Szeged Related Indices of Unilateral Polyomino Chain and Unilateral Hexagonal Chain

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1 Szeged Related Indices of Unilateral Polyomino Chain Unilateral Hexagonal Chain Wei Gao, Li Shi Abstract In theoretical chemistry, the Szeged index Szeged related indices were introduced to measure the stability of alkanes the strain energy of cycloalkanes In this paper, we determine the edge-vertex Szeged index vertex-edge Szeged index of unilateral polyomino chain unilateral hexagonal chain Furthermore, the total Szeged indices, third atom bond connectivity indices, PI polynomials, vertex PI polynomials, Szeged polynomials edge Szeged polynomials of these chemical structures are presented Index Terms molecular graph, edge-vertex Szeged index, vertex-edge Szeged index, total Szeged index, third atom bond connectivity index, unilateral polyomino chain, unilateral hexagonal chain I INTRODUCTION One of the most important applications of chemical graph theory is to measure chemical, physical pharmaceutical properties of molecules which is called alkanes Several indices relied on the graphical structure of the alkanes are defined employed to model both the melting point boiling point of the molecules Molecular graph is a topological representation of a molecule such that each vertex represents an atom of molecule, covalent bounds between atoms are represented by edges between the corresponding vertices Specifically, topologicandex can be regarded as a score function f : G R, with this property that f(g 1 ) = f(g 2 ) if two molecular graphs G 1 G 2 are isomorphic There are several vertex distance-based degree-based indices which are introduced to analyze the chemical properties of molecule graph, such as: Wiener index, PI index, Szeged index atom-bond connectivity index Several papers contributed to determine the indices of special molecular graphs to be applied in engineering (See Yan et al [1] [2], Gao et al [3] [], Gao Shi [5], Xi Gao [6], Gao Wang [7], Reineix et al [8], Liu Zhang [9], Okamoto Itofor [10] for more details) All (molecular) graphs considered in this paper are finite, loopless, have no multiple edges Let G be a (molecular) graph with vertex set V (G) edge set E(G) All graph notations terminologies used but undefined in this paper can be found in [11] Let e = uv be an edge of the molecular graph G The number of vertices of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by n u (e) Analogously, n v (e) is the number of vertices of G whose distance to the vertex v is smaller than the distance to the Manuscript received November 2, 201; revised January 10, 2015 This work was supported in part by NSFC (No ) W Gao is with the School of Information Technology, Yunnan Normal University, Kunming, China gaowei@ynnueducn L Shi is with the Institute of Medical Biology, Chinese Academy of Medical Sciences Peking Union Medical College, Kunming, China shiliimb@gmailcom vertex u The number of edges of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by m u (e) Analogously, m v (e) is the number of edges of G whose distance to the vertex v is smaller than the distance to the vertex u The Szeged index is closely related to the Wiener index defined as Sz(G) = n u (e)n v (e) Some conclusions for Szeged index applied in chemical engineering can refer to Yousefi-Azari et al [12] The edge Szeged index of G is defined as Sz e (G) = m u (e)m v (e) Cai Zhou [1] determined the n-vertex unicyclic graphs with the largest, the second largest, the smallest the second smallest edge Szeged indices Mahmiani Iranmanesh [15] computed the edge-szeged index of HAC5C7 nanotube Chiniforooshan [16] presented the molecular graphs with maximum edge Szeged index Khalifeh et al [17] studied the edge Szeged index of Hamming molecular graphs C - nanotubes Zhan Qiao [18] determined the edge Szeged index of