## 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.

The first degree-based 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:

$$M_{1}(G)=\sum_{a_{1}\in V(G)}\bigl(d_{G}(a_{1}) \bigr)^{2}, \qquad M_{2}(G)=\sum_{a_{1}a_{2}\in E(G)}d_{G}(a_{1})d_{G}(a_{2}).$$

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 (product-connectivity index) by Li and Gutman [4], which is defined as follows:

$$R_{\alpha}(G)=\sum_{a_{1}a_{2}\in E(G)}\bigl(d_{G}(a_{1})d_{G}(a_{2}) \bigr)^{\alpha},$$

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 $$(d_{G}(a_{1})+d_{G}(a_{2}))^{-\frac{1}{2}}$$. This concept was extended to the general sum-connectivity index in 2010 [6], which is defined as follows:

$$\chi_{\alpha}(G)=\sum_{a_{1}a_{2}\in E(G)} \bigl(d_{G}(a_{1})+d_{G}(a_{2}) \bigr)^{\alpha},$$

where α is a real number. Then $$\chi_{-1/2}(G)$$ is the classical sum-connectivity index. The sum-connectivity 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 as follows:

$$H(G)=\sum_{a_{1}a_{2}\in E(G)}\frac{2}{d_{G}(a_{1})+d_{G}(a_{2})}=2 \chi_{-1}(G).$$

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 [810].

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.

For a connected graph G, define four related graphs as follows:

1. 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. 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. 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. 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 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 $$[b_{1}=b_{2}\in V(H)\mbox{ and }a_{1}a_{2}\in E(F(G))]$$.

$$G+_{F}H$$ 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 sum-connectivity index and general sum-connectivity index for trees, unicyclic graphs, 2-connected graphs and bicyclic graphs were given in [1117]. 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

If $$\alpha<0$$, then the lower and upper bounds on the general sum-connectivity index are $$\gamma_{1}\leq\chi_{\alpha}(G+_{S}H)\leq \gamma_{2}$$, where

\begin{aligned}& \gamma_{1}=2^{\alpha}n_{1}m_{2}( \bigtriangleup_{G}+\bigtriangleup _{H})^{\alpha}+2^{\alpha}n_{2}m_{1}(2 \bigtriangleup_{G}+\bigtriangleup _{H})^{\alpha}, \\& \gamma_{2}=2^{\alpha}n_{1}m_{2}( \delta_{G}+\delta_{H})^{\alpha }+2^{\alpha}n_{2}m_{1}(2 \delta_{G}+\delta_{H})^{\alpha}. \end{aligned}

Equality holds if and only if G and H are regular graphs.

### Proof

By the definition of the general sum-connectivity index, we have

\begin{aligned} \chi_{\alpha}(G+_{S}H) =&\sum _{(a_{1},b_{1})(a_{2},b_{2})\in E(G+_{S}H)}\bigl[d_{G+_{S}H}(a_{1},b_{1})+d_{G+_{S}H}(a_{2},b_{2}) \bigr]^{\alpha } \\ =&\sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[d_{G+_{S}H}(a_{1},b_{1})+d_{G+_{S}H}(a_{1},b_{2}) \bigr]^{\alpha} \\ &{}+\sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(S(G))} \bigl[d_{G+_{S}H}(a_{1},b_{1})+d_{G+_{S}H}(a_{2},b_{1}) \bigr]^{\alpha}. \end{aligned}
(1)

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

\begin{aligned}& \sum_{a_{1}\in V(G)}\sum _{b_{1}b_{2}\in E(H)}\bigl[d_{G+_{S}H}(a_{1},b_{1})+d_{G+_{S}H}(a_{1},b_{2}) \bigr]^{\alpha} \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[\bigl(d_{G}(a_{1})+d_{H}(b_{1}) \bigr)+\bigl(d_{G}(a_{1})+d_{H}(b_{2}) \bigr)\bigr]^{\alpha } \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[2d_{G}(a_{1})+d_{H}(b_{1})+d_{H}(b_{2}) \bigr]^{\alpha} \\& \quad \geq 2^{\alpha}n_{1}m_{2}(\bigtriangleup_{G}+ \bigtriangleup_{H})^{\alpha}. \end{aligned}
(2)

