Abstract
The general sumconnectivity index \(\chi_{\alpha}(G)\), for a (molecular) graph G, is defined as the sum of the weights \((d_{G}(a_{1})+d_{G}(a_{2}))^{\alpha}\) of all \(a_{1}a_{2}\in E(G)\), where \(d_{G}(a_{1})\) (or \(d_{G}(a_{2})\)) denotes the degree of a vertex \(a_{1}\) (or \(a_{2}\)) in the graph G; \(E(G)\) denotes the set of edges of G, and α is an arbitrary real number. Eliasi and Taeri (Discrete Appl. Math. 157:794803, 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 sumconnectivity index of the four operations on graphs.
Similar content being viewed by others
1 Introduction
Let \(G=(V,E)\) be a simple connected graph having vertex set \(V(G)=\{ a_{1},a_{2},a_{3},\ldots,a_{n}\}\) and edge set \(E(G)=\{e_{1},e_{2},e_{3},\ldots,e_{m}\}\). The order and size of graph G are denoted by n and m, respectively. The degree of a vertex \(a\in V(G)\) is the number of vertices whose distance from a is exactly one and denoted by \(d_{G}(a)\). The minimum and maximum degrees of graph G are denoted by \(\delta_{G}\) and \(\bigtriangleup_{G}\), respectively. We will use the notations of \(P_{n}\), \(C_{n}\), and \(K_{n}\) for path, cycle, and complete graph with order n, 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 degreebased topological indices are of great importance.
The first degreebased topological indices that were defined by Gutman and Trinajstić [2] in 1972, are the first and second Zagreb indices. These indices were originally defined as follows:
Here \(M_{1}(G)\) and \(M_{2}(G)\) denote the first and second Zagreb indices, respectively. The Randić connectivity index, proposed by Randić in 1975 [3], is the most used molecular descriptor. It is defined as the sum over all the edges of the graph of the terms \((d_{G}(a_{1})d_{G}(a_{2}))^{\frac{1}{2}}\). It has been extended to the general Randić connectivity index (productconnectivity index) by Li and Gutman [4], which is defined as follows:
where α is a real number. The sumconnectivity 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 \((d_{G}(a_{1})+d_{G}(a_{2}))^{\frac{1}{2}}\). This concept was extended to the general sumconnectivity index in 2010 [6], which is defined as follows:
where α is a real number. Then \(\chi_{1/2}(G)\) is the classical sumconnectivity index. The sumconnectivity index and productconnectivity index correlate well with the πelectron energy of benzenoid hydrocarbons [7]. Another variant of the Randić index of G is the harmonic index, denoted by \(H(G)\) and defined as follows:
We have \(H(G)\leq R(G)\) 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 [8–10].
Let G and H be two vertexdisjoint graphs. The cartesian product of G and H, denoted by \(G \mathrel{\square} H\), is a graph with vertex set \(V(G \mathrel{\square} H)=V(G)\times V(H)\) and \((a_{1},b_{1})(a_{2},b_{2})\in E(G\mathrel{\square} H)\) whenever [\(a_{1}=a_{2}\) and \(b_{1}b_{2}\in E(H)\)] or [\(a_{1}a_{2}\in E(G)\) and \(b_{1}=b_{2}\)]. The order and size of \(G\mathrel{\square} H\) are \(n_{1}n_{2}\) and \(m_{1}n_{2}+m_{2}n_{1}\), respectively.
For a connected graph G, define four related graphs as follows:

1.
\(S(G)\) 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.
\(T(G)\) has as its vertices, the edges and vertices of G. Adjacency in \(T(G)\) is defined as adjacency or incidence for the corresponding elements of G. The graph \(T(G)\) 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 1.
Eliasi and Taeri [1] 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. The Fsum, denoted by \(G+_{F}H\) of graphs G and H, is a graph with the set of vertices \(V(G+_{F}H)=(V(G)\cup E(G))\times V(H)\) and \((a_{1},b_{1})(a_{2},b_{2})\in E(G+_{F}H)\), if and only if \([a_{1}=a_{2}\in V(G)\mbox{ and }b_{1}b_{2}\in E(H)]\) or \([b_{1}=b_{2}\in V(H)\mbox{ and }a_{1}a_{2}\in E(F(G))]\).
\(G+_{F}H\) 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 shown in Figure 2.
Several extremal properties of the sumconnectivity index and general sumconnectivity index for trees, unicyclic graphs, 2connected graphs and bicyclic graphs were given in [11–17]. Eliasi and Taeri [1] 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. [18] computed the first and second Zagreb indices for the graph operations \(S(G)\), \(R(G)\), \(Q(G)\), and \(T(G)\). In this paper, we will compute the sharp bounds on the general sumconnectivity index of Fsums of the graphs.