bridge molecular graph Gutman Ashrafi [19] established the basic properties of edge Szeged index Wang Liu [20] proposed a method of calculating the edge- Szeged index of hexagonal chain The definition of edge-vertex Szeged index vertexedge Szeged index are stated as follows: Sz ev (G) = 1 (m u (e)n v (e) m v (e)n u (e)), 2 Sz ve (G) = 1 2 (m u (e)n u (e) m v (e)n v (e)) Results on edge-vertex Szeged index vertex-edge Szeged index of bridge chemical structure can refer to Alaeiyan Asadpour [21] The number of edges vertices of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by t u (e) Analogously, t v (e) is the number of edges vertices of G whose distance to the vertex v is smaller than the distance to the vertex u The total Szeged index of G is defined as Sz T (G) = t u (e)t v (e) Manuel et al, [22] determined the total Szeged index of C -Nanotubes, C -Nanotori Dendrimer Nanostars The third atom bond connectivity index was introduced as ABC 3 (G) = uv E m u (e) m v (e) 2 m u (e)m v (e)

2 The PI index of molecular graph G is denoted by P I(G) = (m v (e) m u (e)) The vertex PI index of molecular graph G is defined as P I v (G) = (n v (e) n u (e)) Khalifeh et al, [23] presented the vertex PI indices of cartesian product molecular graphs Ashrafi et al, [2] studied the vertex PI index of an infinite family of fullerenes Khalifeh et al, [25] raised a matrix method for computing vertex PI index of molecular graphs As the extension of PI index vertex PI index, the PI polynomial vertex PI polynomial are introduced by Ashrafi et al, [26] stated as P I(G, x) = x (mv(e)mu(e)) P I v (G, x) = x (nv(e)nu(e)) Loghman Badakhshiana [27] determined the PI polynomial of T UC C 8 (S) nanotubes nanotorus Alamian et al, [28] presented the PI polynomial of V-phenylenic nanotubes nanotori Loghman Badakhshiana [29] raised the PI polynomial of zig-zag polyhex nanotubes Similarly, like Szeged index edge Szeged index, the Szeged polynomial edge Szeged polynomial are defined as Se v (G, x) = x (mv(e)mu(e)) Se e (G, x) = x (nv(e)nu(e)), respectively More results on Szeged polynomial edge Szeged polynomial can refer to [30], [31] [32] Although there have been several advances in PI index, vertex PI index, Szeged index, edge Szeged index atom bond connectivity index of molecular graphs, the study of edge-vertex Szeged index, vertex-edge Szeged index, total Szeged index, third atom bond connectivity index, PI polynomial, vertex PI polynomial, Szeged polynomial edge Szeged polynomial of special chemical structures has been largely limited In addition, as widespread critical chemical structures, polyomino system, hexagonal system are widely used in medical science pharmaceutical field As an example, polyomino chain is one of the basic chemical structures, exists widely in benzene alkali molecular structures For these reasons, we have attracted tremendous academic industrianterests to research the edge-vertex Szeged index, vertex-edge Szeged index, total Szeged index, third atom bond connectivity index, PI polynomial, vertex PI polynomial, Szeged polynomial edge Szeged polynomial of these molecular structure from a mathematical point of view The contributions of our paper are two-fold First, we compute the edge-vertex Szeged index, vertex-edge Szeged index, total Szeged index, third atom bond connectivity index, PI polynomial, vertex PI polynomial, Szeged polynomial edge Szeged polynomial of unilateral polyomino chain Second, the edge-vertex Szeged index, vertex-edge Szeged index, total Szeged index, third atom bond connectivity index, PI polynomial, vertex PI polynomial, Szeged polynomial edge Szeged polynomial of unilateral hexagonal chain is calculated II SZEGED RELATED INDICES OF UNILATERAL POLYOMINO CHAIN From the view of graph theory, polyomino is a finite 2- connected planar graph each interior face is surrounded by a square with length Polyomino