equal

18 8 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO to MA ξ c c(o m n ) = 34n 56n l n n s=l n l n s= n s=l n [ l n t=s [ n s [ n s t=l n l n {4n 3(s ) t} 08 {3(n t) s } 08 {3(n s) t } 08 [ s {4s t n 3} 08 t=l n [ s {4(n ) s 3t} n s=l n l n [ n n s=l n [ l n t=n s [ n t=s {n 3s t } {n s 3t } {s 3t 4}. Proof. As in Theorem 7, by using Table 4 and equation (8) the result follows. Theorem 9. For every n 4 and n 0( mod ) consider the graph of G = On m, with n = m. Then the modified augmented eccentric connectivity polynomial of G is equal to ( MA ξc(o c n m, x) = l (48x 4n 48x 3n 48x 4n )(l n x x )(l ) 48x (3n) (l x )n 3x 4n ) 48x (7n/) 96x 4n 6x 4n l n n s=l n l n n [ l n t=s [ n s x (4n 3(s ) t) 48 [ n s l n [ s x (4(n) s 3t) x (3(n t)s) 48 t=l n x (3(n s)t) [ s t=l n x (4st n 3) n l n [ n n s=l n [ l n t=n s x (n3s t ) x (n s3t ) [ n x. (s3t 4) t=s

19 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 9 Table 3. Partition of vertices of the type u t s of On m based on degree product and eccentricity of each vertex when n 0( mod ). Representative M u t s eccentricity Range Frequency u t s 4 4n s t =, n ; s = u t s 6 4n s t =, n, n s l n u t s 6 3n s t =, n, n l n s n u t s 4n 3(s ) t s =, l n t l n u t s 7 4n 3(s ) t s l n, l n 8 l 3n 4 s t l n u t s 7 4(n ) s 3t s l n, l n 4 (l n ) t s u t s 7 3n s 3t l n s n, l n 4 (l n ) t n s u t s 7 n 3s t l n s n, l n 4 (l n ) n s t l n u t s 3(n s) t s =, l n l n t n u t s 7 3(n s) t s l n, l (n 4)(n ) 8 l n t n s u t s 7 n s 3t s l n, l n 4 (l n ) n s t n u t s 7 4s n t 3 l n s n, l n 4 (l n ) l n t s u t s 7 s 3t 4 l n s n, l n 4 (l n ) s t n Proof. By using the arguments in the proof of Theorem, the values from Table 3 and equation (8) given below we get MA ξ c c(g, x) = u V (G) M u x ɛu = 4 M u t s x ɛut s u t s V (G)

20 0 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO Table 4. Partition of vertices of the type u t s of On m based on degree product and eccentricity of each vertex when n ( mod ). Representative M u t s eccentricity Range Frequency u t s 4 4n s t =, n ; s = u t s 6 4n s t =, n, n s l n u t s 6 3n s t =, n, n l n s n u t s 4n 3(s ) t = s, (l n ) t l n u t s 7 4n 3(s ) t s l n, l n 3 4 s t l n u t s 7 4(n ) s 3t s l n t s, l n 4 (l n ) (l n ) u t s 7 3n s 3t l n s n, l n n (l ) 4 t n s u t s 7 n 3s t l n s n, l n 4 n s t l n u t s 3(n s) t s =, (l n l n t n u t s 7 n s 3t s l n, l n 3 4 n s t n u t s 7 4s n t 3 l n s n, l n 4 l n t s u t s 7 s 3t 4 l n s n, l n 4 s t n (l n ) ) (l n ) (l n ) (l n ) [ l n MA ξ c (On m, x) = 4 4x 4n 6x (4n s) 4 [ l n l n x (4n t) [ l n t=s n 6x (3ns ) 7x (4n t)

21 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n l n [ s 7x (4(n) s 3t) n l n n [ l n [ n s n s=l n 7x (n3s t ) t=l n 7x (3(n s)t) [ s t=l n 7x (4st n 3) [ n s n 7x (3(n t)s) t=l n x (3(n )t) l n [ n n s=l n t=n s [ n t=s 7x (n s3t ) 7x (s3t 4). After some easy calculations we get ( MA ξc(o c n m, x) = l (48x 4n 48x 3n 48x 4n )(l n x x )(l ) 48x (3n) (l x )n 3x 4n ) 48x (7n/) 96x 4n 6x 4n l n n s=l n l n n [ l n t=s [ n s x (4n 3(s ) t) 48 [ n s l n [ s x (4(n) s 3t) x (3(n t)s) 48 t=l n x (3(n s)t) [ s t=l n x (4st n 3) n l n [ n n s=l n [ l n t=n s [ n t=s x (n3s t ) x (n s3t ) x (s3t 4). Theorem 0. For every n 3 and n ( mod ) consider the graph of G = O m n, with n = m. Then the modified augmented eccentric connectivity polynomial of G is equal to

