The minimum value of geometric-arithmetic index of graphs with minimum degree 2

Abstract

The geometric-arithmetic index was introduced in the chemical graph theory and it has shown to be applicable. The aim of this paper is to obtain the extremal graphs with respect to the geometric-arithmetic index among all graphs with minimum degree 2. Let G(2, n) be the set of connected simple graphs on n vertices with minimum degree 2. We use linear programming formulation and prove that the minimum value of the first geometric-arithmetic $(GA_{1})$ index of G(2, n) is obtained by the following formula:

$$\begin{aligned} GA_1^* = \left\{ \begin{array}{ll} n&{}\quad n \le 24, \\ \mathrm{{24}}\mathrm{{.79}}&{}\quad n = 25, \\ \frac{{4\left( {n - 2} \right) \sqrt{2\left( {n - 2} \right) } }}{n}&{}\quad n \ge 26. \\ \end{array} \right. \end{aligned}$$

Appendix: The expression of \(GA_1^{1}\)

Here, we give more details about the expression of $GA_1^{1}$. Note that

$$\begin{aligned} GA_1^1 \left( G \right)= & {} \frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}x_{2,n - 2} + \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{n + 1}}n_2 + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}n_{n - 2} \\&+ \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} x_{2,i} + \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{n - 1 + i}}} n_i + \sum \limits _{3 \le i \le j \le n - 2} {\frac{{2\sqrt{ij} }}{{i + j}}} x_{i,j}. \\ \end{aligned}$$

Substituting (9) and (10), we have

$$\begin{aligned}&GA_1^1\left( G \right) \\&\quad =\, \frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}\left[ \frac{{\left( {n - 1} \right) \left( {n - 3} \right) }}{{n - 2}} - \sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{{n - i - 2}}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) } {x_{2,i}}\right. \\&\quad \left. \qquad \qquad \qquad \qquad \qquad - \sum \limits _{3 \le i \le j \le n - 2} {\left( {\frac{{n - 3}}{{i - 1}} + \frac{{n - 3}}{{j - 1}}} \right) \frac{{{x_{i,j}}}}{{n - 2}}} \right] \\&\qquad + \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{n + 1}}\left[ \frac{{\left( {n - 1} \right) \left( {n - 3} \right) }}{{n - 2}} + \frac{{2{x_{2,2}}}}{{n - 2}} - \sum \limits _{i = 3}^{n - 3} {\frac{{\left( {n - i - 2} \right) {x_{2,i}}}}{{\left( {i - 1} \right) \left( {n - 2} \right) }}}\right. \\&\quad \qquad \qquad \qquad \qquad \qquad \left. - \sum \limits _{3 \le i \le j \le n - 2} {\left( {\frac{{n - 3}}{{i - 1}} + \frac{{n - 3}}{{j - 1}}} \right) \frac{{{x_{i,j}}}}{{n - 2}}} \right] \\&\qquad + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\left[ \frac{{n - 1}}{{n - 2}} + \frac{{2{x_{n - 2,n - 2}}}}{{n - 2}} - \sum \limits _{i = 3}^{n - 3} {\left( {\frac{1}{{i - 1}} - 1} \right) \frac{{{x_{i,n - 2}}}}{{n - 2}}} \right. \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. - \sum \limits _{2 \le i \le j \le n - 3} {\left( {\frac{1}{{i - 1}} + \frac{1}{{j - 1}}} \right) \frac{{{x_{i,j}}}}{{n - 2}}} \right] \\&\qquad + \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{n - 1 + i}}} \left( {\frac{{{x_{2,i}} + {x_{3,i}} + \cdots + {x_{i - 1,i}} + 2{x_{i,i}} + {x_{i,i + 1}} + \cdots + {x_{i,n - 2}}}}{{i - 1}}} \right) \\&\qquad + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}} + \sum \limits _{3 \le i \le j \le n - 2} {\frac{{2\sqrt{ij} }}{{i + j}}} {x_{i,j}}. \end{aligned}$$

