The sharp bounds on general sum-connectivity index of four operations on graphs
- Shehnaz Akhter^{1} and
- Muhammad Imran^{1, 2}Email author
https://doi.org/10.1186/s13660-016-1186-x
© Akhter and Imran 2016
Received: 26 May 2016
Accepted: 21 September 2016
Published: 30 September 2016
Abstract
The general sum-connectivity 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. 157:794-803, 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.
Keywords
MSC
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 degree-based topological indices are of great importance.
Let G and H be two vertex-disjoint 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.
- 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.
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 F-sum, denoted by \(G+_{F}H\) of graphs G and H, is a graph with the set of vertices \(V(G+_{F}H)=(V(G)\cup E(G))\times V(H)\) 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 \([b_{1}=b_{2}\in V(H)\mbox{ and }a_{1}a_{2}\in E(F(G))]\).
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 [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 sum-connectivity index of F-sums of the graphs.
2 The general sum-connectivity index of 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 \(F=S\).
Theorem 2.1
Proof
Example 1
Example 2
Theorem 2.2
Proof
Example 3
Example 4
Theorem 2.3
Proof
Example 5
Example 6
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
Example 7
Example 8
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 \(\alpha<0\). 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 sum-connectivity index for all values of α and this still remains an open and challenging problem for researchers.
Declarations
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. 20-367/NRPU/R&D/HEC/12/831.
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.
Authors’ Affiliations
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