Since $$|E(S(G))|=2|E(G)|$$ and $$\bigtriangleup_{S(G)}=\bigtriangleup _{G}$$, we have

\begin{aligned}& \sum_{b_{1}\in V(H)}\sum _{a_{1}a_{2}\in E(S(G))}\bigl[d_{G+_{S}H}(a_{1},b_{1})+d_{G+_{S}H}(a_{2},b_{1}) \bigr]^{\alpha} \\& \quad = \sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(S(G))} \bigl[d_{S(G)}(a_{1})+d_{H}(b_{1})+d_{S(G)}(a_{2}) \bigr]^{\alpha} \\& \quad \geq n_{2}|E\bigl(S(G)\bigr)|(2\bigtriangleup_{S(G)}+ \bigtriangleup_{H}) \\& \quad = 2n_{2}m_{1}(2\bigtriangleup_{G}+ \bigtriangleup_{H}). \end{aligned}
(3)

Using equations (2) and (3) in equation (1), we get

$$\chi_{\alpha}(G+_{S}H)\geq2^{\alpha}n_{1}m_{2}( \bigtriangleup _{G}+\bigtriangleup_{H})^{\alpha}+2n_{2}m_{1}(2 \bigtriangleup _{G}+\bigtriangleup_{H}).$$

Similarly we can compute

$$\chi_{\alpha}(G+_{S}H)\leq2^{\alpha}n_{1}m_{2}( \delta_{G}+\delta _{H})^{\alpha}+2n_{2}m_{1}(2 \delta_{G}+\delta_{H}).$$

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 sum-connectivity index of $$P_{n}+_{S}P_{m}$$ are

\begin{aligned}& \gamma_{1}=mn\bigl(8^{\alpha}+2\times6^{\alpha} \bigr)-8^{\alpha}n-2\times 6^{\alpha}m, \\& \gamma_{2}=mn \bigl(4^{\alpha}+2\times3^{\alpha}\bigr)-4^{\alpha}n-2 \times3^{\alpha}m. \end{aligned}

### Example 2

The general sum-connectivity index of $$C_{n}+_{S}C_{m}$$ and $$K_{n}+_{S}K_{m}$$ is

\begin{aligned}& \chi_{\alpha}(C_{n}+_{S}C_{m})= mn \bigl(8^{\alpha}+2\times6^{\alpha}\bigr), \\& \chi_{\alpha}(K_{n}+_{S}K_{m})= mn \bigl[2^{\alpha-1}(m-1) (m+n-2)^{\alpha }+(n-1) (2n+m-3)^{\alpha}\bigr]. \end{aligned}

### Theorem 2.2

If $$\alpha<0$$, then the lower and upper bounds on the general sum-connectivity index are $$\gamma_{1}\leq\chi_{\alpha}(G+_{R}H)\leq \gamma_{2}$$, where

\begin{aligned}& \gamma_{1}= 2^{\alpha}(n_{1}m_{2}+n_{2}m_{1}) (2\bigtriangleup _{G}+\bigtriangleup_{H})^{\alpha}+2n_{2}m_{1}(2 \bigtriangleup _{G}+\bigtriangleup_{H}+2)^{\alpha}, \\& \gamma_{2}= 2^{\alpha}(n_{1}m_{2}+n_{2}m_{1}) (2\delta_{G}+\delta _{H})^{\alpha}+2n_{2}m_{1}(2 \delta_{G}+\delta_{H}+2)^{\alpha}. \end{aligned}

Equality holds if and only if G and H are regular graphs.

