On the general sum-connectivity index of tricyclic graphs

Abstract

The general sum-connectivity index of a graph G is a molecular descriptor defined as $\chi _{\alpha }(G)=\sum _{uv\in E(G)}(d_G(u)+d_G(v))^\alpha$, where $d_G(u)$ denotes the degree of vertex u in G and $\alpha$ is a real number. In this paper, we obtain the first third graphs with maximum general sum-connectivity index among the connected tricyclic graphs of order n for $\alpha \ge 1$ by four transformations which increase the general sum-connectivity index.

1 Introduction

Let $G=(V(G),E(G))$ be a connected simple graph with $|V(G)|=n$ and $|E(G|=m$. If $m=n+c-1$, then G is called a c-cyclic graph. Specially, if $c=0,1,2$ and 3, then G is called a tree, a unicyclic graph, a bicyclic graph and a tricyclic graph, respectively. Let $P_n$ and $S_n$ be respectively the path and the star with n vertices. Let $N_G(v)$ denote the neighbor set of vertex v in G, then $d_{G}(v)=|N_G(v)|$ is the degree of v in G. A pendent path in G is a path having one end vertex of degree at least 3, the other is of degree 1 and the intermediate vertices are of degree 2. An internal path of G is defined as a walk $v_0v_1,\ldots ,v_s(s\ge 1)$ such that the vertices $v_0,v_1,\ldots ,v_s$ are distinct, $d_G(v_0)>2, d_G(v_s)>2$ and $d_G(v_i)=2$, whenever $0<i<s$. Other undefined notation may refer to [1].

The well-known Randić index R(G) of G, is defined as:

$$\begin{aligned} R(G)=\sum _{uv\in E(G)}(d_{G}(u)d_{G}(v))^{-\frac{1}{2}}, \end{aligned}$$

which is proposed by Randić in 1975 [2], has received intensive attention since its successful applications in QSPR and QSAR [3]. Later, Bollobás and Erdös [4] generalized this index to

$$\begin{aligned} R_{\alpha }(G)=\sum _{uv\in E(G)}(d_{G}(u)d_{G}(v))^{\alpha } \end{aligned}$$

for every graph G and an arbitrary value of $\alpha$. The mathematical properties of R(G) as well as those of its generalization $R_{\alpha }(G)$ have been studied extensively as summarized in the books [5, 6].

Recently, a closely related variant of Randić index called the sum-connectivity index [7], denoted by $\chi (G)$, is defined as:

$$\begin{aligned} \chi (G)=\sum _{uv\in E(G)}(d_{G}(u)+d_{G}(v))^{-\frac{1}{2}}. \end{aligned}$$

It has been found that $\chi (G)$ and R(G) correlate well among themselves and with $\pi$-electronic energy of benzenoid hydrocarbons [7]. Similarly, the general sum-connectivity index [8] is defined as:

$$\begin{aligned} \chi _{\alpha }(G)=\sum _{uv\in E(G)}(d_{G}(u)+d_{G}(v))^{\alpha }. \end{aligned}$$

Several extremal properties of the general sum-connectivity index have already been established for general graphs [8], multigraphs [6], trees [6, 7, 9], unicyclic graphs [10, 11] and bicyclic graphs [12].

In this paper we want to extend the extremal study of the general sum-connectivity index to tricyclic graphs (connected graphs with n vertices and $n+2$ edges). More precisely, we will find the graphs with the first fourth largest value of $\chi _{\alpha }(G)$ among the tricyclic graphs of order n for $\alpha \ge 1$ by four transformations which increase the general sum-connectivity index.

2 Preliminaries

In this section, we introduce some graphic transformations and lemmas, which will be used to prove our main results.

Transformation I

[10] Let u and v be two adjacent vertices of a graph G such that $N_G(u)=\{v,z_1,\ldots ,z_p\}, N_G(v)=\{u,w_1,\ldots ,w_s\}$, where $\{z_1,\ldots ,z_p\}\cap \{w_1,\ldots ,w_s\}=\emptyset$, $p\ge 0, s\ge 1$. Let $T_1(G)=G-vw_1-vw_2-\cdots -vw_s+uw_1+uw_2+\cdots +uw_s$. We say that $T_1(G)$ is a $T_1$-transform of G (Fig. 1).

