The sharp bounds on general sum-connectivity index of four operations on graphs

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1 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 DOI 0.86/s x R E S E A R C H Open Access The sharp bounds on general sum-connectivity index of four operations on graphs Shehnaz Akhter and Muhammad Imran,2* * Correspondence: imrandhab@gmail.com Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-2, Islamabad, Pakistan 2 Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 555, Al Ain, United Arab Emirates Abstract The general sum-connectivity index χ α (G), for a (molecular) graph G,isdefinedasthe sum of the weights (d G (a )+d G (a 2 )) α of all a a 2 E(G), where d G (a )(ord G (a 2 )) denotes the degree of a vertex a (or a 2 ) in the graph G; E(G) denotes the set of edges of G, andα is an arbitrary real number. Eliasi and Taeri (Discrete Appl. Math. 57: , 2009)introduced four new operations based on the graphs S(G),R(G), Q(G), and T(G), and they also computed the Wiener index of these graph operations in terms of W(F(G))and W(H),where F is one of the symbols S, R, Q, T. The aim of this paper is to obtain sharp bounds on the general sum-connectivity index of the four operations on graphs. MSC: 05C2; 05C90 Keywords: general sum-connectivity index; operation on graphs; cartesian product; total graph Introduction Let G =(V, E) be a simple connected graph having vertex set V(G) ={a, a 2, a 3,...,a n } and edge set E(G)={e, e 2, e 3,...,e m }. The order and size of graph G are denoted by n and m,respectively.thedegreeofavertexa V(G)isthenumberofverticeswhosedistance from a is exactly one and denoted by d G (a). The minimum and maximum degrees of graph G are denoted by δ G and G, respectively. We will use the notations of P n, C n,andk n for path,cycle,andcompletegraphwithordern,respectively. A topological index is a mathematical measure which correlates to the chemical structures of any simple finite graph. They are invariant under the graph isomorphism. They play an important role in the study of QSAR/QSPR. There are numerous topological descriptors that have some applications in theoretical chemistry. Among these topological descriptors the degree-based topological indices are of great importance. The first degree-based topological indices that were defined by Gutman and Trinajstić 2] in 972, are the first and second Zagreb indices. These indices were originally defined as follows: M (G)= a V (G) ( dg (a ) ) 2, M2 (G)= a a 2 E(G) d G (a )d G (a 2 ). 206 Akhter and Imran. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 2 of 0 Here M (G) andm 2 (G) denote the first and second Zagreb indices, respectively. The Randić connectivity index, proposed by Randić in 975 3], is the most used moleculardescriptor.itisdefinedasthesumoveralltheedgesofthegraphoftheterms (d G (a )d G (a 2 )) 2. It has been extended to the general Randić connectivity index (productconnectivity index) by Li and Gutman 4],which is defined asfollows: R α (G)= a a 2 E(G) ( dg (a )d G (a 2 ) ) α, where α is a real number. The sum-connectivity index was proposed by Zhou and Trinajstić 5] in 2009, which is defined as the sum over all the edges of the graph of the terms (d G (a )+d G (a 2 )) 2. This concept was extended to the general sum-connectivity index in 200 6],whichisdefinedasfollows: χ α (G)= a a 2 E(G) ( dg (a )+d G (a 2 ) ) α, where α is a real number. Then χ /2 (G) is the classical sum-connectivity index. The sumconnectivity index and product-connectivity index correlate well with the π-electron energy of benzenoid hydrocarbons 7]. Another variant of the Randić index of G is the harmonic index, denoted by H(G) and defined asfollows: H(G)= a a 2 E(G) 2 d G (a )+d G (a 2 ) =2χ (G). We have H(G) R(G) by the inequality between arithmetic means and geometric means, with equality if and only if G is a regular graph. For more details of these topological indices we refer the reader to 8 0]. Let G and H be two vertex-disjoint graphs. The cartesian product of G and H,denoted by G H,isagraphwithvertexsetV(G H)=V(G) V(H)and(a, b )(a 2, b 2 ) E(G H)whenevera = a 2 and b b 2 E(H)] or a a 2 E(G)andb = b 2 ]. The order and size of G H are n n 2 and m n 2 + m 2 n,respectively. For a connected graph G, define four related graphs as follows:. S(G) is the graph obtained by inserting an additional vertex in each edge of G. Equivalently, each edge of G is replaced by a path of length 2. The graph S(G) is called the subdivision graph of G. 2. R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge. 3. Q(G) is obtained from G by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G. 4. T(G) hasasitsvertices,theedgesandverticesofg. Adjacency in T(G) is defined as adjacency or incidence for the corresponding elements of G.ThegraphT(G) is called the total graph of G. The four operations on graph S(C 4 ), R(C 4 ), Q(C 4 ), T(C 4 ) are depicted in Figure. Eliasi and Taeri ] introduced four new operations that are based on S(G), R(G), Q(G), T(G), as follows: Let F be one of the symbols S, R, Q, T.TheF-sum, denoted by G+ F H of graphs G and H, is a graph with the set of vertices V(G + F H)=(V(G) E(G)) V(H)and(a, b )(a 2, b 2 )