### Proof

By the definition of the general sum-connectivity index, we have

\begin{aligned} \chi_{\alpha}(G+_{R}H) =&\sum _{(a_{1},b_{1})(a_{2},b_{2})\in E(G+_{R}H)}\bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{2}) \bigr]^{\alpha } \\ =&\sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{1},b_{2}) \bigr]^{\alpha} \\ &{}+\sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(R(G))} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha}. \end{aligned}
(4)

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

\begin{aligned}& \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{1},b_{2}) \bigr]^{\alpha} \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[d_{R(G)}(a_{1})+d_{H}(b_{1})+d_{R(G)}(a_{1})+d_{H}(b_{2}) \bigr]^{\alpha} \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[2d_{R(G)}(a_{1})+\bigl(d_{H}(b_{1})+d_{H}(b_{2}) \bigr)\bigr]^{\alpha} \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[4d_{G}(a_{1})+\bigl(d_{H}(b_{1})+d_{H}(b_{2}) \bigr)\bigr]^{\alpha} \\& \quad \geq 2^{\alpha}n_{1}m_{2}(2\bigtriangleup_{G}+ \bigtriangleup_{H}), \end{aligned}
(5)
\begin{aligned}& \sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(R(G))} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha} \\& \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(R(G))\\ a_{1},a_{2}\in V(G)}} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha} \\& \qquad {} +\sum_{b_{1}\in V(H)}\sum _{\substack{a_{1}a_{2}\in E(R(G))\\a_{1}\in V(G),a_{2}\in V(R(G))-V(G)}}\bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha}. \end{aligned}
(6)

(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

\begin{aligned}& \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(R(G))\\ a_{1},a_{2}\in V(G)}} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha} \\& \quad =\sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(G)} \bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha} \\& \quad =\sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(G)} \bigl[\bigl(d_{R(G)}(a_{1})+d_{H}(b_{1}) \bigr)+\bigl(d_{R(G)}(a_{2})+d_{H}(b_{1}) \bigr)\bigr]^{\alpha } \\& \quad =\sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(G)} \bigl[2\bigl(d_{G}(a_{1})+d_{G}(a_{2}) \bigr)+2d_{H}(b_{1})\bigr]^{\alpha} \\& \quad \geq 2^{\alpha}n_{2}m_{1}(2\bigtriangleup_{G}+ \bigtriangleup _{H})^{\alpha}. \end{aligned}
(7)

Note that $$|E(R(G))|=2|E(G)|$$, 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

\begin{aligned}& \sum_{b_{1}\in V(H)}\sum _{\substack{a_{1}a_{2}\in E(R(G))\\a_{1}\in V(G),a_{2}\in V(R(G))-V(G)}}\bigl[d_{G+_{R}H}(a_{1},b_{1})+d_{G+_{R}H}(a_{2},b_{1}) \bigr]^{\alpha } \\& \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(R(G))\\a_{1}\in V(G),a_{2}\in V(R(G))-V(G)}} \bigl[\bigl(d_{R(G)}(a_{1})+d_{H}(b_{1}) \bigr)+d_{R(G)}(a_{2})\bigr]^{\alpha } \\& \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(R(G))\\a_{1}\in V(G),a_{2}\in V(R(G))-V(G)}} \bigl[\bigl(d_{R(G)}(a_{1})+d_{R(G)}(a_{2}) \bigr)+d_{H}(b_{1})\bigr]^{\alpha } \\& \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(R(G))\\a_{1}\in V(G),a_{2}\in V(R(G))-V(G)}} \bigl[2\bigl(d_{G}(a_{1})+1\bigr)+d_{H}(b_{1}) \bigr]^{\alpha} \\& \quad \geq n_{2}\bigl|E\bigl(R(G)\bigr)\bigr|(2\bigtriangleup_{G}+ \bigtriangleup_{H}+2)^{\alpha} \\& \quad = 2n_{2}m_{1}(2\bigtriangleup_{G}+ \bigtriangleup_{H}+2)^{\alpha}. \end{aligned}
(8)

Using equations (5)-(8) in equation (4), we get the required result,

$$\chi_{\alpha}(G+_{R}H)\geq2^{\alpha }(n_{1}m_{2}+n_{2}m_{1}) (2\bigtriangleup_{G}+\bigtriangleup_{H})^{\alpha }+2n_{2}m_{1}( 2\bigtriangleup_{G}+\bigtriangleup_{H}+2)^{\alpha}.$$