chain is one class of polyomino such that the connection of centres for adjacent squares constitute a path c 1 c 2 c n, where c i is the centre of i-th square Polyomino chain H n is called a linear chain if the subgraph induced by all 3-degree vertices is a graph with n 2 squares Furthermore, polyomino chain H n is called a Zig-zag chain if the subgraph induced by all vertices with degree > 2 is path with n 1 edges In what follows, we use L n Z n to denote linear polyomino chain Zigzag polyomino chain, respectively The structure of L n Z n can refer to Figure 1 Linear polyomino chain Zig-zag polyomino chain Fig1 The structure of L n Z n Use the similar technology raised in Gutman Klavzar [13], we define elementary cut as follows Choose an edge e of the polyomino system draw a straight line through the center of e, orthogonal on e This line wilntersect the perimeter in two end points P 1 P 2 The straight line segment C whose end points are P 1 P 2 is the elementary cut, intersecting the edge e A fragment S in polyomino chain is just maximal linear chain which includes the squares in start end vertices Let l(s) be the length of fragment which denotes the number of squares it is contained Let H n be a polyomino chain with n squares consist of fragment sequence S 1, S 2,, S m (m 1) Denote l(s i ) = (i = 1,, m) It is not difficult to verify that l 1 l 2 l m = n m 1 V (H n) = 2n 2, E(H n) = 3n 1 For the k-th fragment of polyomino chain, the cut of this fragment is the cut which intersects with l k 1 parallel edges of squares in this fragment A fragment called horizontal fragment if its cut parallels to the horizontal direction, called vertical fragment if its cut parallels to the vertical direction Unilateral polyomino chain is a special kind of polyomino chain such that for each vertical fragment, two horizontal fragments (if exists) adjacent it appear in the left right sides, respectively

3 I II II P 1 P 2 I C II II I II II I Fig2 I-type cut II-type cut of unilateral polyomino chain Our main result in this section stated as follows presents the edge-vertex Szeged index, vertex-edge Szeged index total Szeged index of unilateral polyomino chain Theorem 1: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment = = Sz ev (Hn) l 1 1 m [(12j ) (12j )m (12j )(l 1 j) (20j 8)] l m [(12l m 12j 8) I (12l m 12j 8)m (12l m 12j 8)j (8l m 8j ))] l k 1 m [(2 2k 2j 2)(3 3(m k) 3(l k j) 1) (3 3k 3j 1)(2 1 2 m 2(m k) 2(l k j) 2)] m (l k 1)[(2 2k l k 3)(3 3(m k) l k ) (3 3k l k 3)(2 m 2(m k) l k 1)] Sz ve (Hn) l 1 1 m m [(2 2m 2(l 1 j) )(3 m 3m 3(l 1 j) ) 6j 2 j] l m [(2 2m 2j 2)(3 3m 3j 1) (2l m 2j 2)(3l m 3j 1)] l k 1 [(2 3j 1) (3 1)(2 1 2 m 2k 2j 2)(3 m 3k 3(m k) 3(l k j) 2(m k) 2(l k j) 2)] m (l k 1)[(2 2k l k 3)(3 3k l k 3) (3 l k )(2 m m 3(m k) 2(m k) l k 1)] Sz T (Hn) l 1 1 m = 2 [(5j 2)(5 5m 5(l 1 j) 8)] l m [(5 5m 5j 3)(5l m 5j 3)] l k 1 [(5 5k 5j 3)(5 5(m k) 5(l k j) 3)] m (l k 1)[(5 5k 2l k 6)(5 5(m k) 2l k 1)] m m Proof The cuts in H n are divided into two types: I-type II-type (see Figure 2) An edge is called I-type (or, II-type) if it intersects with I-type (or, II-type) cut Now, we consider the following two cases Case 1 If edge e is I-type in j-th square of k-th fragment (ie, e is edge which is passed by dotted line in Figure 2) We observe that there l k 1 such edges in k-th fragment Subcase 11 If k = 1 n 1 (e) = l 1 1, m n 2 (e) = 2 2m l 1 3, m 2 (e) = 3 m 1 (e) = l 1, m 3m l 1 3 Subcase 12 If k = m Then, we obtain n 1 (e) = 2 2m l m 3, n 2 (e) = l m 1, m 1 (e) = 3 3m l m 3,