3 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 3 of 0 Figure The graphs C 4, S(C 4 ), R(C 4 ), Q(C 4 ), T(C 4 ). Figure 2 The graphs P 4 + S P 4, P 4 + R P 4, P 4 + Q P 4 and P 4 + T P 4. E(G + F H), if and only if a = a 2 V(G)andb b 2 E(H)] or b = b 2 V(H)anda a 2 E(F(G))]. G + F H is consists of n 2 copies of the graph F(G), and we label these copies by vertices of H. The vertices in each copy have two types, the vertices in V(G) (black vertices) and the vertices in E(G) (white vertices). Now we join only black vertices with the same name in F(G) in which their corresponding labels are adjacent in H. The graphs P 4 + F P 4 are showninfigure2. Several extremal properties of the sum-connectivity index and general sum-connectivity index for trees, unicyclic graphs, 2-connected graphs and bicyclic graphs were given in 7]. Eliasi and Taeri ] computed the expression for the Wiener index of four graph operations which are based on these graphs S(G), R(G), Q(G), and T(G), in terms of W(F(G)) and W(H). Deng et al. 8] computed the first and second Zagreb indices for the graph operations S(G), R(G), Q(G), and T(G). In thispaper, we will computethe sharp bounds on the general sum-connectivity index of F-sums of the graphs. 2 Thegeneralsum-connectivity indexof F-sum of graphs In this section, we derive the sharp bounds on the general sum-connectivity index of four operations on graphs. First we compute the case F = S.

4 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 4 of 0 Theorem 2. If α <0,then the lower and upper bounds on the general sum-connectivity index are γ χ α (G + S H) γ 2, where γ =2 α n m 2 ( G + H ) α +2 α n 2 m (2 G + H ) α, γ 2 =2 α n m 2 (δ G + δ H ) α +2 α n 2 m (2δ G + δ H ) α. Equality holds if and only if G and H are regular graphs. Proof By the definition of the general sum-connectivity index, we have χ α (G + S H)= (a,b )(a 2,b 2 ) E(G+ S H) = + b V (H) a a 2 E(S(G)) dg+s H(a, b )+d G+S H(a 2, b 2 ) ] α dg+s H(a, b )+d G+S H(a, b 2 ) ] α dg+s H(a, b )+d G+S H(a 2, b ) ] α. () Note that d G (a) G and d G (a) δ G, equality holds if and only if G is a regular graph, and similarly d H (b) H and d H (b) δ G, equality holds if and only if H is a regular graph. We have dg+s H(a, b )+d G+S H(a, b 2 ) ] α = = ( dg (a )+d H (b ) ) + ( d G (a )+d H (b 2 ) )] α 2dG (a )+d H (b )+d H (b 2 ) ] α 2 α n m 2 ( G + H ) α. (2) Since E(S(G)) =2 E(G) and S(G) = G,wehave dg+s H(a, b )+d G+S H(a 2, b ) ] α b V (H) a a 2 E(S(G)) = b V (H) a a 2 E(S(G)) n 2 E ( S(G) ) (2 S(G) + H ) ds(g) (a )+d H (b )+d S(G) (a 2 ) ] α =2n 2 m (2 G + H ). (3) Using equations (2)and(3)in equation(), we get χ α (G + S H) 2 α n m 2 ( G + H ) α +2n 2 m (2 G + H ). Similarly we can compute χ α (G + S H) 2 α n m 2 (δ G + δ H ) α +2n 2 m (2δ G + δ H ). Equality holds if and only if G and H are regular graphs. This completes the proof.