Similarly, we can compute

$$\chi_{\alpha}(G+_{R}H)\leq2^{\alpha}(n_{1}m_{2}+n_{2}m_{1}) (2\delta _{G}+\delta_{H})^{\alpha}+2n_{2}m_{1}(2 \delta_{G}+\delta_{H}+2)^{\alpha}.$$

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 sum-connectivity index of $$P_{n}+_{R}P_{m}$$ are

\begin{aligned}& \gamma_{1}=2^{2\alpha+1}mn\bigl(3^{\alpha}+2^{\alpha} \bigr)-12^{\alpha }n-2^{2\alpha}m\bigl(3^{\alpha}+2^{\alpha+1} \bigr), \\& \gamma_{2}=2mn\bigl(6^{\alpha}+5^{\alpha}\bigr)-6^{\alpha}n-m\bigl(6^{\alpha}+2\times 5^{\alpha}\bigr). \end{aligned}

### Example 4

The general sum-connectivity index of $$C_{n}+_{R}C_{m}$$ and $$K_{n}+_{R}K_{m}$$ is

\begin{aligned}& \chi_{\alpha}(C_{n}+_{R}C_{m})=2^{2\alpha+1}mn \bigl(3^{\alpha}+2^{\alpha}\bigr), \\& \chi_{\alpha}(K_{n}+_{R}K_{m})=2^{\alpha-1}mn(m+n-2) (m+2n-3)^{\alpha }+mn(n-1) (2n+m-1)^{\alpha}. \end{aligned}

### Theorem 2.3

If $$\alpha<0$$, then the lower and upper bounds on the general sum-connectivity index are $$\gamma_{1}\leq\chi_{\alpha}(G+_{Q}H)\leq \gamma_{2}$$, where

\begin{aligned}& \gamma_{1}= 2^{\alpha}n_{1}m_{2}( \bigtriangleup_{G}+\bigtriangleup _{H})^{\alpha}+2n_{2}m_{1}(3 \bigtriangleup_{G}+\bigtriangleup _{H})^{\alpha}+ 4^{\alpha}n_{2}\bigtriangleup^{\alpha}_{G}\biggl( \frac{1}{2}M_{1}(G)-m_{1}\biggr), \\& \gamma_{2}= 2^{\alpha}n_{1}m_{2}( \delta_{G}+\delta_{H})^{\alpha }+2n_{2}m_{1}(3 \delta_{G}+\delta_{H})^{\alpha}+ 4^{\alpha}n_{2} \delta^{\alpha}_{G}\biggl(\frac{1}{2}M_{1}(G)-m_{1} \biggr). \end{aligned}

Equality holds if and only if G and H are regular graphs.

### Proof

By the definition of the general sum-connectivity index, we have

\begin{aligned} \chi_{\alpha}(G+_{Q}H) =& \sum_{(a_{1},b_{1})(a_{2},b_{2})\in E(G+_{Q}H)} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{2},b_{2}) \bigr]^{\alpha } \\ =& \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{1},b_{2}) \bigr]^{\alpha} \\ &{} +\sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(Q(G))} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{1},b_{2}) \bigr]^{\alpha}. \end{aligned}
(9)