Fig. 1

The graphs G and $T_1(G)$ in Transformation I

Lemma 2.1

[10] Let G and $T_1(G)$ be the graphs in Transformation I, if $\alpha < 0$, then $\chi _{\alpha }(G)>\chi _{\alpha }(T_1(G))$ and if $\alpha > 0$, then $\chi _{\alpha }(G)<\chi _{\alpha }(T_1(G))$.

Lemma 2.2

[12] The real function defined by $f_{\alpha ,a}(x)=(x+a)^{\alpha }-x^{\alpha }$ is strictly increasing for all $\alpha >1$, $a>0$.

Transformation II

Let G be a graph as shown in Fig. 2, and

$$\begin{aligned} N_G(u)=\{u_1,u_2,\ldots ,u_k,u',v,w_1,w_2,\ldots ,w_s\}, \end{aligned}$$

where $u_1,u_2,\ldots ,u_k$ are all the pendent vertices which are adjacent to u, integers $k\ge 1,s\ge 0$, then $d_G(u)-k=s+2\ge 2$. Let $v_1,v_2,\ldots ,v_t$ are all the pendent vertices which are adjacent to v and $d_G(v)=t+2$. Define $T_2(G)$ as the graph obtained from G by deleting $vv_1,vv_2,\ldots ,vv_t$ and adding $uv_1,uv_2,\ldots ,uv_t$.

Fig. 2

The graphs G and $T_2(G)$ in Transformation II

Lemma 2.3

Let G and $T_2(G)$ be the graphs in Transformation II, if $uv\in E(G),$ integers $k\ge 1,s\ge 0,$ $d_G(v)=t+2, t\ge 1$ and $d_G(u')\ge d_G(v')$, then $\chi _{\alpha }(G)<\chi _{\alpha }(T_2(G))$ for $\alpha > 1$.

Proof

By direct calculation, we have

$$\begin{aligned}&\chi _{\alpha }(T_2(G))-\chi _{\alpha }(G)=\left[ (d_G(u')+k+t+s+2)^{\alpha }-(d_G(u')+k+s+2)^{\alpha }\right] \\&\quad +\,\left[ (d_G(v')+2)^{\alpha }-(d_G(v')+t+2)^{\alpha }\right] \\&\quad +\,\left[ (k+t+s+3)^{\alpha }-(k+s+3)^{\alpha }]+t[(k+t+s+3)^{\alpha }-(t+3)^{\alpha }\right] \\&\quad +\,\sum _{i=1}^s\left[ (d_G(w_i)+k+t+s+2)^{\alpha }-(d_G(w_i)+k+s+2)^{\alpha }\right] . \end{aligned}$$

Furthermore, let

$$\begin{aligned} f_{\alpha ,t}(d_G(u')+k+s+2)= & {} (d_G(u')+k+t+s+2)^{\alpha } -(d_G(u')+k+s+2)^{\alpha }]\\ f_{\alpha ,t}(d_G(v')+2)= & {} (d_G(v')+t+2)^{\alpha }-(d_G(v')+2)^{\alpha }. \end{aligned}$$

Note that $d_G(u')\ge d_G(v')$, by Lemma 2.2, we have $f_{\alpha ,t}(d_G(u')+k+s+2)\ge f_{\alpha ,t}(d_G(v')+2)$. Hence we have the desired results. $\square$

Remark 1

Lemma 2.3 is a generalization of Lemma 3 from [12].

For a graph G, the base of G, denoted by $\widehat{G}$, is defined by the unique subgraph of G containing no pendent vertex. Obviously, for a graph G, by repeated Transformations I and II, finally, we can obtain a graph, denoted by $G_0$, which cannot carry on further the Transformations I and II. Then by Lemmas 2.1 and 2.3, we have the following results.