5 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 5 of 0 Example The lower and upper bounds on the general sum-connectivity index of P n + S P m are γ = mn ( 8 α +2 6 α) 8 α n 2 6 α m, γ 2 = mn ( 4 α +2 3 α) 4 α n 2 3 α m. Example 2 The general sum-connectivity index of C n + S C m and K n + S K m is χ α (C n + S C m )=mn ( 8 α +2 6 α), χ α (K n + S K m )=mn 2 α (m )(m + n 2) α +(n )(2n + m 3) α]. Theorem 2.2 If α <0,then the lower and upper bounds on the general sum-connectivity index are γ χ α (G + R H) γ 2, where γ =2 α (n m 2 + n 2 m )(2 G + H ) α +2n 2 m (2 G + H +2) α, γ 2 =2 α (n m 2 + n 2 m )(2δ G + δ H ) α +2n 2 m (2δ G + δ H +2) α. Equality holds if and only if G and H are regular graphs. Proof By the definition of the general sum-connectivity index, we have χ α (G + R H)= dg+r H(a, b )+d G+R H(a 2, b 2 ) ] α (a,b )(a 2,b 2 ) E(G+ R H) = + dg+r H(a, b )+d G+R H(a, b 2 ) ] α. (4) Note that d G (a) G and d G (a) δ G, equality holds if and only if G is a regular graph, and similarly d H (b) H and d H (b) δ G, equality holds if and only if H is a regular graph. We have dg+r H(a, b )+d G+R H(a, b 2 ) ] α = = = dr(g) (a )+d H (b )+d R(G) (a )+d H (b 2 ) ] α 2dR(G) (a )+ ( d H (b )+d H (b 2 ) )] α 4dG (a )+ ( d H (b )+d H (b 2 ) )] α 2 α n m 2 (2 G + H ), (5)

6 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 6 of 0 = b V (H) + a a 2 E(R(G)) a,a 2 V (G) a V (G),a 2 V (R(G)) V (G). (6) (i) a a 2 E(R(G)) and a, a 2 V(G) ifandonlyifa a 2 E(G), (ii) d R(G) (a )=2d G (a ), we have b V (H) = a a 2 E(R(G)) a,a 2 V (G) b V (H) a a 2 E(G) = b V (H) a a 2 E(G) = b V (H) a a 2 E(G) ( dr(g) (a )+d H (b ) ) + ( d R(G) (a 2 )+d H (b ) )] α 2 ( dg (a )+d G (a 2 ) ) +2d H (b ) ] α 2 α n 2 m (2 G + H ) α. (7) Note that E(R(G)) =2 E(G), andifa V(G) thend R(G) (a )=2d G (a )andifa 2 V(R(G)) V(G)thend R(G) (a 2 )=2,wehave a V (G),a 2 V (R(G)) V (G) = a V (G),a 2 V (R(G)) V (G) = a V (G),a 2 V (R(G)) V (G) = a V (G),a 2 V (R(G)) V (G) ( dr(g) (a )+d H (b ) ) + d R(G) (a 2 ) ] α ( dr(g) (a )+d R(G) (a 2 ) ) + d H (b ) ] α 2 ( dg (a )+ ) + d H (b ) ] α n 2 E R(G) (2 G + H +2) α =2n 2 m (2 G + H +2) α. (8) Using equations (5)-(8) in equation(4), we get the required result, χ α (G + R H) 2 α (n m 2 + n 2 m )(2 G + H ) α +2n 2 m (2 G + H +2) α. Similarly, we can compute χ α (G + R H) 2 α (n m 2 + n 2 m )(2δ G + δ H ) α +2n 2 m (2δ G + δ H +2) α. Equality holds if and only if G and H are regular graphs. This completes the proof.