With simple calculations, we conclude that

$$\begin{aligned}&\sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{n - 1 + i}}} \left( {\frac{{{x_{2,i}} + {x_{3,i}} + \cdots + {x_{i - 1,i}} + 2{x_{i,i}} + {x_{i,i + 1}} + \cdots + {x_{i,n - 2}}}}{{i - 1}}} \right) \\&\quad =\, \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{2,i}}}\\&\qquad + \,\sum \limits _{3 \le i \le j \le n - 3} {\left( {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }} + \frac{{2\sqrt{j\left( {n - 1} \right) } }}{{\left( {n - 1 + j} \right) \left( {j - 1} \right) }}} \right) {x_{i,j}}} \\&\qquad +\, \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{i,n - 2}}}. \end{aligned}$$

So,

$$\begin{aligned}&GA_1^1\left( G \right) \\&\quad = \frac{{2\left( {n - 1} \right) \left( {n - 3} \right) \sqrt{2\left( {n - 2} \right) } }}{{n\left( {n - 2} \right) }} + \frac{{2\left( {n - 1} \right) \left( {n - 3} \right) \sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) \left( {n - 2} \right) }}\\&\qquad + \,\frac{{2\left( {n - 1} \right) \sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 2} \right) }}\\&\qquad - \,\frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}\sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{{n - i - 2}}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) } {x_{2,i}} \\&\qquad +\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }}\left( {\frac{{2{x_{2,2}}}}{{n - 2}} - \sum \limits _{i = 3}^{n - 3} {\frac{{\left( {n - i - 2} \right) {x_{2,i}}}}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} } \right) \end{aligned}$$

$$\begin{aligned}&\qquad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{1}{{i - 1}}} \right) \frac{{{x_{2,i}}}}{{n - 2}}}\\&\qquad + \,\sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{2,i}}} + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}}\\&\qquad -\, \frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}\left[ {\sum \limits _{3 \le i \le j \le n - 2} {\left( {\frac{{n - 3}}{{i - 1}} + \frac{{n - 3}}{{j - 1}}} \right) \frac{{{x_{i,j}}}}{{n - 2}}} } \right] \\&\qquad -\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{n + 1}}\left[ {\sum \limits _{3 \le i \le j \le n - 2} {\left( {\frac{{n - 3}}{{i - 1}} + \frac{{n - 3}}{{j - 1}}} \right) \frac{{{x_{i,j}}}}{{n - 2}}} } \right] \\&\qquad +\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\left[ {\frac{{2{x_{n - 2,n - 2}}}}{{n - 2}}} \right] \\&\qquad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}} \left[ {\sum \limits _{i = 3}^{n - 3} {\left( {\frac{1}{{i - 1}} - 1} \right) \frac{{{x_{i,n - 2}}}}{{n - 2}}} } \right] \\&\qquad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\left[ {\sum \limits _{3 \le i \le j \le n - 3} {\left( {\frac{1}{{i - 1}} + \frac{1}{{j - 1}}} \right) \frac{{{x_{i,j}}}}{{n - 2}}} } \right] \\&\qquad +\, \sum \limits _{3 \le i \le j \le n - 3} {\left( {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }} + \frac{{2\sqrt{j\left( {n - 1} \right) } }}{{\left( {n - 1 + j} \right) \left( {j - 1} \right) }}} \right) {x_{i,j}}} \\&\qquad +\, \sum \limits _{3 \le i \le j \le n - 2} {\frac{{2\sqrt{ij} }}{{i + j}}} {x_{i,j}}+ \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{i,n - 2}}}. \end{aligned}$$

