SOME VARIANTS OF THE SZEGED INDEX UNDER ROOTED PRODUCT OF GRAPHS

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1 Miskolc Mathematical Notes HU e-issn Vol. 17 (017), No., pp DOI: /MMN.017. SOME VARIANTS OF THE SZEGED INDE UNDER ROOTED PRODUCT OF GRAPHS MAHDIEH AZARI Received 10 November, 015 Abstract. The Szeged index S.G/ of a connected graph G is defined as the sum of the terms n u.ejg/n v.ejg/ over all edges e D uv of G, where n u.ejg/ is the number of vertices of G lying closer to u than to v and n v.ejg/ is the number of vertices of G lying closer to v than to u. In this paper, some variants of the Szeged index such as the edge PI index, edge Szeged index, edge-vertex Szeged index, vertex-edge Szeged index, and revised edge Szeged index are studied under rooted product of graphs. Results are applied to compute these graph invariants for some chemical graphs by specializing components in rooted products. 010 Mathematics Subject Classification: 05C1; 05C76 Keywords: edge PI index, edge Szeged index, edge-vertex Szeged index, vertex-edge Szeged index, revised edge Szeged index, rooted product of graphs 1. INTRODUCTION In this paper, we consider connected finite graphs without loops or multiple edges. Let G be such a graph with vertex set V.G/ and edge set E.G/. We denote by d.u;vjg/ the distance between the vertices u and v in G which is the length of any shortest path in G connecting u and v. Let e D uv be the edge of G connecting the vertices u and v. The quantities n 0.ejG/, n u.ejg/, and n v.ejg/ are defined to be the number of vertices of G equidistant from u and v, the number of vertices of G whose distance to u is smaller than the distance to v, and the number of vertices of G whose distance to v is smaller than the distance to u, respectively, i.e., n 0.ejG/ D jf V.G/ W d. ;ujg/ D d. ;vjg/gj; n u.ejg/ D jf V.G/ W d. ;ujg/ < d. ;vjg/gj; n v.ejg/ D jf V.G/ W d. ;vjg/ < d. ;ujg/gj: For an edge e D uv E.G/ and a vertex V.G/, the distance between and e is defined as d. ;ejg/ D minfd. ;ujg/;d. ;vjg/g. The quantities m 0.ejG/, m u.ejg/, and m v.ejg/ are defined to be the number of edges of G equidistant from u and v, the number of edges of G whose distance to u is smaller than the distance c 017 Miskolc University Press

2 76 MAHDIEH AZARI to v, and the number of edges of G whose distance to v is smaller than the distance to u, respectively, i.e., m 0.ejG/ D jff E.G/ W d.u;f jg/ D d.v;f jg/gj; m u.ejg/ D jff E.G/ W d.u;f jg/ < d.v;f jg/gj; m v.ejg/ D jff E.G/ W d.v;f jg/ < d.u;f jg/gj: For the vertex V.G/, we define m.g/ D jfe D uv E.G/ W d.u; jg/ d.v; jg/gj: Chemical graphs, particularly molecular graphs, are graph-based descriptions of molecules, with vertices representing the atoms and edges representing the bonds. A numerical invariant associated with a chemical graph is called topological index or graph invariant. Topological indices are used in theoretical chemistry for the design of chemical compounds with given physicochemical properties or given pharmacologic and biological activities [6, 0]. The Wiener index [1], defined as the sum of distances between all pairs of vertices in a chemical graph, is the oldest and the most thoroughly studied topological index from both theoretical and practical point of view. Motivated by the original definition of the Wiener index, the Szeged index [11] was introduced in 199 which coincides with the Wiener index for a tree. It found applications in quantitative structure-property-activity-toxicity modeling [16]. The Szeged index of a graph G is defined as S.G/ D n u.e jg /n v.e jg /: eduve.g/ In recent years, some variants of the Szeged index such as the vertex PI index [17], edge PI index [15], edge Szeged index [1], edge-vertex Szeged index [18], vertexedge Szeged index [9], revised Szeged index [19], and revised edge Szeged index [7] have attracted much attention in both chemistry and mathematics. These indices are defined for a graph G as follows: PI v.g/ D nu.ejg/ C n v.ejg/ ; PI e.g/ D S e.g/ D S ev.g/ D 1 eduve.g/ eduve.g/ eduve.g/ eduve.g/ mu.ejg/ C m v.ejg/ ; m u.ejg/m v.ejg/; nu.ejg/m v.ejg/ C n v.ejg/m u.ejg/ ;