22 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO ( MA ξc(o c n m (, x) = l 48x 3n 48x 4n 48x 4n) (l x ) 48x (l 7n ) l 96x 4n 6x 4n x )(l n l 3 ) 48x (3n) (l x )n 3x 4n n s=l n l n n s=l n l n s= [ s [ l n t=s [ n s [ n s x (4n 3(s ) t) 08 x (3(n t)s) 08 t=l n x (3(n s)t) t=l n x (4st n 3) 08 l n 08 n [ s x (4(n) s 3t) n s=l n l n s=l n [ n [ n [ l n t=n s t=s x (n3s t ) x (n s3t ) x (s3t 4). Proof. As in Theorem 9, by using Table 4 and the equation (8) the result follows. 4. Conclusions and comparison between the indices High discriminating power and extremely low degeneracy are desirable properties of an ideal topological index, which researchers in theoretical chemistry are striving to achieve. The values of MA ξ c (G) were computed for all the possible structure of three and four vertices. The values and the structures have been presented in Table 5 and their comparison is presented in Table 6. Modified augmented eccentric connectivity index demonstrate exceptionally high discriminating power, defined as the ratio of the highest to lowest value for all possible structures with the same number of vertices. This is evident from the fact that the ratio of the highest to lowest value for all possible structure containing three and four vertices is very high in contrast to ξ(g) and ξ c (G). The ratio of the highest to lowest value for all possible structures containing four vertices for MA ξ c (G) is 6.75 in comparison to.78 and.7 for ξ(g) and ξ c (G), respectively. The exceptionally high discriminating power of the proposed indices makes them extremely sensitive towards minor change(s) in molecular structure. This extreme sensitivity towards branching and the discriminating power of proposed indices are clearly evident from the respective index values of all the possible structures with four vertices. Degeneracy: the number of compounds having identical values/the total number of compounds with the same number of vertices.

23 NEW AND MODIFIED ECCENTRIC INDICES OF OCTAGONAL GRID O m n 3 Table 5. Values of eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index for all possible structures with three and four. S.N Structure ξ(g) MA ξ c (G) ξ c (G) Table 6. Comparison of the discriminating power and degeneracy of eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index using all possible structures with three and four vertices. ξ(g) MA ξ c (G) ξ c (G) For three vertices Minimum value Maximum value 6 Ratio : :.34 :. Degeneracy / 0/ 0/ For four vertices Minimum value 9 6 Maximum value Ratio :.78 :6.75 :.7 Degeneracy /6 /6 /6 Degeneracy is a measure of the ability of an index to differentiate between the relative positions of atom in a molecule. MA ξ c (G) did not exhibit any degeneracy for all possible structures with three vertices whereas MA ξ c (G) had a very low degeneracy of one in the

24 4 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO case of all possible structures with four vertices (Table 6). ξ(g) had one identical values out of 6 structures with only four vertices. Extremely low degeneracy indicates the enhanced capability of these indices to differentiate and demonstrate slight variations in the molecular structure, which clearly reveals the remote chance of different structures having the same value. The Table 7 shows a comparison between the eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index for octagonal grid O n m for finite n = 3,..., 0. Table 7. comparison of ξ c (O n m), E ξ c (O n m) and MA ξ c (O n m) for O n m, when m = n. [n, m ξ c (O n m) E ξ c (O n m) MA ξ c (O n m) [3,3 888 [4, [5, [6,6 97 [7, [8, [9, [0, Acknowledgements The third author of this work were partially supported by MINECO grant number MTM P, Fundación Séneca de la Región de Murcia grant number 99/PI/4. The third and four authors of this work were partially supported by National Nature Science Foundation of China (Western Region Funds) grant number

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26 6 M. NAEEM, M.K. SIDDIQUI, J.L.G. GUIRAO AND W. GAO [3 M. H. Sunilkumar, B. K. Bhagyashri, G. B. Ratnamma, and M. G. Vijay, QSPR analysis of certain graph theocratical matrices and their corresponding energy, Applied Mathematics and Nonlinear Sciences, vol., no., pp. 3-50, 07. [4 H. Wiener, Correlation of Heats of Isomerization, and Differences in Heats of Vaporization of Isomers, Among the Paraffin Hydrocarbons J. Am. Chem. Soc., 69 (947), [5 H. Wiener, Influence of interatomic forces on paraffin properties. J. Chem. Phys., 5 (947), [6 H. Wiener, Vapour-pressure-temperature relations among the branched paran hydrocarbons. J. Chem. Phys., 5 (948), [7 B. Zhou, Z. Du, On eccentric connectivity index. MATCH Commun Math Comput Chem,, 63 (00), Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal, Pakistan. address: naeempkn@gmail.com, kamransiddiqui75@gmail.com Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena, Hospital de Marina, 3003-Cartagena, Región de Murcia, Spain. Corresponding Author address: juan.garcia@upct.es 3 School of Information Science and Technology, Yunnan Normal University, Kunming, China address: gaowei@ynnu.edu.cn