On the other hand

$$\begin{aligned}&\quad -\, \frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}\sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{{n - i - 2}}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) } {x_{2,i}}\\&\quad +\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }}\left( {\frac{{2{x_{2,2}}}}{{n - 2}} - \sum \limits _{i = 3}^{n - 3} {\frac{{\left( {n - i - 2} \right) {x_{2,i}}}}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} } \right) \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}} \sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{1}{{i - 1}}} \right) \frac{{{x_{2,i}}}}{{n - 2}}} \\&\quad +\,\sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{2,i}}} + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}}\\&= - \frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}\sum \limits _{i = 2}^{n - 3} {\left( {\frac{{i\left( {n - 3} \right) }}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) } {x_{2,i}} + \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }} \left( {\frac{{2{x_{2,2}}}}{{n - 2}}} \right) \\&\quad -\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }} \left( {\sum \limits _{i = 3}^{n - 3} {\left( { - 1 + 1 + \frac{{\left( {n - i - 2} \right) }}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) {x_{2,i}}} } \right) \\&\quad +\, \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{2,i}}} \end{aligned}$$

$$\begin{aligned}&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}} \sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{1}{{i - 1}}} \right) \frac{{{x_{2,i}}}}{{n - 2}}} + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}}\\&= - \frac{{2\sqrt{2\left( {n - 2} \right) } }}{n}\sum \limits _{i = 2}^{n - 3} {\left( {\frac{{\left( {i - 1} \right) \left( {n - 3} \right) + \left( {n - 3} \right) }}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) } {x_{2,i}}\\&\quad +\, \frac{{2\sqrt{2\left( {n - 1}\right) } }}{{\left( {n + 1} \right) }}\left( {\frac{{2{x_{2,2}}}}{{n - 2}}} \right) \\&\quad +\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }}\sum \limits _{i = 3}^{n - 3} {{x_{2,i}}} - \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }}\left( {\sum \limits _{i = 3}^{n - 3} {\left( {\frac{{i\left( {n - 3} \right) }}{{\left( {i - 1} \right) \left( {n - 2} \right) }}} \right) {x_{2,i}}} } \right) \\&\quad +\, \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{2,i}}} \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{1}{{i - 1}}} \right) \frac{{{x_{2,i}}}}{{n - 2}}} + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}}\\ \end{aligned}$$

$$\begin{aligned}&= - \frac{{2\sqrt{2\left( {n - 2} \right) } }}{{n\left( {n - 2} \right) }}\sum \limits _{i = 2}^{n - 3} {\left( {\frac{{n - 3}}{1} + \frac{{n - 3}}{{i - 1}}} \right) } {x_{2,i}}\\&\quad + \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }}\left( {\frac{{2{x_{2,2}}}}{{n - 2}}} \right) \\&\quad +\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }}\sum \limits _{i = 3}^{n - 3} {{x_{2,i}}} - \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) \left( {n - 2} \right) }} \left( {\sum \limits _{i = 3}^{n - 3} {\left( {\frac{{n - 3}}{1} + \frac{{n - 3}}{{i - 1}}} \right) {x_{2,i}}} } \right) \\&\quad +\, \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{2,i}}} \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{1}{{i - 1}}} \right) \frac{{{x_{2,i}}}}{{n - 2}}} + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}}\\&= \sum \limits _{i = 2}^{n - 3} {\left( {\frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) }} + \frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}} \right) {x_{2,i}}}\\&\quad -\, \frac{{2\sqrt{2\left( {n - 2} \right) } }}{{n\left( {n - 2} \right) }}\sum \limits _{i = 2}^{n - 3} {\left( {\frac{{n - 3}}{1} + \frac{{n - 3}}{{i - 1}}} \right) } {x_{2,i}}\\&\quad -\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) \left( {n - 2} \right) }} \left( {\sum \limits _{i = 2}^{n - 3} {\left( {\frac{{n - 3}}{1} + \frac{{n - 3}}{{i - 1}}} \right) {x_{2,i}}} } \right) \\&\quad - \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\sum \limits _{i = 2}^{n - 3} {\left( {1 + \frac{1}{{i - 1}}} \right) \frac{{{x_{2,i}}}}{{n - 2}}} + \sum \limits _{i = 2}^{n - 3} {\frac{{2\sqrt{2i} }}{{i + 2}}} {x_{2,i}}, \end{aligned}$$