4 m 2 (e) = l m Subcase 13 If 2 k m 1 Then, we get n 1 (e) = 2 2k l k 3, n 2 (e) = 2 m 2(m k) l k 1, m 1 (e) = 3 3k l k 3, m 2 (e) = 2 m 3(m k) l k Case 2 If edge e is II-type in j-th square of k-th fragment (ie, e is edge which is passed by real line in Figure 2) Subcase 21 If k = 1 Then, we yield n 2 (e) = 2 m 2 (e) = 3 n 1 (e) = 2j, m 2m 2(l 1 j), m 1 (e) = 3j 2, m 3m 3(l 1 j) Subcase 22 If k = m Then, we infer n 1 (e) = 2 2m 2j 2, n 2 (e) = 2l m 2j 2, m 1 (e) = 3 3m 3j 1, m 2 (e) = 3l m 3j 1 Subcase 23 If 2 k m 1 Then, we deduce n 2 (e) = 2 m 2 (e) = 3 n 1 (e) = 2 2k 2j 2, m 2(m k) 2(l k j) 2, m 1 (e) = 3 3k 3j 1, m 3(m k) 3(l k j) 1 Hence, by combining the above cases the definition of the edge-vertex Szeged index, vertex-edge Szeged index total Szeged index, we obtain Sz ev (Hn) = 1 l {2 m [2j(3 3m 3(l 1 j) ) (3j 2)(2 l m (3 m 2m 2(l 1 j) )] [(2 2m 2j 2)(3l m 3j 1) 3m 3j 1)(2l m 2j 2)] l k 1 [(2 2k 2j 2)(3 3(m k) 3(l k j) 1) (3 3k 3j 1)(2 m 2(m k) 2(l k j) 2)] m (l k 1)[(2 2k l k 3)(3 3(m k) l k ) (3 3k l k 3)(2 l k 1)]} m m m 2(m k) Sz ve (Hn) = 1 l {2 m [2j(3j 2) (2 2m 2(l 1 j) )(3 m 3m 3(l 1 j) )] l m [(2 2m 2j 2)(3 3m 3j 1) (2l m 2j 2)(3l m 3j 1)] l k 1 [(2 3k 3j 1) (3 1)(2 m 2k 2j 2)(3 m 3(m k) 3(l k j) 2(m k) 2(l k j) 2)] m (l k 1)[(2 2k l k 3)(3 3k l k 3) (3 l k )(2 m m 3(m k) 2(m k) l k 1)]}

5 Sz T (Hn) l 1 1 m = 2 [(2j 3j 2) (2 2m 2(l 1 j) 3 m 3m 3(l 1 j) )] l m [(2 2m 2j 2 3 3j 1) (2l m 2j 2 3l m 3j 1)] l k 1 [(2 2k 2j 2 3 3k 3j 1) m (3 3(m k) 3(l k j) 1 m 2(m k) 2(l k j) 2)] m (l k 1)[(2 2k l k 3 3 (3 3k l k 3) m 3(m k) l k 2 2(m k) l k 1)] m 3m The desired conclusion is yielded by arranging above formulas Corollary 2: Let L n be the linear chain with n squares Sz ev (L n) = 3n 3 n 2 7n, Sz ve (L n) = 5n 3 2n 2 3n 2, Sz T (L n) = 37 3 n3 13n n 13 Proof Using the definition of linear chain, we have m = 1, l 1 = n, l 2 = = l m = 0 In terms of Theorem 1, we immediately get the result Corollary 3: Let Zn be the Zig-zag chain with n squares Then, we get Sz ev (Z n) = 3m m2 1 2 m 8, Sz ve (Z n) = 6m m m 8, Sz T (Z n) = 25 2 m m2 85m 36 Proof By virtue of the definition of Zig-zag chain, we have m = n 1, l 1 = l 2 = = l m = 2 In view of Theorem 1, the result is immediately obtained By what we obtained in the proving procedures, we infer the following conclusions on the third atom bond connectivity index Theorem : Let Hn l(s i ) = (i = 1,, m) be the length of each fragment = 2 ABC 3 (Hn) l 1 1 l m 3 m 3m (3j 2)(3 m {(3 m 3m 3j ) 3m)/((3 3m 3j 1)(3l m 3j 1))} 1 2 l k 1 m {(3 3m)/((3 3k 3j 1)(3 m 3(m k j) 1))} 1 2 m (l k 1){(3 2l k 3 1)/((3 3k 3 l k )(3 l k ))} 1 2 m m 3m 3(m k) Corollary 5: Let L n be the linear chain with n squares n 1 ABC 3 (L 3n 3 n) = 2 (3j 2)(3n 3j 1) n 3n 3 (3j 2)(3n 3j 1) (n 1) 2n 2 n Corollary 6: Let Zn be the Zig-zag chain with n squares 3m ABC 3 (Zn) = 2 3m 1 m 3m 1 3 (3k 1)(3(m k) 2) Furthermore, we get the following conclusions on PI Szeged polynomials Theorem 7: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment P I(H n) = 2(l 1 l m 2)(3 m 3m 2) l k 1 m (3 3m 2) m (l k 1)(3 2l k 3 m 3m 3)

6 Corollary 8: Let L n be the linear chain with n squares P I(L n) = 1n 2 1n Corollary 9: Let Z n be the Zig-zag chain with n squares P I(Z n) = 9m 2 15m 8 Theorem 10: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment P I v (Hn) = 2(l 1 l m 2)(2n 2) l k 1 m 2 (2n 2) (l k 1)(2n 2) Corollary 11: Let L n be the linear chain with n squares P I v (L n) = 10n 2 n 6 Corollary 12: Let Z n be the Zig-zag chain with n squares P I v (Z n) = 6m 2 20m 16 Theorem 13: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment P I(H n, x) = 2(l 1 l m 2)x 3 m li 3m m l k 1 x 3 m li 3m (l k 1)x 3 lil k3 m li 3m3 Corollary 1: Let L n be the linear chain with n squares P I(L n, x) = (n 1)x 3n 1 (n 1)x 2n Corollary 15: Let Z n be the Zig-zag chain with n squares P I(Z n, x) = x 3m 3mx 3m1 Theorem 16: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment P I v (H n, x) = 2(l 1 l m 2)x (2n) 2 l k 1 x (2n) m (l k 1)x (2n) Corollary 17: Let L n be the linear chain with n squares P I v (L n, x) = (5n 3)x 2n Corollary 18: Let Z n be the Zig-zag chain with n squares P I v (Z n, x) = (3m )x 2m Theorem 19: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment Se v (Hn, x) l 1 1 = 2 l m x 2j(2 m li 2m 2j) x (2 m li 2m)(2lm 2j) l k 1 {x 2 m li 2kj x 2 m m (l k 1){x 2 x 2 m li 2(m kj) } lil k 2k3 li 2(m k)l k1 } Corollary 20: Let L n be the linear chain with n squares n 1 Se v (L n, x) = 2 2 n x 2j(2 m li 2j) x (2 m li)(2n 2j) (n 1)x 2n Corollary 21: Let Z n be the Zig-zag chain with n squares Se v (Zn, x) = 2x 2(2m) 2x 6(2m) m 3x (2k1)(2m 2) Theorem 22: Let Hn l(s i ) = (i = 1,, m) be the length of each fragment Se e (Hn) l 1 1 = 2 l m x (3j 2)(3 m li 3m 3j) x ((3 li 3m3j1)(3lm 3j1)) l k 1 x 3 li 3k3j1 x 3 m li 3(m kj)1 m x 3 m (l k 1)x 3 li 3k3l k li 3(m k)l k

7 Corollary 23: Let L n be the linear chain with n squares n 1 Se e (L n, x) = 2 x (3j 2)(3n 3j1) n x (3j 2)(3n 3j1) (n 1)x 2n Corollary 2: Let Z n be the Zig-zag chain with n squares Se e (Z n, x) = x 3m1 m 3x (3)(3m 3k ) III SZEGED RELATED INDICES OF UNILATERAL HEXAGONAL CHAIN Hexagonal chain is one class of hexagonal system which is made up of hexagonal In hexagonal chain, each two hexagonals has one common edge or no common vertex Two hexagonals are adjacented if they have common edge No three or more hexagonals share one vertex Each hexagonal has two adjacent hexagonals except hexagonals in terminus, each hexagonal chain has two hexagonals in terminus It is easy to verify that the hexagonal chain with n hexagonals has n 2 vertices 5n 1 edges Let L 6 n Zn 6 be the linear hexagonal chain Zig-zag hexagonal chain, respectively The chemical structure of L 6 n Zn 6 can refer to Figure 3 for more details Again, we use the similar trick which was presented in Gutman Klavzar [13], we define elementary cut as follows Choose an edge e of the hexagonal system draw a straight line through the center of e, orthogonal on e This line wilntersect the perimeter in two end points P 1 P 2 The straight line segment C whose end points are P 1 P 2 is the elementary cut, intersecting the edge e A fragment S in hexagonal chain is just maximal linear chain which include the hexagonals in start end vertices Let l (S) be the length of fragment which denotes the number of hexagonals it is contained Let Hn 6 be a hexagonal chain with n hexagonals consist of fragment sequence S 1, S 2,, S m (m 1) Denote l (S i ) = (i = 1,, m) Then, we verify that l 1 l 2 l m = n m 1 since each two adjacent fragment have one common hexagonal For the k-th fragment of hexagonal chain, the cut of this fragment is the cut which intersects with l k 1 parallel edges of hexagonals in this fragment A fragment called horizontal fragment if its cut parallels to the horizontal direction, otherwise called inclined fragment Unilateral hexagonal chain is a special class of hexagonal chain such that the cut for each inclined fragment at the same angle with a horizontal direction As an example, Figure shows a structure of unilateral hexagonal chain Clearly, linear hexagonal chain L 6 n is a unilateral hexagonal chain with one fragment, Zig-zag is a unilateral hexagonal chain with n 1 fragments We now show the edge-vertex Szeged index, vertex-edge Szeged index total Szeged index of unilateral hexagonal chain Theorem 25: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment Sz ev (H 6 n) l 1 1 = 2 {(j 1)(5( m (5( k 1) 2l k)(( m k) 2l k ) m m 1) l 1 j 3)} k {(( k 1) 1)(5( m l k) (5( k 1) 2l k)(( m k) 3)} (5( m l k)(( l k 1 m k) m {(( k 1) j 1) m k) 2l k) (5( k 1) m m k) l k j 3)} l m {(( m 1) j 1)(5( m k) 2l k) (5( k 1) l k)(l m j 3)} 1 (l k 1){((( k 1) 2l k 2 m 1)(5( m k) 5j 2) (5( k 1) 5j 3)(( 1))} Sz ve (H 6 n) m m m k) 2l k l 1 1 = 2 {(j 1)(5( k 1) 2l k) m (( m 1) l 1 j 3)(5( m k) 2l k)} k {(( l k) (( k 1) 1)(5( m m k) 3)(5( m k 1) m m