7 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 7 of 0 Example 3 The lower and upper bounds on the general sum-connectivity index of P n + R P m are γ =2 2α+ mn ( 3 α +2 α) 2 α n 2 2α m ( 3 α +2 α+), γ 2 =2mn ( 6 α +5 α) 6 α n m ( 6 α +2 5 α). Example 4 The general sum-connectivity index of C n + R C m and K n + R K m is χ α (C n + R C m )=2 2α+ mn ( 3 α +2 α), χ α (K n + R K m )=2 α mn(m + n 2)(m +2n 3) α + mn(n )(2n + m ) α. Theorem 2.3 If α <0,then the lower and upper bounds on the general sum-connectivity index are γ χ α (G + Q H) γ 2, where γ =2 α n m 2 ( G + H ) α +2n 2 m (3 G + H ) α +4 α n 2 α G 2 M (G) m, γ 2 =2 α n m 2 (δ G + δ H ) α +2n 2 m (3δ G + δ H ) α +4 α n 2 δg α 2 M (G) m. Equality holds if and only if G and H are regular graphs. Proof By the definition of the general sum-connectivity index, we have χ α (G + Q H)= (a,b )(a 2,b 2 ) E(G+ Q H) = + dg+q H(a, b )+d G+Q H(a 2, b 2 ) ] α dg+q H(a, b )+d G+Q H(a, b 2 ) ] α dg+q H(a, b )+d G+Q H(a, b 2 ) ] α. (9) Note that d G (a) G and d G (a) δ G, equality holds if and only if G is a regular graph, and similarly d H (b) H and d H (b) δ G, equality holds if and only if H is a regular graph. We have dg+q H(a, b )+d G+Q H(a, b 2 ) ] α = = = ( dq(g) (a )+d H (b ) ) + ( d Q(G) (a )+d H (b 2 ) )] α 2dQ(G) (a )+ ( d H (b )+d H (b 2 ) )] α 2dG (a )+ ( d H (b )+d H (b 2 ) )] α 2 α n m 2 ( G + H ) α, (0) dg+q H(a, b )+d G+Q H(a 2, b 2 ) ] α

8 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 8 of 0 = dg+q H(a, b )+d G+Q H(a 2, b ) ] α a V (G),a 2 V (Q(G)) V (G) + a,a 2 V (Q(G)) V (G) a V (G),a 2 V (Q(G)) V (G) = a V (G),a 2 V (Q(G)) V (G) = a V (G),a 2 V (Q(G)) V (G) dg+q H(a, b )+d G+Q H(a 2, b ) ] α, () dg+q H(a, b )+d G+Q H(a 2, b ) ] α dq(g) (a )+d H (b )+d Q(G) (a 2 ) ] α dg (a )+d H (b )+d Q(G) (a 2 ) ] α. (2) Note that d Q(G) (a 2 )=d G (w i )+d G (w j )fora 2 V(Q(G)) V(G), a 2 is the vertex inserted into the edge w i w j of G.Thenwehave dg (a )+d H (b )+ ( d G (w i )+d G (w j ) )] α a V (G),a 2 V (Q(G)) V (G) 2n 2 m (3 G + H ) α, dg+q H(a, b )+d G+Q H(a 2, b ) ] α a,a 2 V (Q(G)) V (G) = a,a 2 V (Q(G)) V (G) dq(g) (a )+d Q(G) (a 2 ) ] α. (3) Since a is the vertex inserted into the edge w i w j of G and a 2 is the vertex inserted into the edge w j w k of G, dg (w i )+d G (w j )+d G (w j )+d G (w k ) ] α b V (H) w i w j E(G) w j w k E(G) = b V (H) w i w j E(G) w j w k E(G) ( 4 α α G n 2 2 M (G)+m dg (w i )+d G (w k )+2d G (w j ) ] α ). (4) Therefore, using equations (0)-(4)in equation(9), we get the required result, χ α (G + Q H) 2 α n m 2 ( G + H ) α +2n 2 m (3 G + H ) α +4 α n 2 α G Similarly, we can compute χ α (G + Q H) 2 α n m 2 (δ G + δ H ) α +2n 2 m (3δ G + δ H ) α +4 α n 2 δ α G 2 M (G) m. 2 M (G) m. Equality holds if and only if G and H are regular graphs. This completes the proof.

9 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 9 of 0 Example 5 The lower and upper bounds on the general sum-connectivity index of P n + Q P m are γ =2 3α (4mn n 4m), γ 2 =2 2α (4mn n 4m). Example 6 The general sum-connectivity index of C n + Q C m and K n + Q K m is χ α (C n + Q C m )=2 3α+2 mn, χ α (K n + Q K m )=2 α mn(m )(m + n 2) α + mn(n )(3n + m 4) α +2 2α mn(n 2)(n ) α. Since deg G+T H (a, b)=deg G+ R H (a, b)fora V(G)andb V(H), deg G+ T H (a, b)=deg G+ Q H (a, b) fora V(T(G)) V(G) andb V(H), we can get the following result by the proofs of Theorems 2.2and 2.3. Theorem 2.4 If α <0,then the lower and upper bounds on the general sum-connectivity index are γ χ α (G + T H) γ 2, where γ =2 α (n m 2 + n 2 m )(2 G + H ) α +2n 2 m (4 G + H ) α +4 α n 2 α G 2 M (G) m, γ 2 =2 α (n m 2 + n 2 m )(2δ G + δ H ) α +2n 2 m (4δ G + δ H ) α +4 α n 2 δg α 2 M (G) m. Equality holds if and only if G and H are regular graphs. Example 7 The lower and upper bounds on the general sum-connectivity index of P n + T P m are γ =2 α mn ( 2 6 α +4 α +2 5 α) 2 α n 2 α m ( 6 α +2 4 α +2 5 α), γ 2 = mn ( 2 6 α +4 α +2 5 α) 6 α n m ( 6 α +2 4 α +2 5 α). Example 8 The general sum-connectivity index of C n + T C m and K n + T K m is χ α (C n + T C m )=2 α mn ( 2 6 α +4 α +2 5 α), χ α (K n + T K m )=2 α mn(m + n 2)(m +2n 3) α + mn(n )(4n + m 5) α +2 2α mn(n 2)(n ) α+. 3 Conclusion The sharp bounds on the general sum-connectivity index of the new four sums of the graphs were computed in this paper, for α <0.However,ifα > 0 then these bounds will become γ 2 χ α (G + F H) γ. These results can be extended for a tenser product and the normal product of the graphs with respect to the general sum-connectivity index for all values of α and this still remains an open and challenging problem for researchers.