3 S ve.g/ D 1 S.G/ D S e.g/ D SOME VARIANTS OF THE SZEGED INDE 76 eduve.g/ eduve.g/ eduve.g/ nu.ejg/m u.ejg/ C n v.ejg/m v.ejg/ ; nu.ejg/ C n 0.ejG/ nv.ejg/ C n 0.ejG/ ; mu.ejg/ C m 0.ejG/ mv.ejg/ C m 0.ejG/ : We refer the reader to [1, 10, 1] for more information on these indices. The rooted product G 1 fg g of a graph G 1 and a rooted graph G is the graph obtained by taking one copy of G 1 and jv.g 1 /j copies of G, and by identifying the root vertex of the i-th copy of G with the i-th vertex of G 1, for i D 1;;:::;jV.G 1 /j. In this paper, we study the edge PI index, edge Szeged index, edge-vertex Szeged index, vertex-edge Szeged index, and revised edge Szeged index under rooted product of graphs. Results are applied to compute these invariants for some chemical graphs by specializing components in rooted products. For more information on computing topological indices of rooted product see [ 5, 8, 1, ].. RESULTS AND DISCUSSION Let G 1 and G be two connected graphs with vertex sets V.G 1 / and V.G / and edge sets E.G 1 / and E.G /, respectively. In this section, we compute some variants of the Szeged index for rooted product of G 1 and G. Throughout this section, the graph G is assumed to be rooted on the vertex x V.G / and the degree of x in G is denoted by ı. Also, we denote by n i and m i, the order and size of the graph G i, respectively, where i f1;g. In addition, for notational convenience, we define N D n v.ejg /; N D n u.ejg /; M D d.u;xjg /<d.v;xjg / d.u;xjg /<d.v;xjg / m v.ejg /; M D d.u;xjg /<d.v;xjg / d.u;xjg /<d.v;xjg / m u.ejg /: Theorem 1. The edge PI index of the rooted product G 1 fg g is given by PI e.g 1 fg g/ D PI e.g 1 / C m PI v.g 1 / C n 1 PI e.g / C n 1 m 1 C.n 1 1/m mx.g /: (.1) Proof. From the definition of the edge PI index, we have PI e.g 1 fg g/ D mu.ejg 1 fg g/ C m v.ejg 1 fg g/ : eduve.g 1 fg g/ We partition the above sum into two sums as follows:

4 76 MAHDIEH AZARI The first sum S 1 consists of contributions to PI e.g 1 fg g/ of edges from G 1, S 1 D mu.ejg 1 fg g/ C m v.ejg 1 fg g/ : eduve.g 1 / S 1 D mu.ejg 1 / C m n u.ejg 1 / C m v.ejg 1 / C m n v.ejg 1 / D eduve.g 1 / eduve.g 1 / C m eduve.g 1 / mu.ejg 1 / C m v.ejg 1 / D PI e.g 1 / C m PI v.g 1 /: nu.ejg 1 / C n v.ejg 1 / The second sum S consists of contributions to PI e.g 1 fg g/ of edges from n 1 copies of G, S D n 1 mu.ejg 1 fg g/ C m v.ejg 1 fg g/ : eduve.g / S D n 1 m1 C.n 1 C n 1 D n 1 h C d.u;xjg / d.v;xjg / d.u;xjg /Dd.v;xjG / d.u;xjg / d.v;xjg / d.u;xjg /Dd.v;xjG / C n 1 m 1 C.n 1 1/m C m u.ejg / C m v.ejg / mu.ejg / C m v.ejg / mu.ejg / C m v.ejg / mu.ejg / C m v.ejg / i 1/m jfe D uv E.G / W d.u;xjg / d.v;xjg /gj D n 1 PI e.g / C n 1 m 1 C.n 1 1/m mx.g /: Eq. (.1) is obtained by adding the quantities S 1 and S. Theorem. The edge Szeged index of the rooted product G 1 fg g is given by S e.g 1 fg g/ D S e.g 1 / C m S.G 1/ C m S ev.g 1 / C n 1 S e.g / C n 1 m 1 C.n 1 1/m M : (.)