8 Linear hexagonal chain Zig-zag hexagonal chain Fig3 The structure of L 6 n Z 6 n I II I I II II II II I II II II P 1 P 2 C II I II Fig I-type cut II-type cut of unilateral hexagonal chain k) 2l k)} l k 1 {(( 1) 2l k) (( 3)(5( m k 1) j 1)(5( m m k) l k j m k) 2l k)} l m {(( m 1) j 1)(5( k 1) 2l k) (l m j 3)(5( m m k) 2l k)} 1 (l k 1){(( k 1) 2l k 2 1)(5( k 1) 5j 3) (( m k m m k) 2l k 1)(5( m k) 5j 2)} Sz T (H 6 n) l 1 1 m = (5( k 1) 2l k j 1) ( 5 m j 7 9m 5k 2l k) (9 9k 8 6l k) (9 9m 9k 3 2l k) (9( l k 1 m (9( k 1) j 1 2l k) m l m ( m k) 6l k j 3) 5 l k)(9l m j 3 5 (9 m j 5k 8 5m 5k 2l k) (l k 1)(9 9k 2l k 7 5j) m 9m 9k 6l k 3 5j)

9 Proof The cuts in H 6 n are divided into two types: I-type II-type (see Figure ) An edge is called I-type if it intersects with I-type cut Also, an edge is called II-type if it intersects with II-type cut In what follows, we consider two situations according to the type of edge Case 1 If edge e is I-type in j-th square of k-th fragment (ie, e is edge which is passed by dotted line in Figure ) Then, the value of m 1 (e) m 2 (e) is calculated as follows: m 1 (e) = 5( k 1) 5j 3, m m 2 (e) = 5( m k) 5j 2 Subcase 11 If k = 1 n 2 (e) = n 1 (e) = 2l 1 1 m ( m 1) 2l 1 1 Subcase 12 If k = m Then, we obtain n 1 (e) = ( m 1) 2l k 1 n 2 (e) = 2l m 1 Subcase 13 If 2 k m 1 Then, we get n 1 (e) = ( k 1) 2l k 1 n 2 (e) = m ( m k) 2l k 1 Case 2 If edge e is II-type in j-th square of k-th fragment (ie, e is edge which is passed by real line in Figure ) Then, we deduce m 1 (e) = 5( k 1) 2l k, m 2 (e) = 5( m m k) 2l k Subcase 21 If k = 1 Then, we yield n 2 (e) = n 1 (e) = j 1 m l 1 j 3 Subcase 22 If k = m Then, we infer n 1 (e) = ( k 1) j 1 n 2 (e) = l m j 3 Subcase 23 If 2 k m 1 Then, we deduce n 1 (e) = ( k 1) j 1 n 2 (e) = m (l k j) 3 In particular, if j = l k in Subcase 23, we have n 1(e) = k ( k1) 1 n 2(e) = m ( mk)3 Hence, by combining the above cases the definition of the the edge-vertex Szeged index, vertex-edge Szeged index total Szeged index, we get Sz ev (H 6 n) = 1 l1 1 2 { {(j 1)[5( m (5( k 1) 2l k)[( m k) 2l k] m m 1) l 1 j 3]} k {[( k 1) 1][5( m k) 2l k] (5( k 1) 2l k)[( m k) 3]} [5( m l k)[( l k 1 m {[( k 1) j 1] m m k) 2l k] (5( k 1) m m k) l k j 3]} l m {[( m 1) j 1][5( m m k) 2l k] (5( k 1) 2l k)(l m j 3)} (l k 1){[( k 1) 2l k 1] m (5( m k) 5j 2) (5( k 1) 5j 3)[( Sz ve (H 6 n) m m k) 2l k 1]}} = 1 l1 1 2 { {(j 1)(5( k 1) 2l k)

10 m [( m 1) l 1 j 3][5( m k) 2l k]} 2 {[( m k k 1) 1](5( k 1) 2l k) [( m k) 3][5( l k 1 m {[( k 1) 2l k) [( 3][5( m m m k) 2l k]} k 1) j 1](5( m m k) 2l k]} m k) l k j l m {[( m 1) j 1](5( k 1) 2l k) (l m j 3)[5( m m k) 2l k]} (l k 1){[( k 1) 2l k 1] (5( k 1) 5j 3) [( m m l k 1](5( m k) 5j 2)}} Sz T (H 6 n) l 1 1 = {(j 1 5( k 1) 2l k) m [( m 1) l 1 j 3 5( m k) 2l k]} k {(( k 1) 1 5( k 1) 2l k) [( 5( m m m k) 2l k]} l k 1 {(( k 1) 2l k) (( m k) m k) 3 m k 1) j 1 5( m m k) l k j 3 5( m m k) 2l k)} l m {(( m 1) j 1 5( k 1) 2l k) (l m j 3 5( m m k) 2l k)} (l k 1){(( k 1) 2l k 1 5( k 1) 5j 3) (( m m m k) 2l k 1 5( m k) 5j 2)} The desired conclusion is yielded by arranging above formulas As we presented in Corollary 2 Corollary 3, we can calculate the the edge-vertex Szeged index, vertex-edge Szeged index total Szeged index of linear chain L 6 n with n hexagonals Zig-zag chain Zn 6 with n hexagonals by virtue of the same tricks These computations are left to the readers According to what we obtained in the proving procedures, we deduce the following results on third atom bond connectivity index Theorem 26: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment ABC 3 (Hn) 6 5 m = 5m (5( m m 1) 3) m k {(5 5m 2)/((5( k 1) 3)(5( m m k) 2))} 1 2 l m k 1 m {(5 5m 2)/((5( k 1) m 5j 3)(5( m k) 5j 2))} 1 2 m (l k 1){(5 5 m 3)/((5( k 1) 2l k)(5( l k))} 1 2 l k 5m m m k) Corollary 27: Let L 6 n be the linear chain with n hexago-