Theorem 2.4

For a graph G, let $G_0$ be the graph defined as above, then

• (i) All the cut-edges of $G_0$ are pendent edges;

• (ii) Let $U=\{u|d_{\widehat{G_0}}(u)\ge 3\}$, all bunches of pendent edges to $V(\widehat{G_0})-U$ are situated at distances of at least two one from another;

• (iii) $\chi _{\alpha }(G)<\chi _{\alpha }(G_0)$.

Transformation III

For a graph G, let $G_0$ be the graph defined as above. Let $U=\{u|d_{\widehat{G_0}}(u)\ge 3\}$, $v\in V(\widehat{G_0})-U$ and $v_1,v_2,\ldots ,v_t(t\ge 1)$ are all the pendent vertices which are adjacent to v in $G_0$. For a vertex $u\in U,$ if $uv\in E(G_0)$ or P(u, v) is an internal path from u to v in $G_0$, let $T_3(G_0)=G_0-\{vv_1,vv_2,\ldots ,vv_t\}+\{uv_1,uv_2,\ldots ,uv_t\}.$

Lemma 2.5

Let $G_0, T_3(G_0)$ be the graphs in Transformation III, then $\chi _{\alpha }(G_0)<\chi _{\alpha }(T_3(G_0))$.

Proof

Case 1 If $uv\in E(G_0)$, by the definition of $G_0$, we know that $d_{\widehat{G_0}}(v)=2$. Let $N_{\widehat{G_0}}(v)=\{u,w\}$ and $u_1,\ldots ,u_k(k\ge 0)$ be the all pendent vertices which are adjacent to u in $G_0$.

Subcase 1.1 If $w\in U$ and $uw\in E(G_0)$, let $N_{G_0}(u)=\{v,w, u_1,\ldots ,u_k, w_1,\ldots , w_s\}(s\ge 1)$. By direct calculation, we have

$$\begin{aligned}&\chi _{\alpha }(T_3(G_0))-\chi _{\alpha }(G_0)\\&=\left[ (k+t+s+2+d_{G_0}(w))^{\alpha }-(k+s+2+d_{G_0}(w))^{\alpha }\right] \\&\quad +\,k\left[ (k+t+s+3)^{\alpha }-(k+s+3)^{\alpha }\right] \\&\quad +\,\left[ (d_{G_0}(w)+2)^{\alpha }-(t+2+d_{G_0}(w))^{\alpha }\right] +t\left[ (k+t+s+3)^{\alpha }-(t+3)^{\alpha }\right] \\&\quad +\,\sum _{i=1}^s\left[ (d_{G_0}(w_i)+k+t+s+2)^ {\alpha }-(d_{G_0}(w_i)+k+s+2)^{\alpha }\right] \\&\ge \left[ (k+t+s+2+d_{G_0}(w))^{\alpha }-(k+s+2+d_{G_0}(w))^{\alpha }\right] \\&\quad -\,\left[ (t+2+d_{G_0}(w))^{\alpha }-(d_{G_0}(w)+2)^{\alpha }\right] >0. \end{aligned}$$

Subcase 1.2 If $w\in U$ and $uw\notin E(G_0)$, without loss of generality, let $d_{G_0}(w)\le d_{G_0}(u)$(otherwise, we add the edges to w). Let $N_{G_0}(u)=\{v, u_1,\ldots ,u_k, w_1,\ldots , w_s\}(s\ge 2)$, then $d_{G_0}(u)=k+s+1$. Note that $d_{G_0}(w_i)\ge 2$ and $t\ge 1$. By direct calculation, we have