5 SOME VARIANTS OF THE SZEGED INDE 765 Proof. From the definition of the edge Szeged index, we have S e.g 1 fg g/ D m u.ejg 1 fg g/m v.ejg 1 fg g/: eduve.g 1 fg g/ We partition the above sum into two sums as follows: The first sum S 1 consists of contributions to S e.g 1 fg g/ of edges from G 1, S 1 D m u.ejg 1 fg g/m v.ejg 1 fg g/: eduve.g 1 / S 1 D mu.ejg 1 / C m n u.ejg 1 / m v.ejg 1 / C m n v.ejg 1 / D eduve.g 1 / eduve.g 1 / C m eduve.g 1 / m u.ejg 1 /m v.ejg 1 / C m eduve.g 1 / n u.ejg 1 /n v.ejg 1 / nu.ejg 1 /m v.ejg 1 / C n v.ejg 1 /m u.ejg 1 / D S e.g 1 / C m S.G 1/ C m S ev.g 1 /: The second sum S consists of contributions to S e.g 1 fg g/ of edges from n 1 copies of G, S D n 1 m u.ejg 1 fg g/m v.ejg 1 fg g/: eduve.g / S D n 1 m1 C.n 1 1/m C m u.ejg / m v.ejg / C n 1 D n 1 h C d.u;xjg /<d.v;xjg / d.u;xjg /Dd.v;xjG / d.u;xjg /<d.v;xjg / d.u;xjg /Dd.v;xjG / m u.ejg /m v.ejg / m u.ejg /m v.ejg / i m u.ejg /m v.ejg / C n 1 m 1 C.n 1 1/m d.u;xjg /<d.v;xjg / D n 1 S e.g / C n 1 m 1 C.n 1 1/m M : m v.ejg /

6 766 MAHDIEH AZARI Eq. (.) is obtained by adding the quantities S 1 and S. Theorem. The edge-vertex Szeged index of the rooted product G 1 fg g is given by S ev.g 1 fg g/ D n S ev.g 1 / C n m S.G 1 / C n 1 S ev.g / C 1 n 1n.n 1 1/M C 1 n 1 m 1 C m.n 1 1/ N : (.) Proof. From the definition of the edge-vertex Szeged index, we have S ev.g 1 fg g/ D 1 nu.ejg 1 fg g/m v.ejg 1 fg g/ eduve.g 1 fg g/ C n v.ejg 1 fg g/m u.ejg 1 fg g/ : We partition the above sum into two sums as follows: The first sum S 1 consists of contributions to S ev.g 1 fg g/ of edges from G 1, S 1 D 1 nu.ejg 1 fg g/m v.ejg 1 fg g/cn v.ejg 1 fg g/m u.ejg 1 fg g/ : eduve.g 1 / S 1 D 1 hn n u.ejg 1 / m v.ejg 1 / C m n v.ejg 1 / eduve.g 1 / C n n v.ejg 1 / m u.ejg 1 / C m n u.ejg 1 / i D 1 n C n m eduve.g 1 / eduve.g 1 / nu.ejg 1 /m v.ejg 1 / C n v.ejg 1 /m u.ejg 1 / n u.ejg 1 /n v.ejg 1 / D n S ev.g 1 / C n m S.G 1 /: The second sum S consists of contributions to S ev.g 1 fg g/ of edges from n 1 copies of G, S D 1 n 1 nu.ejg 1 fg g/m v.ejg 1 fg g/ eduve.g / C n v.ejg 1 fg g/m u.ejg 1 fg g/ : S D 1 n h nu 1.ejG / C n.n 1 1/ m v.ejg / d.u;xjg /<d.v;xjg /