10 Akhter and Imran Journal of Inequalities and Applications (206) 206:24 Page 0 of 0 Competing interests The authors declare that they have no competing interests. Authors contributions The idea to obtain the bounds for the four operations of the graphs for general sum-connectivity index of the graphs was proposed by MI. After several discussions, SA obtained some sharp bounds for four operations. MI checked these bounds and suggested improvements. The first draft was prepared by SA which was verified and improved by MI. The final version was prepared by SA. Both authors read and approved the final manuscript. Acknowledgements The authors are very grateful to the referees for their constructive suggestions and useful comments, which improved this work very much. This research is supported by the grant of Higher Education Commission of Pakistan Ref. No /NRPU/R&D/HEC/2/83. Received: 26 May 206 Accepted: 2 September 206 References. Eliasi, M, Taeri, B: Four new sums of graphs and their Wiener indices. Discrete Appl. Math. 57, (2009) 2. Gutman, I, Trinajstić, N: Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 7, (972) 3. Randić, M: On characterization of molecular branching. J. Am. Chem. Soc. 97, (975) 4. Li, X, Gutman, I: Mathematical Aspects of Randić Type Molecular Structure Description. University of Kragujevac, Kragujevac (2006) 5. Zhou, B, Trinajstić, N: On a novel connectivity index. J. Math. Chem. 46, (2009) 6. Zhou, B, Trinajstić, N: On general sum-connectivity index. J. Math. Chem. 47, (200) 7. Lučić, B, Trinajstić, N, Zhou, B: Comparison between the sum-connectivity index and product-connectivity index for benzenoid hydrocarbons. Chem. Phys. Lett. 475, (2009) 8. Akhter, S, Imran, M: On degree based topological descriptors of strong product graphs. Can. J. Chem. 94(6), (206) 9. Khalifeh, MH, Azari, HY, Ashrafi, AR: The first and second Zagreb indices of some graph operations. Discrete Appl. Math. 57, (2009) 0. Zhou, B: Zagreb indices. MATCH Commun. Math. Comput. Chem. 52, 3-8 (2004). Du, Z, Zhou, B, Trinajstić, N: On the general sum-connectivity index of trees. Appl. Math. Lett. 24, (20) 2. Du, Z, Zhou, B, Trinajstić, N: Minimum general sum-connectivity index of unicyclic graphs. J. Math. Chem. 48, (200) 3. Tache, R-M: General sum-connectivity index with α for bicyclic graphs. MATCH Commun. Math. Comput. Chem. 72, (204) 4. Tomescu, I, Kanwal, S: Ordering trees having small general sum-connectivity index. MATCH Commun. Math. Comput. Chem. 69, (203) 5. Tomescu, I, Kanwal, S: Unicyclic graphs of given girth k 4 having smallest general sum-connectivity index. Discrete Appl. Math. 64, (204) 6. Tomescu, I: 2-connected graphs with minimum general sum-connectivity index. Discrete Appl. Math. 78, 35-4 (204) 7. Tomescu, I, Kanwal, S: On the general sum-connectivity index of connnected unicyclic graphs with k pendant vertices. Discrete Appl. Math. 8, (205) 8. Deng, H, Sarala, D, Ayyaswamy, SK, Balachandran, S: The Zagreb indices of four operations on graphs. Appl. Math. Comput. 275, (206)