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

\begin{aligned}& \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{1},b_{2}) \bigr]^{\alpha} \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[\bigl(d_{Q(G)}(a_{1})+d_{H}(b_{1}) \bigr)+\bigl(d_{Q(G)}(a_{1})+d_{H}(b_{2}) \bigr)\bigr]^{\alpha } \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[2d_{Q(G)}(a_{1})+\bigl(d_{H}(b_{1})+d_{H}(b_{2}) \bigr)\bigr]^{\alpha} \\& \quad = \sum_{a_{1}\in V(G)}\sum_{b_{1}b_{2}\in E(H)} \bigl[2d_{G}(a_{1})+\bigl(d_{H}(b_{1})+d_{H}(b_{2}) \bigr)\bigr]^{\alpha} \\& \quad \geq 2^{\alpha}n_{1}m_{2}(\bigtriangleup_{G}+ \bigtriangleup_{H})^{\alpha}, \end{aligned}
(10)
\begin{aligned}& \sum_{b_{1}\in V(H)}\sum_{a_{1}a_{2}\in E(Q(G))} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{2},b_{2}) \bigr]^{\alpha} \\& \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(Q(G))\\a_{1}\in V(G),a_{2}\in V(Q(G))-V(G)}} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{2},b_{1}) \bigr]^{\alpha } \\ & \qquad {}+\sum_{b_{1}\in V(H)}\sum _{\substack{a_{1}a_{2}\in E(Q(G))\\ a_{1},a_{2}\in V(Q(G))-V(G)}}\bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{2},b_{1}) \bigr]^{\alpha}, \end{aligned}
(11)
\begin{aligned}& \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(Q(G))\\a_{1}\in V(G),a_{2}\in V(Q(G))-V(G)}} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{2},b_{1}) \bigr]^{\alpha} \\ & \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(Q(G))\\a_{1}\in V(G),a_{2}\in V(Q(G))-V(G)}} \bigl[d_{Q(G)}(a_{1})+d_{H}(b_{1})+d_{Q(G)}(a_{2}) \bigr]^{\alpha} \\ & \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(Q(G))\\a_{1}\in V(G),a_{2}\in V(Q(G))-V(G)}} \bigl[d_{G}(a_{1})+d_{H}(b_{1})+d_{Q(G)}(a_{2}) \bigr]^{\alpha}. \end{aligned}
(12)

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

\begin{aligned} &\sum_{b_{1}\in V(H)}\sum _{\substack{a_{1}a_{2}\in E(Q(G))\\a_{1}\in V(G),a_{2}\in V(Q(G))-V(G)}}\bigl[d_{G}(a_{1})+d_{H}(b_{1})+ \bigl(d_{G}(w_{i})+d_{G}(w_{j})\bigr) \bigr]^{\alpha} \\ &\quad \geq 2n_{2}m_{1}(3\bigtriangleup_{G}+ \bigtriangleup_{H})^{\alpha}, \\ &\sum_{b_{1}\in V(H)}\sum_{\substack{a_{1}a_{2}\in E(Q(G))\\ a_{1},a_{2}\in V(Q(G))-V(G)}} \bigl[d_{G+_{Q}H}(a_{1},b_{1})+d_{G+_{Q}H}(a_{2},b_{1}) \bigr]^{\alpha} \\ &\quad = \sum_{b_{1}\in V(H)}\sum _{\substack{a_{1}a_{2}\in E(Q(G))\\ a_{1},a_{2}\in V(Q(G))-V(G)}}\bigl[d_{Q(G)}(a_{1})+d_{Q(G)}(a_{2}) \bigr]^{\alpha}. \end{aligned}
(13)

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,

\begin{aligned}& \sum_{b_{1}\in V(H)}\sum _{\substack{w_{i}w_{j}\in E(G)\\ w_{j}w_{k}\in E(G)}}\bigl[d_{G}(w_{i})+d_{G}(w_{j})+d_{G}(w_{j})+d_{G}(w_{k}) \bigr]^{\alpha} \\ & \quad = \sum_{b_{1}\in V(H)}\sum_{\substack{w_{i}w_{j}\in E(G)\\ w_{j}w_{k}\in E(G)}} \bigl[d_{G}(w_{i})+d_{G}(w_{k})+2d_{G}(w_{j}) \bigr]^{\alpha} \\ & \quad \geq 4^{\alpha}\bigtriangleup_{G}^{\alpha}n_{2} \biggl(\frac{1}{2}M_{1}(G)+m_{1}\biggr). \end{aligned}
(14)

Therefore, using equations (10)-(14) in equation (9), we get the required result,

$$\chi_{\alpha}(G+_{Q}H)\geq2^{\alpha}n_{1}m_{2}( \bigtriangleup _{G}+\bigtriangleup_{H})^{\alpha}+2n_{2}m_{1}(3 \bigtriangleup _{G}+\bigtriangleup_{H})^{\alpha} + 4^{\alpha}n_{2}\bigtriangleup_{G}^{\alpha}\biggl( \frac{1}{2}M_{1}(G)-m_{1}\biggr).$$