11 nals ABC 3 (L 6 5n 1 n) = (5n 3) n 1 5n 3 (5j 3)(5n 5j 2) (n 1) 2n 2 n Corollary 28: Let Zn 6 be the Zig-zag chain with n hexagonals ABC 3 (Zn) 6 5m = (5m 2) 5m 2 (5k 2)(5m 5k 2) 3 m 5m 1 (5k 1)(5m 5k ) At last, we present the following conclusions Theorem 29: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment P I(Hn) 6 m m = (5 5m 6) 2 (5 5m ) l m k 1 m (5 5m ) m (l k 1)(5 5 m l k 5m 5) Corollary 30: Let L 6 n be the linear chain with n hexagonals P I(L 6 n) = 2n 2 12n Corollary 31: Let Z 6 n be the Zig-zag chain with n hexagonals P I(Z 6 n) = 25m 2 27m 26 Theorem 32: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment P I v (H 6 n) = (l 1 l m 2m 10)(n 2) l k 1 (n 2) (l k 1)(n 2) Corollary 33: Let L 6 n be the linear chain with n hexagonals P I v (L 6 n) = 32n 2 16n 16 Corollary 3: Let Z 6 n be the Zig-zag chain with n hexagonals P I v (Z 6 n) = 20n 2 10n 10 Theorem 35: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment P I(H 6 n, x) = x (5 m 5m6) 2 m l k 1 m x (5 m 5m) (l k 1)x (5 5 m x (5 m 5m) l k 5m5) Corollary 36: Let L 6 n be the linear chain with n hexagonals P I(L 6 n, x) = (n 1)x 5n1 (n 1)x n Corollary 37: Let Z 6 n be the Zig-zag chain with n hexagonals P I(Z 6 n, x) = x 5m6 2(m 1)x 5m 3mx 5m3 Theorem 38: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment P I v (H 6 n, x) = (l 1 l m 2m 10)x n l k 1 x n (l k 1)x n Corollary 39: Let L 6 n be the linear chain with n hexagonals P I v (L 6 n, x) = 8(n 1)x n Corollary 0: Let Z 6 n be the Zig-zag chain with n hexagonals P I v (Z 6 n, x) = 5(n 1)x n Theorem 1: Let Hn l (S i ) = (i = 1,, m) be the length of each fragment

12 Se v (H 6 n, x) l 1 1 = x (j 1)[( m m1) j3] x [( k k1) 1][( m mk)3] l k 1 {x ( k1)j 1 x ( m mk) j3 } l m x [( m1)j 1](l m j3) (l k 1){x ( k1)l k 1 x ( m mk)l k 1 } Corollary 2: Let L 6 n be the linear chain with n hexagonals n 1 Se v (L 6 n, x) = x (j 1)(n j3) n x (j 1)(n j3) Corollary 3: Let Z 6 n be the Zig-zag chain with n hexagonals Se v (Zn, 6 x) = 8x 3(n 1) 2 3 x (k1)(m k5) x (k3)(m k3) Theorem : Let Hn l (S i ) = (i = 1,, m) be the length of each fragment Se e (H 6 n, x) = x (5( m m1) 3) l m k 1 x ((5( k k1) 3)(5( m mk))) {x 5( k1)5j 3 x 5( m mk) 5j } m (l k 1){x 5( k1)l k x 5( m mk)l k } Corollary 5: Let L 6 n be the linear chain with n hexagonals n 1 Se e (L 6 n, x) = x (5n 3) (n 1)x n2 x (5j 3)(5n 5j) Corollary 6: Let Z 6 n be the Zig-zag chain with n hexagonals Se e (Zn, 6 x) = x (5m) 2 m 3x (5)(5m 5k) ACKNOWLEDGMENT x ((5k)(5m 5k)) We thank the reviewers for their constructive comments in improving the quality of this paper REFERENCES [1] L Yan, Y Li, W Gao, J Li, PI Index for Some Special Graphs, Journal of Chemical Pharmaceutical Research, vol 5, no 11, pp , 2013 [2] L Yan, Y Li, W Gao, J Li, On the Extremal Hyper-wiener Index of Graphs, Journal of Chemical Pharmaceutical Research, vol 6, no 3, pp 77-81, 201 [3] W Gao, L Liang, Y Gao, Some Results on Wiener Related Index Shultz Index of Molecular Graphs, Energy Education Science Technology: Part A, vol 32, no 6, pp , 201 [] W Gao, L Liang, Y Gao, Total Eccentricity, Adjacent Eccentric Distance Sum Gutman Index of Certain Special Molecular Graphs, The BioTechnology: An Indian Journal, vol 10, no 9, pp , 201 [5] W Gao L Shi, Wiener Index of Gear Fan Graph Gear Wheel Graph, Asian Journal of Chemistry, vol 26, no 11, pp , 201 [6] W F Xi W Gao, Geometric-Arithmetic Index Zagreb Indices of Certain Special Molecular Graphs, Journal of Advances in Chemistry, vol 10, no 2, pp , 201 [7] W Gao, W F Wang, Second Atom-Bond Connectivity Index of Special Chemical Molecular Structures, Journal of Chemistry, Vol 201, Article ID 90625, 8 pages, [8] A Reineix, P Dur, F Dubois, Kron s Method Cell Complexes for Magnetomotive Electromotive Forces Olivier Maurice, IAENG International Journal of Applied Mathematics, vol, no, pp , 201 [9] J Liu, X D Zhang, Cube-Connected Complete Graphs, IAENG International Journal of Applied Mathematics, vol, no 3, pp , 201 [10] S Okamoto, A Ito, Effect of Nitrogen Atoms Grain Boundaries on Shear Properties of Graphene by Molecular Dynamics Simulations, Engineering Letters, vol 22, no 3, pp, 12-18, 201 [11] J A Bondy