$$\begin{aligned}&\chi _{\alpha }(T_3(G_0))-\chi _{\alpha }(G_0)\\&\quad =k\left[ (k+t+s+2)^{\alpha }-(k+s+2)^{\alpha }\right] + t\left[ (k+t+s+2)^{\alpha }-(t+3)^{\alpha }\right] \\&\qquad +\left[ (d_{G_0}(w)+2)^{\alpha }-(t+2+d_{G_0}(w))^{\alpha }\right] \\&\qquad +\sum _{i=1}^s\left[ (d_{G_0}(w_i)+k+t+s+1)^{\alpha }- (d_{G_0}(w_i)+k+s+1)^{\alpha }\right] \\&\quad =k\left[ (k+t+s+2)^{\alpha }-(k+s+2)^{\alpha }\right] +t\left[ (k+t+s+2) ^{\alpha }-(t+3)^{\alpha }\right] \\&\qquad +\left[ (d_{G_0}(w)+2)^{\alpha }-(t+2+d_{G_0}(w))^{\alpha }\right] \\&\qquad +\sum _{i=1}^s\left[ (d_{G_0}(w_i)+t+d_{G_0}(u))^ {\alpha }-(d_{G_0}(w_i)+d_{G_0}(u))^{\alpha }\right] \\&\quad >\left[ (t+d_{G_0}(w_i)+d_{G_0}(u))^{\alpha }-(d_{G_0} (w_i)+d_{G_0}(u))^{\alpha }\right] \\&\qquad -\left[ (t+d_{G_0}(w)+2)^{\alpha }-(d_{G_0}(w)+2)^{\alpha }\right] \ge 0. \end{aligned}$$

Subcase 1.3 If $w\notin U$, then $d_{G_0}(w)=2$. let $N_{G_0}(u)=\{v,u_1,\ldots ,u_k, w_1,\ldots , w_s\}(s\ge 1)$. If $k=0$, by Transformation I, we can obtain a graph $T_1(G)$ with $\chi _\alpha (G)<\chi _\alpha (T_1(G))$. So we can assume that $k\ge 1$. Then

$$\begin{aligned}&\chi _{\alpha }(T_3(G_0))-\chi _{\alpha }(G_0)\\&\quad =\left[ 4^{\alpha }-(t+4)^{\alpha }]+k[(k+t+s+2)^{\alpha }-(k+s+2)^{\alpha }\right] \\&\qquad +\,\,t\left[ (k+t+s+2)^{\alpha }-(t+3)^{\alpha }\right] \\&\qquad +\sum _{i=1}^s\left[ (d_{G_0}(w_i)+k+t+s+1)^{\alpha }- (d_{G_0}(w_i)+k+s+1)^{\alpha }\right] \\&\quad \ge \left[ 4^{\alpha }-(t+4)^{\alpha }\right] +\left[ (d_{G_0}(w_1)+k+t+s+1)^{\alpha }- (d_{G_0}(w_1)+k+s+1)^{\alpha }\right] . \end{aligned}$$

Now let

$$\begin{aligned} f_{\alpha ,t}(d_{G_0}(w_1)+k+s+1)= & {} (d_{G_0}(w_1)+k+t+s+1)^{\alpha } -(d_{G_0}(w_1)+k+s+1)^{\alpha },\\ f_{\alpha ,t}(4)= & {} (t+4)^{\alpha }-4^{\alpha }, \end{aligned}$$

Obviously, $f_{\alpha ,t}(d_{G_0}(w_1)+k+s+1)\ge f_{\alpha ,t}(4)$ since $d_{G_0}(w_1)+k+s+1\ge 4$. Further by Lemma 2.2, we have $\chi _{\alpha }(G_0)<\chi _{\alpha }(T_3(G_0))$.

Case 2 If P(u, v) is an internal path from u to v in $G_0$, by Case 1, we can assume that all the neighbors of u and v situated on the base of $G_0$ have degree 2 in $G_0$. Let $N_{G_0}(u)=\{u_1,\ldots ,u_k, w_1,\ldots , w_s\}(k\ge 0,s\ge 3)$, where $u_1,\ldots ,u_k(k\ge 0)$ are the all pendent vertices which are adjacent to u in $G_0$. Then

$$\begin{aligned}&\chi _{\alpha }(T_3(G_0))-\chi _{\alpha }(G_0)\\&\quad =2[4^{\alpha }-(t+4)^{\alpha }]+k[(k+t+s+1)^{\alpha }-(k+s+1)^{\alpha }]\\&\qquad +\,\, t[(k+t+s+1)^{\alpha }-(t+3)^{\alpha }]+s[(k+t+s+2)^{\alpha }-(k+s+2) ^{\alpha }]\\&\quad >2[4^{\alpha }-(t+4)^{\alpha }]+s[(k+t+s+2)^{\alpha }-(k+s+2)^{\alpha }]. \end{aligned}$$