Similarly, we can compute

$$\chi_{\alpha}(G+_{Q}H)\leq2^{\alpha}n_{1}m_{2}( \delta_{G}+\delta _{H})^{\alpha}+2n_{2}m_{1}(3 \delta_{G}+\delta_{H})^{\alpha}+ 4^{\alpha}n_{2} \delta_{G}^{\alpha}\biggl(\frac{1}{2}M_{1}(G)-m_{1} \biggr).$$

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 sum-connectivity index of $$P_{n}+_{Q}P_{m}$$ are

$$\gamma_{1}=2^{3\alpha}(4mn-n-4m), \qquad \gamma_{2}=2^{2\alpha}(4mn-n-4m).$$

### Example 6

The general sum-connectivity index of $$C_{n}+_{Q}C_{m}$$ and $$K_{n}+_{Q}K_{m}$$ is

\begin{aligned}& \chi_{\alpha}(C_{n}+_{Q}C_{m}) = 2^{3\alpha+2}mn, \\& \chi_{\alpha}(K_{n}+_{Q}K_{m}) = 2^{\alpha-1}mn(m-1) (m+n-2)^{\alpha }+mn(n-1) (3n+m-4)^{\alpha}\\& \hphantom{\chi_{\alpha}(K_{n}+_{Q}K_{m}) ={}}{}+2^{2\alpha-1}mn(n-2) (n-1)^{\alpha}. \end{aligned}

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 sum-connectivity index are $$\gamma_{1}\leq\chi_{\alpha}(G+_{T}H)\leq \gamma_{2}$$, where

\begin{aligned}& \gamma_{1} = 2^{\alpha}(n_{1}m_{2}+n_{2}m_{1}) (2\bigtriangleup _{G}+\bigtriangleup_{H})^{\alpha}+2n_{2}m_{1}(4 \bigtriangleup _{G}+\bigtriangleup_{H})^{\alpha} \\& \hphantom{\gamma_{1} ={}}{}+ 4^{\alpha}n_{2}\bigtriangleup_{G}^{\alpha}\biggl( \frac{1}{2}M_{1}(G)-m_{1}\biggr), \\& \gamma_{2} = 2^{\alpha}(n_{1}m_{2}+n_{2}m-{1}) (2\delta_{G}+\delta _{H})^{\alpha}+2n_{2}m_{1}(4 \delta_{G}+\delta_{H})^{\alpha} \\& \hphantom{\gamma_{2} ={}}{}+ 4^{\alpha}n_{2} \delta_{G}^{\alpha}\biggl(\frac{1}{2}M_{1}(G)-m_{1} \biggr). \end{aligned}

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

\begin{aligned}& \gamma_{1} = 2^{\alpha}mn\bigl(2\times6^{\alpha}+4^{\alpha}+2 \times5^{\alpha}\bigr)-12^{\alpha}n-2^{\alpha}m \bigl(6^{\alpha}+2\times4^{\alpha}+2\times5^{\alpha}\bigr), \\& \gamma_{2} = mn\bigl(2\times6^{\alpha}+4^{\alpha}+2 \times5^{\alpha}\bigr)-6^{\alpha }n-m\bigl(6^{\alpha}+2 \times4^{\alpha}+2\times5^{\alpha}\bigr). \end{aligned}

### Example 8

The general sum-connectivity index of $$C_{n}+_{T}C_{m}$$ and $$K_{n}+_{T}K_{m}$$ is

\begin{aligned}& \chi_{\alpha}(C_{n}+_{T}C_{m}) = 2^{\alpha}mn\bigl(2\times6^{\alpha}+4^{\alpha }+2 \times5^{\alpha}\bigr), \\& \chi_{\alpha}(K_{n}+_{T}K_{m}) = 2^{\alpha-1}mn(m+n-2) (m+2n-3)^{\alpha }+mn(n-1) (4n+m-5)^{\alpha}\\& \hphantom{\chi_{\alpha}(K_{n}+_{T}K_{m}) ={}}{}+2^{2\alpha-1}mn(n-2) (n-1)^{\alpha+1}. \end{aligned}

## 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.