7 SOME VARIANTS OF THE SZEGED INDE 767 C n v.ejg / m 1 C m.n 1 1/ C m u.ejg / i C 1 n 1 nu.ejg /m v.ejg / C n v.ejg /m u.ejg / D 1 n 1 C h d.u;xjg /Dd.v;xjG / d.u;xjg /<d.v;xjg / d.u;xjg /Dd.v;xjG / C 1 n 1n.n 1 1/ nu.ejg /m v.ejg / C n v.ejg /m u.ejg / nu.ejg /m v.ejg / C n v.ejg /m u.ejg / i d.u;xjg /<d.v;xjg / C 1 n 1 m 1 C m.n 1 1/ m v.ejg / d.u;xjg /<d.v;xjg / n v.ejg / D n 1 S ev.g / C 1 n 1n.n 1 1/M C 1 n 1 m 1 C m.n 1 1/ N : Eq. (.) is obtained by adding the quantities S 1 and S. Let G be a graph with edge set E.G/. We define the second vertex PI index of G as PI v./.g/ D nu.ejg/ C n v.ejg/ : eduve.g/ Theorem ( [9]). Let G be a graph of order n and size m. Then S.G/ D mn 1./ PI v.g/ C 1 S.G/: (.) Theorem 5. The vertex-edge Szeged index of the rooted product G 1 fg g is given by S ve.g 1 fg g/ D n S ve.g 1 / C 1 n m PI./ v.g 1/ C n 1 S ve.g / C 1 n 1n.n 1 1/ m 1 C m.n 1 1/ m x.g / (.5) C 1 n 1 m 1 C m.n 1 1/ N C 1 n 1n.n 1 1/M :

8 768 MAHDIEH AZARI Proof. From the definition of the vertex-edge Szeged index, we have S ve.g 1 fg g/ D 1 h n u.ejg 1 fg g/m u.ejg 1 fg g/ eduve.g 1 fg g/ i C n v.ejg 1 fg g/m v.ejg 1 fg g/ : We partition the above sum into two sums as follows: The first sum S 1 consists of contributions to S ve.g 1 fg g/ of edges from G 1, S 1 D 1 h i n u.ejg 1 fg g/m u.ejg 1 fg g/cn v.ejg 1 fg g/m v.ejg 1 fg g/ : eduve.g 1 / S 1 D 1 hn n u.ejg 1 / m u.ejg 1 / C m n u.ejg 1 / eduve.g 1 / C n n v.ejg 1 / m v.ejg 1 / C m n v.ejg 1 / i D 1 n C 1 n m eduve.g 1 / eduve.g 1 / nu.ejg 1 /m u.ejg 1 / C n v.ejg 1 /m v.ejg 1 / nu.ejg 1 / C n v.ejg 1 / D n S ve.g 1 / C 1 n m PI./ v.g 1/: The second sum S consists of contributions to S ve.g 1 fg g/ of edges from n 1 copies of G, S D 1 n h 1 n u.ejg 1 fg g/m u.ejg 1 fg g/ eduve.g / i C n v.ejg 1 fg g/m v.ejg 1 fg g/ : S D 1 n h nu 1.ejG / C n.n 1 1/ m 1 C m.n 1 1/ d.u;xjg /<d.v;xjg / C m u.ejg / i C n v.ejg /m v.ejg / C 1 n 1 d.u;xjg /Dd.v;xjG / nu.ejg /m u.ejg / C n v.ejg /m v.ejg /