And let

$$\begin{aligned} f_{\alpha ,t}(k+s+2)= & {} (k+t+s+2)^{\alpha }-(k+s+2)^{\alpha },\\ f_{\alpha ,t}(4)= & {} (t+4)^{\alpha }-4^{\alpha }, \end{aligned}$$

then $f_{\alpha ,t}(k+s+2)\ge f_{\alpha ,t}(4)$ since $k+s+2\ge 5$. Hence $\chi _{\alpha }(G_0)<\chi _{\alpha }(T_3(G_0))$. $\square$

Remark 2

By repeated Transformation III, all the pendent edges which are adjacent to a vertex in $V(\widehat{G_0})-U$ can move to a vertex in U, denote the resulted graph by $TT_3(G_0)$, then $\chi _{\alpha }(G_0)<\chi _{\alpha }(TT_3(G_0))$. Furthermore, for any $u,u'\in U$, we have $uu'\in E(TT_3(G_0))$ or $P(u,u')=ux_1x_2\cdots x_tu'(t\ge 1)$ is an internal path from u to $u'$ in $TT_3(G_0).$

Transformation IV

If $P(u,u')=ux_1x_2\cdots x_tu'$ is an internal path from u to $u'$ in $TT_3(G_0),$ let $T_4(TT_3(G_0))=TT_3(G_0)-x_1x_2+ux_2$.

Similar to the proof of Lemma 2.5, we have

Lemma 2.6

Let $TT_3(G_0), T_4(TT_3(G_0))$ be the graphs in Transformation IV, then $\chi _{\alpha }(TT_3(G_0))<\chi _{\alpha }(T_4(TT_3(G_0)))$.

3 Main results

The base of a tricyclic graph G, denoted by $\widehat{G}$, is the minimal tricyclic subgraph of G. Obviously, $\widehat{G}$ is the unique tricyclic subgraph of G containing no pendent vertex, and G can be obtained from $\widehat{G}$ by planting trees to some vertices of $\widehat{G}$. By [13], we know that tricyclic graphs have the following four types of bases (as shown in Figs. 3, 4, 5): $G_j^3 (j=1,\ldots , 7),G_j^4 (j=1,\ldots , 4), G_j^6 (j=1,\ldots , 3)$ and $G_1^7$. Let

$$\begin{aligned}&{\fancyscript{G}}_{n,n+2}^3=\{G|\widehat{G}\cong G_j^3, j\in \{1,\ldots , 7\}\};\ \ \ \ {\fancyscript{G}}_{n,n+2}^4=\{G|\widehat{G}\cong G_j^4, j\in \{1,\ldots , 4\}\}; \\&{\fancyscript{G}}_{n,n+2}^6=\{G|\widehat{G}\cong G_j^6, j\in \{1,\ldots , 3\}\}; \quad {\fancyscript{G}}_{n,n+2}^7=\{G|\widehat{G}\cong G_1^7\}. \end{aligned}$$

Then ${\fancyscript{G}}_{n,n+2}={\fancyscript{G}}_{n,n+2}^3\cup {\fancyscript{G}}_{n,n+2}^4\cup {\fancyscript{G}}_{n,n+2}^6\cup {\fancyscript{G}}_{n,n+2}^7.$

Fig. 3

The graphs $G_i^3(i=1,2,\ldots ,7)$

Fig. 4

The graphs $G_i^4(i=1,2,\ldots ,4)$

Fig. 5

The graphs $G_i^6(i=1,2,3)$ and $G_1^7$

Lemma 3.1

Let $T_a^3, T_b^3$ be the graphs as shown in Fig. 6, $s\ge t\ge 1$, then $\chi _{\alpha }(T_a^3)<\chi _{\alpha }(T_b^3)$.