U S R Mutry, Graph Theory, Berlin: Spring, 2008 [12] H Yousefi-Azari, B Manoochehrian, A R Ashrafi, Szeged Index of Some Nanotubes, Current Applied Physics, vol 8, no 6, pp , 2008 [13] I Gutman S Klavzar, An Algorithm for the Calculation of the Szeged Index of Benzenoid Hydrocarbons, Journal of Chemical Information Computer Sciences, vol 35, pp , 1995 [1] X Cai B Zhou, Edge Szeged Index of Unicyclic Graphs, MATCH Commun Math Comput Chem, vol 63, pp 133-1, 2010 [15] A Mahmiani A Iranmanesh, Edge-Szeged Index of HAC 5 C 7 [r, p] Nanotube, MATCH Commun Math Comput Chem, vol 62, , 2009 [16] E Chiniforooshan, Maximum Values of Szeged Index Edge- Szeged Index of Graphs, Electronic Notes in Discrete Mathematics, vol 3, no 1, pp 05-09, 2009 [17] M H Khalifeh, H Yousefi-Azari, A R Ashrafi, I Gutman, The Edge Szeged Index of Product Graphs, CCACAA, vol 81, no 2, pp , 2008 [18] F Zhan Y Qiao, On Edge Szeged Index of Bridge Graphs, Lecture Notes in Electrical Engineering, vol 206, pp , 2013 [19] I Gutman A R Ashrafi, The Edge Version of the Szeged Index, CCACAA, vol 81, no 2, pp , 2008

13 [20] S Wang B Liu, A Method of Calculating the Edge-Szeged Index of Hexagonal Chain, MATCH Commun Math Comput Chem, vol 68, pp 91-96, 2012 [21] M Alaeiyan J Asadpour, The Vertex-edge Szeged Index of Bridge Graphs, World Applied Sciences Journal, vol 1, no 8, pp , 2011 [22] P Manuel, I Rajasingh, M Arockiaraj, Total-Szeged Index of C -Nanotubes, C -Nanotori Dendrimer Nanostars, Journal of Computational Theoretical Nanoscience, vol 10, pp 05-11, 2013 [23] M H Khalifeh, H Yousefi-Azari, A R Ashrafi, Vertex edge PI indices of cartesian product graphs, Discrete Appl Math, vol 156, pp , 2008 [2] A R Ashrafi, M Ghorbani, M Jalali, The vertex PI Szeged indices of an infinite family of fullerenes, J Theor Comput Chem, vol 7, no 2, pp , 2008 [25] M H Khalifeh, H Yousefi-Azari, A R Ashrafi, A matrix method for computing Szeged vertex PI indices of join composition of graphs, Linear Alg Appl, vol 29, no (11-12), pp , 2008 [26] A R Ashrafi, B Manoochehrian, H Yousefi-Azari, On the PI polynomial of a graph, Util Math, vol 71, pp , 2006 [27] A Loghman L Badakhshiana, PI polynomial of T UC C 8 (S) nanotubes nanotorus, Digest Journal of Nanomaterials Biostructures, vol, no, pp , 2009 [28] V, Alamian, A Bahrami, B Edalatzadeh, PI polynomial of V- phenylenic nanotubes nanotori, International Journal of Molecular Sciences, vol 9, pp , 2008 [29] A Loghman L Badakhshiana, PI polynomial of zig-zag polyhex nanotubes, Digest Journal of Nanomaterials Biostructures, vol 3, no, pp , 2008 [30] A R Ashrafi M Mirzargar, The edge Szeged polynomial of graphs, MATCH Commun Math Comput Chem, vol 60 pp , 2008 [31] M Mirzargar, PI, szeged edge szeged polynomials of a dendrimer nanostar, MATCH Commun Math Comput Chem, vol 62, pp , 2009 [32] M Ghorbani M Jalali, The vertex PI, szeged omega polynomials of carbon nanocones CNC [n], MATCH Commun Math Comput Chem, vol 62 pp, , 2009 Wei Gao, male, was born in the city of Shaoxing, Zhejiang Province, China on Feb 13, 1981 He got two bachelor degrees on computer science from Zhejiang industrial university in 200 mathematics education from College of Zhejiang education in 2006 Then, he was enrolled in department of computer science information technology, Yunnan normal university, got Master degree there in 2009 In 2012, he got PhD degree in department of Mathematics, Soochow University, China Now, he acts as lecturer in the department of information, Yunnan Normal University As a researcher in computer science mathematics, his interests are covering two disciplines: Graph theory, Statistical learning theory, Information retrieval, Artificial Intelligence Li Shi, female, is professor in department of research development management, institute of medical biology, Chinese academy of medical sciences (CAMS) Peking union medical college (PUMC)