9 D 1 n 1 C h SOME VARIANTS OF THE SZEGED INDE 769 d.u;xjg /<d.v;xjg / d.u;xjg /Dd.v;xjG / nu.ejg /m u.ejg / C n v.ejg /m v.ejg / nu.ejg /m u.ejg / C n v.ejg /m v.ejg / i C 1 n 1n.n 1 1/ m 1 C m.n 1 1/ ˇˇˇfe D uv E.G / W d.u;xjg / < d.v;xjg /gˇ C 1 n 1 m 1 C m.n 1 1/ n u.ejg / C 1 n 1n.n 1 1/ d.u;xjg /<d.v;xjg / d.u;xjg /<d.v;xjg / m u.ejg / D n 1 S ve.g / C 1 n 1n.n 1 1/ m 1 C m.n 1 1/ m x.g / C 1 n 1 m 1 C m.n 1 1/ N C 1 n 1n.n 1 1/M : Eq. (.5) is obtained by adding the quantities S 1 and S. Using Eq. (.), we can get an alternative formula for the vertex-edge Szeged index of the rooted product of G 1 and G. Corollary 1. The vertex-edge Szeged index of the rooted product G 1 fg g is given by S ve.g 1 fg g/ D n S ve.g 1 / C n m S.G 1 / n m S.G 1 / C n 1 S ve.g / Proof. From Eq. (.), we get C 1 n 1 n m 1 m C 1 n 1n.n 1 1/ m 1 C m.n 1 1/ m x.g / C 1 n 1n.n 1 1/M C 1 n 1 m 1 C m.n 1 1/ N : (.6) PI./ v.g 1/ D m 1 n 1 S.G 1 / C S.G 1 /: Eq. (.6) is obtained by applying the above equation in Eq. (.5) and simplifying the resulting expression. Let G be a graph with edge set E.G/. We define the second edge PI index of G as.g/ D mu.ejg/ C m v.ejg/ : PI./ e eduve.g/

10 770 MAHDIEH AZARI Theorem 6 ([9]). Let G be a graph of size m. Then S e.g/ D m 1./ PI e.g/ C 1 S e.g/: (.7) Lemma 1. The second edge PI index of G 1 fg g is given by PI e./.g 1 fg g/ D PI e./.g 1 / C m./ PI v.g 1/ C m S ve.g 1 / C n 1 PI./ e.g / C n 1 m 1 C.n 1 1/m mx.g / C n 1 m 1 C.n 1 1/m M : (.8) Proof. From the definition of the second edge PI index, we have.g 1 fg g/ D mu.ejg 1 fg g/ C m v.ejg 1 fg g/ : PI./ e eduve.g 1 fg g/ We partition the above sum into two sums as follows: The first sum S 1 consists of contributions to PI e./.g 1 fg g/ of edges from G 1, S 1 D mu.ejg 1 fg g/ C m v.ejg 1 fg g/ : eduve.g 1 / S 1 D mu.ejg 1 / C m n u.ejg 1 / C mv.ejg 1 / C m n v.ejg 1 / D eduve.g 1 / eduve.g 1 / C m C m eduve.g 1 / mu.ejg 1 / C m v.ejg 1 / eduve.g 1 / nu.ejg 1 / C n v.ejg 1 / nu.ejg 1 /m u.ejg 1 / C n v.ejg 1 /m v.ejg 1 / D PI e./.g 1 / C m./ PI v.g 1/ C m S ve.g 1 /: The second sum S consists of contributions to PI e./.g 1 fg g/ of edges from n 1 copies of G, S D n 1 mu.ejg 1 fg g/ C m v.ejg 1 fg g/ : eduve.g / S D n 1 m1 C.n 1 d.u;xjg /<d.v;xjg / 1/m C m u.ejg / C mv.ejg /