2 The general sumconnectivity index of Fsum of graphs
In this section, we derive the sharp bounds on the general sumconnectivity index of four operations on graphs. First we compute the case \(F=S\).
Theorem 2.1
If \(\alpha<0\), then the lower and upper bounds on the general sumconnectivity index are \(\gamma_{1}\leq\chi_{\alpha}(G+_{S}H)\leq \gamma_{2}\), where
Equality holds if and only if G and H are regular graphs.
Proof
By the definition of the general sumconnectivity index, we have
Note that \(d_{G}(a)\leq\bigtriangleup_{G}\) and \(d_{G}(a)\geq\delta_{G}\), equality holds if and only if G is a regular graph, and similarly \(d_{H}(b)\leq\bigtriangleup_{H}\) and \(d_{H}(b)\leq\delta_{G}\), equality holds if and only if H is a regular graph. We have
Since \(E(S(G))=2E(G)\) and \(\bigtriangleup_{S(G)}=\bigtriangleup _{G}\), we have
Using equations (2) and (3) in equation (1), we get
Similarly we can compute
Equality holds if and only if G and H are regular graphs. This completes the proof. □
Example 1
The lower and upper bounds on the general sumconnectivity index of \(P_{n}+_{S}P_{m}\) are
Example 2
The general sumconnectivity index of \(C_{n}+_{S}C_{m}\) and \(K_{n}+_{S}K_{m}\) is
Theorem 2.2
If \(\alpha<0\), then the lower and upper bounds on the general sumconnectivity index are \(\gamma_{1}\leq\chi_{\alpha}(G+_{R}H)\leq \gamma_{2}\), where
Equality holds if and only if G and H are regular graphs.
Proof
By the definition of the general sumconnectivity index, we have
Note that \(d_{G}(a)\leq\bigtriangleup_{G}\) and \(d_{G}(a)\geq\delta_{G}\), equality holds if and only if G is a regular graph, and similarly \(d_{H}(b)\leq\bigtriangleup_{H}\) and \(d_{H}(b)\leq\delta_{G}\), equality holds if and only if H is a regular graph. We have
(i) \(a_{1}a_{2}\in E(R(G))\) and \(a_{1},a_{2}\in V(G)\) if and only if \(a_{1}a_{2}\in E(G)\), (ii) \(d_{R(G)}(a_{1})=2d_{G}(a_{1})\), we have
Note that \(E(R(G))=2E(G)\), and if \(a_{1}\in V(G)\) then \(d_{R(G)}(a_{1})=2d_{G}(a_{1})\) and if \(a_{2}\in V(R(G))V(G)\) then \(d_{R(G)}(a_{2})=2\), we have
Using equations (5)(8) in equation (4), we get the required result,
Similarly, we can compute
Equality holds if and only if G and H are regular graphs. This completes the proof. □
Example 3
The lower and upper bounds on the general sumconnectivity index of \(P_{n}+_{R}P_{m}\) are
Example 4
The general sumconnectivity index of \(C_{n}+_{R}C_{m}\) and \(K_{n}+_{R}K_{m}\) is
Theorem 2.3
If \(\alpha<0\), then the lower and upper bounds on the general sumconnectivity index are \(\gamma_{1}\leq\chi_{\alpha}(G+_{Q}H)\leq \gamma_{2}\), where
Equality holds if and only if G and H are regular graphs.
Proof
By the definition of the general sumconnectivity index, we have
Note that \(d_{G}(a)\leq\bigtriangleup_{G}\) and \(d_{G}(a)\geq\delta_{G}\), equality holds if and only if G is a regular graph, and similarly \(d_{H}(b)\leq\bigtriangleup_{H}\) and \(d_{H}(b)\leq\delta_{G}\), equality holds if and only if H is a regular graph. We have
Note that \(d_{Q(G)}(a_{2})=d_{G}(w_{i})+d_{G}(w_{j})\) for \(a_{2}\in V(Q(G))V(G)\), \(a_{2}\) is the vertex inserted into the edge \(w_{i}w_{j}\) of G. Then we have
Since \(a_{1}\) 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,
Therefore, using equations (10)(14) in equation (9), we get the required result,
Similarly, we can compute
Equality holds if and only if G and H are regular graphs. This completes the proof. □
Example 5
The lower and upper bounds on the general sumconnectivity index of \(P_{n}+_{Q}P_{m}\) are
Example 6
The general sumconnectivity index of \(C_{n}+_{Q}C_{m}\) and \(K_{n}+_{Q}K_{m}\) is
Since \(\operatorname{deg}_{G+_{T}H}(a,b)=\operatorname{deg}_{G+_{R}H}(a,b)\) for \(a\in V(G)\) and \(b\in V(H)\), \(\operatorname{deg}_{G+_{T}H}(a,b)=\operatorname{deg}_{G+_{Q}H}(a, b)\) for \(a\in V(T(G))V(G)\) and \(b\in V(H)\), we can get the following result by the proofs of Theorems 2.2 and 2.3.