and

$$\begin{aligned}&\sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{i,n - 2}}} + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\left[ {\frac{{2{x_{n - 2,n - 2}}}}{{n - 2}}} \right] \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}} \left[ {\sum \limits _{i = 3}^{n - 3} {\left( {\frac{1}{{i - 1}} - 1} \right) \frac{{{x_{i,n - 2}}}}{{n - 2}}} } \right] \\&= \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{i,n - 2}}} + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\left[ {\frac{{2{x_{n - 2,n - 2}}}}{{n - 2}}} \right] \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 2} \right) }}\left[ {\sum \limits _{i = 3}^{n - 3} {\left( {\frac{{{x_{i,n - 2}}}}{{i - 1}}} \right) } } \right] + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 3} \right) }}\left[ {\sum \limits _{i = 3}^{n - 3} {\frac{{n - 3}}{{n - 2}}{x_{i,n - 2}}} } \right] \\&= \sum \limits _{i = 3}^{n - 3} {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }}{x_{i,n - 2}}} + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{2n - 3}}\left[ {\frac{{2{x_{n - 2,n - 2}}}}{{n - 2}}} \right] \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 2} \right) }}\left[ {\sum \limits _{i = 3}^{n - 3} {\left( {\frac{{{x_{i,n - 2}}}}{{i - 1}}} \right) } } \right] \\&\quad +\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 3} \right) }}\left[ {\sum \limits _{i = 3}^{n - 3} {\left( {1 - \frac{1}{{n - 2}}} \right) {x_{i,n - 2}}} } \right] \\&= \sum \limits _{i = 3}^{n - 2} {\left( {\frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }} + \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 3} \right) }}} \right) {x_{i,n - 2}}} \\&\quad -\, \frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 2} \right) }}\sum \limits _{i = 3}^{n - 2} {\left( {\frac{1}{{i - 1}} + \frac{1}{{n - 3}}} \right) {x_{i,n - 2}}}. \end{aligned}$$

Therefore,

$$\begin{aligned} GA_1^1\left( G \right)= & {} \frac{{2\left( {n - 1} \right) \left( {n - 3} \right) \sqrt{2\left( {n - 2} \right) } }}{{n\left( {n - 2} \right) }} + \frac{{2\left( {n - 1} \right) \left( {n - 3} \right) \sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) \left( {n - 2} \right) }}\\&+ \frac{{2\left( {n - 1} \right) \sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 2} \right) }} + \sum \limits _{i = 2}^{n - 3} {{f_{2,i}}{x_{2,i}}} + \sum \limits _{3 \le i \le j \le n - 2} {{f_{i,j}}{x_{i,j}}}, \end{aligned}$$

where

$$\begin{aligned} {f_{i,j}}= & {} \frac{{2\sqrt{i\left( {n - 1} \right) } }}{{\left( {n - 1 + i} \right) \left( {i - 1} \right) }} + \frac{{2\sqrt{j\left( {n - 1} \right) } }}{{\left( {n - 1 + j} \right) \left( {j - 1} \right) }}\\&\quad - \,\frac{{2\sqrt{\left( {n - 1} \right) \left( {n - 2} \right) } }}{{\left( {2n - 3} \right) \left( {n - 2} \right) }}\left[ {\frac{1}{{i - 1}} + \frac{1}{{j - 1}}} \right] \\&\quad + \,\frac{{2\sqrt{ij} }}{{i + j}} - \frac{{2\sqrt{2\left( {n - 2} \right) } }}{{n\left( {n - 2} \right) }}\left[ {\frac{{n - 3}}{{i - 1}} + \frac{{n - 3}}{{j - 1}}} \right] \\&\quad -\, \frac{{2\sqrt{2\left( {n - 1} \right) } }}{{\left( {n + 1} \right) \left( {n - 2} \right) }}\left[ {\frac{{n - 3}}{{i - 1}} + \frac{{n - 3}}{{j - 1}}} \right] . \end{aligned}$$