11 C n 1 D n 1 h C d.u;xjg /Dd.v;xjG / d.u;xjg /<d.v;xjg / d.u;xjg /Dd.v;xjG / SOME VARIANTS OF THE SZEGED INDE 771 mu.ejg / C m v.ejg / mu.ejg / C m v.ejg / mu.ejg / C m v.ejg / i C n 1 m 1 C.n 1 1/m jfe D uv E.G / W d.u;xjg / < d.v;xjg /gj C n 1 m 1 C.n 1 1/m m u.ejg / d.u;xjg /<d.v;xjg / D n 1 PI e./ mx.g / C n 1 m 1 C.n 1 1/m.G / C n 1 m 1 C.n 1 1/m M : Eq. (.8) is obtained by adding the quantities S 1 and S. Theorem 7. The revised edge Szeged index of the rooted product G 1 fg g is given by S e.g 1fG g/ D S e.g 1/ C m S.G 1 / C m S ev.g 1 / S ve.g 1 / Proof. By Eq. (.7), C n 1 S e.g / C 1 n 1.m 1 C.n 1 1/m /.M M / 1 n 1 m 1 C.n 1 1/m mx.g / C 1 n 1m m1.m 1 C n 1 m / C m.n 1 1/ : (.9) S e.g 1fG g/ D 1.m 1 C n 1 m / 1./ PI e.g 1 fg g/ C 1 S e.g 1 fg g/: Now using Eq. (.) and Eq. (.8), we get S e.g 1fG g/ D 1 m 1 C m 1 n 1m C m 1 n 1 m C 1 n 1 m PI./ e.g 1 / C m./ PI v.g 1/ C m S ve.g 1 / C n 1 PI e./.g / mx C n 1 m 1 C.n 1 1/m.G / C n 1 m 1 C.n 1 1/m M C 1 S e.g 1 / C m S.G 1/ C m S ev.g 1 / C n 1 S e.g / C n 1 m 1 C.n 1 1/m M : Eq. (.9) is obtained by simplifying the above expression.

12 77 MAHDIEH AZARI. EAMPLES AND COROLLARIES In this section, we apply the results of the previous section to compute the edge PI index, edge Szeged index, edge-vertex Szeged index, vertex-edge Szeged index, and revised edge Szeged index of some graphs by specializing components in rooted product. Let P n, S n, and C n denote the n-vertex path, star, and cycle, respectively. Some Szeged-related topological indices of these graphs have been given in Table 1. Graph P n S n C n, n is even C n, n is odd PI v n.n 1/ n.n 1/ n n.n 1/ PI e.n 1/.n /.n 1/.n / n.n / n.n 1/ nc1 S.n 1/ n n.n 1/ n 1 n.n / S e 0 n.n 1/ n.n 1/.n / S ev S ve n.n 1/.n / S nc1.n 1/.n S e nc/.n 1/.n / 1 n.n / n.n /.n 1/ n TABLE 1. Some topological indices of path, star, and cycle. n n.n 1/ n.n 1/ n n As the first example, consider the rooted product of P n and P m, where the root vertex of P m is assumed to be on one of its pendant vertices (vertices of degree one). This molecular graph is called the comb lattice graph. Using Eqs. (.1) (.), (.6), (.9), and Table 1, we easily arrive at: Corollary. Let G D P n fp m g, where the root vertex of P m is assumed to be on one of its pendant vertices. Then (i) PI e.g/ D.nm 1/.nm /; (ii) S e.g/ D nm nm.n / C 6 6.n 9n C / C 11nm 6 (iii) S ev.g/ D nm nm.n / C 6 6.n 6n C / C nm ; (iv) S ve.g/ D nm nm nm 6.n n C /.n C 1/.n C / C 6 ; nm nm (v) S e.g/ D.n / C 6 6.n 6n C / C 7nm 1 1 : Let Pn.m/ denote the m thorn path which is the graph obtained by attaching m pendant vertices to each vertex of the path P n. This graph can be viewed as the rooted 1;