Theorem 2.4
If \(\alpha<0\), then the lower and upper bounds on the general sumconnectivity index are \(\gamma_{1}\leq\chi_{\alpha}(G+_{T}H)\leq \gamma_{2}\), where
Equality holds if and only if G and H are regular graphs.
Example 7
The lower and upper bounds on the general sumconnectivity index of \(P_{n}+_{T}P_{m}\) are
Example 8
The general sumconnectivity index of \(C_{n}+_{T}C_{m}\) and \(K_{n}+_{T}K_{m}\) is
3 Conclusion
The sharp bounds on the general sumconnectivity index of the new four sums of the graphs were computed in this paper, for \(\alpha<0\). However, if \(\alpha>0\) then these bounds will become \(\gamma_{2}\leq\chi _{\alpha}(G+_{F}H)\leq\gamma_{1}\). These results can be extended for a tenser product and the normal product of the graphs with respect to the general sumconnectivity index for all values of α and this still remains an open and challenging problem for researchers.
References
Eliasi, M, Taeri, B: Four new sums of graphs and their Wiener indices. Discrete Appl. Math. 157, 794803 (2009)
Gutman, I, Trinajstić, N: Graph theory and molecular orbitals. Total πelectron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535538 (1972)
Randić, M: On characterization of molecular branching. J. Am. Chem. Soc. 97, 66096615 (1975)
Li, X, Gutman, I: Mathematical Aspects of Randić Type Molecular Structure Description. University of Kragujevac, Kragujevac (2006)
Zhou, B, Trinajstić, N: On a novel connectivity index. J. Math. Chem. 46, 12521270 (2009)
Zhou, B, Trinajstić, N: On general sumconnectivity index. J. Math. Chem. 47, 210218 (2010)
Lučić, B, Trinajstić, N, Zhou, B: Comparison between the sumconnectivity index and productconnectivity index for benzenoid hydrocarbons. Chem. Phys. Lett. 475, 146148 (2009)
Akhter, S, Imran, M: On degree based topological descriptors of strong product graphs. Can. J. Chem. 94(6), 559565 (2016)
Khalifeh, MH, Azari, HY, Ashrafi, AR: The first and second Zagreb indices of some graph operations. Discrete Appl. Math. 157, 804811 (2009)
Zhou, B: Zagreb indices. MATCH Commun. Math. Comput. Chem. 52, 113118 (2004)
Du, Z, Zhou, B, Trinajstić, N: On the general sumconnectivity index of trees. Appl. Math. Lett. 24, 402405 (2011)
Du, Z, Zhou, B, Trinajstić, N: Minimum general sumconnectivity index of unicyclic graphs. J. Math. Chem. 48, 697703 (2010)
Tache, RM: General sumconnectivity index with \(\alpha\geq1\) for bicyclic graphs. MATCH Commun. Math. Comput. Chem. 72, 761774 (2014)
Tomescu, I, Kanwal, S: Ordering trees having small general sumconnectivity index. MATCH Commun. Math. Comput. Chem. 69, 535548 (2013)
Tomescu, I, Kanwal, S: Unicyclic graphs of given girth \(k\geq4\) having smallest general sumconnectivity index. Discrete Appl. Math. 164, 344348 (2014)
Tomescu, I: 2connected graphs with minimum general sumconnectivity index. Discrete Appl. Math. 178, 135141 (2014)
Tomescu, I, Kanwal, S: On the general sumconnectivity index of connnected unicyclic graphs with k pendant vertices. Discrete Appl. Math. 181, 306309 (2015)
Deng, H, Sarala, D, Ayyaswamy, SK, Balachandran, S: The Zagreb indices of four operations on graphs. Appl. Math. Comput. 275, 422431 (2016)
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. 20367/NRPU/R&D/HEC/12/831.
Author information
Authors and Affiliations
Corresponding author
Additional information
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 sumconnectivity 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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Cite this article
Akhter, S., Imran, M. The sharp bounds on general sumconnectivity index of four operations on graphs. J Inequal Appl 2016, 241 (2016). https://doi.org/10.1186/s136600161186x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s136600161186x