13 SOME VARIANTS OF THE SZEGED INDE 77 product of P n and the star graph on m C 1 vertices, where the root vertex of S mc1 is assumed to be on its central vertex (vertex of degree m). Using Eqs. (.1) (.), (.6), (.9), and Table 1, we easily arrive at: Corollary. The following equalities hold: (i) PI e.pn.m// D n m C nm.n / C.n 1/.n /;! (ii) S e.pn nm.m// D 6.n 1/.n C 1/ C nm n 1.n 1/.n / C ;! (iii) S ev.pn nm.m// D 6.n C n 1/ C nm 6.n 5/ C n ; (iv) S ve.pn.m// D n m C nm 6.8n 1n C 1/ C nm.n 1/.7n 11/! 6 C n ; (v) S e.p n nm.m// D 6.n C n 1/ C nm 1.n 7/ C n 1 1.n 6n C 7/ : Let Cn.m/ denote the m thorn cycle which is the graph obtained by attaching m pendant vertices to each vertex of the cycle C n. This graph can be seen as the rooted product of C n and the star graph on m C 1 vertices, where S mc1 is assumed to be rooted on its central vertex. Using Eqs. (.1) (.), (.6), (.9), and Table 1, we easily arrive at: Corollary. The following equalities hold: (i) PI e.c n.m// D n m C nm.n 1/ C n.n / n is even; n m C nm.n 1/ C n.n 1/ n is odd; (ii) S e.c n.m// D ( n.nmcn / n is even; n.n 1/.mC1/ n is odd; ( n (iii) S ev.cn m.m// D.n C / C nm.n 1/ C n.n / n is even; nm.n C 1/ C n m n.n 1/.n 1/ C n is odd; 8 n m C n m.5n / n is even ˆ< (iv) S ve.cn.m// D ˆ: (v) S e.c n.m// D n m C nm.n n C 1/ C n.n / ; n m C nm.5n 6n C 1/ n is odd Cnm.n 1/ C n.n 1/ ;.n C / C nm.n C n 1/ C n :

14 77 MAHDIEH AZARI Finally, consider the rooted product of P n and C m. Note that because of the symmetry of C m any vertex of this graph can be considered at its root vertex. Using Eqs. (.1) (.), (.6), (.9), and Table 1, we easily arrive at: Corollary 5. Let G D P n fc m g. Then n (i) PI e.g/ D m C nm.n / C.n 1/.n / m is even; n m C nm.n /.n 1/ m is odd; 8 nm nm.n 1/ C 6.n C 1/.n / m is even ˆ< C (ii) S e.g/ D nm n 1.n /.n / C ; nm nm ˆ:.n 1/ C 6.n n 1/ m is odd C nm 1.n 18n C 17/ C n 1 6.n n C 6/; 8 n ˆ< m C nm 1.n n 11/ C m n m is even; (iii) S ev.g/ D n m C ˆ: nm 1.n 9n 5/ m is odd C nm 6.n n.n 1/ n C 5/ C ; 8 nm ˆ<.n n C 1/ C nm 1.10n 1n 1/ m is even Cm n (iv) S ve.g/ D ; nm ˆ:.n n C 1/ C nm 1.n 15n 1/ m is odd nm 6.n n.n 1/ / C ; 8 nm nm ˆ<.n 1/ C 6.n 1/.n C / m is even (v) S e.g/ D C nm 6.n n C 1/ C n 1 1.n n C /; nm 5m nc1.n 1/ C C nm 6 ˆ:.5n 9n C / m is odd C 1 1.5n 1n C 10n /: ACKNOWLEDGEMENT The author would like to thank the referee for helpful comments and valuable suggestions. REFERENCES [1] A. R. Ashrafi and M. Mirzargar, The study of an infinite class of dendrimer nanostars by topological index approaches. J. Appl. Math. Comput., vol. 1, no. 1-, pp. 89 9, 009, doi: /s [] M. Azari, Sharp lower bounds on the Narumi-Katayama index of graph operations. Appl. Math. Comput., vol. 9, pp. 09 1, 01, doi: /j.amc [] M. Azari and A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices. MATCH Commun. Math. Comput. Chem., vol. 70, no., pp , 01. [] M. Azari and A. Iranmanesh, Computing the eccentric-distance sum for graph operations. Discrete Appl. Math., vol. 161, no. 18, pp , 01, doi: /j.dam [5] M. Azari and A. Iranmanesh, Clusters and various versions of Wiener-type invariants. Kragujevac J. Math., vol. 9, no., pp , 015, doi: /KgJMath150155A.

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