Fig. 6

The graphs $T_a^3$ and $T_b^3$

Proof

By direct calculation, we have

$$\begin{aligned}&\chi _{\alpha }(T_b^3)-\chi _{\alpha }(T_a^3)\\&\quad =s[(s+6)^\alpha -(s+5)^\alpha ]-t[(t+5)^\alpha -(t+4)^\alpha ]+[(t+5)^\alpha -(t+4)^\alpha ]\\&\qquad +\,[(s+6)^\alpha -(t+5)^\alpha ]+3[(s+7)^\alpha -(s+6) ^\alpha ]-3[(t+6)^\alpha -(t+5)^\alpha ]. \end{aligned}$$

By Lemma 2.2, $\chi _{\alpha }(T_a^3)<\chi _{\alpha }(T_b^3)$. $\square$

Lemma 3.2

Let $T_1^3, T_2^3$ be the graphs as shown in Fig. 7, then $\chi _{\alpha }(T_1^3)<\chi _{\alpha }(T_2^3)$.

Fig. 7

The graphs $T_1^3, T_2^3$

Proof

By direct calculation, we have

$$\begin{aligned} \chi _{\alpha }(T_1^3)= & {} (n-7)(n-2)^\alpha +(n+1)^\alpha +3(n-1)^\alpha +3\cdot 6^\alpha +2\cdot 4^\alpha ,\\ \chi _\alpha (T_2^3)= & {} (n-7)n^\alpha +6(n+1)^\alpha +3\cdot 4^\alpha \end{aligned}$$

Then

$$\begin{aligned}&\chi _\alpha (T_2^3)-\chi _{\alpha }(T_1^3)\\&\quad =[(n-7)n^\alpha +6(n+1)^\alpha +3\cdot 4^\alpha ]-[(n-7)(n-2)^\alpha \\&\qquad +\,(n+1)^\alpha +3(n-1)^\alpha +3\cdot 6^\alpha +2\cdot 4^\alpha ]\\&\quad =(n-7)(n^\alpha -(n-2)^\alpha )+3[(n+1)^\alpha \\&\qquad -\,(n-1)^\alpha ]+2[(n+1)^\alpha -6^\alpha ]+4^\alpha -6^\alpha \\&\quad >2[(n-5+6)^\alpha -6^\alpha ]+4^\alpha -6^\alpha \\&\quad >[(2+6)^\alpha -6^\alpha ]-[(2+4)^\alpha -4^\alpha ]. \end{aligned}$$

Let

$$\begin{aligned} f_{\alpha ,2}(6)= & {} (2+6)^\alpha -6^\alpha ,\\ f_{\alpha ,2}(4)= & {} (2+4)^\alpha -4^\alpha , \end{aligned}$$

then by Lemma 2.2, we have $f_{\alpha ,2}(6)>f_{\alpha ,2}(4).$ Hence $\chi _{\alpha }(T_1^3)<\chi _{\alpha }(T_2^3)$. $\square$

By repeated translations I–IV and Lemmas 2.1, 2.3, 2.5, 2.6, 3.1 and 3.2, we have

Theorem 3.3

Let $G\in {\fancyscript{G}}_{n,n+2}^3$ and $G\ne T_1^3,T_2^3$, $\chi _{\alpha }(G)<\chi _{\alpha }(T_1^3)<\chi _{\alpha }(T_2^3)$.

Lemma 3.4

Let $T_1^4, T_2^4$ be the graphs as shown in Fig. 8, then $\chi _{\alpha }(T_1^4)<\chi _{\alpha }(T_2^4)$.

Fig. 8

The graphs $T_1^4, T_2^4$

Proof

By direct calculation, we have

$$\begin{aligned} \chi _{\alpha }(T_1^4)= & {} (n-6)(n-1)^\alpha +2 (n+1)^\alpha +2n^\alpha +6^\alpha +4^\alpha +2\cdot 5^\alpha ,\\ \chi _{\alpha }(T_2^4)= & {} (n-6)n^\alpha +(n+2) ^\alpha +4(n+1)^\alpha +4^\alpha +2\cdot 5^\alpha . \end{aligned}$$

Then

$$\begin{aligned}&\chi _{\alpha }(T_2^4)-\chi _{\alpha }(T_1^4)\\&\quad =(n-6)[n^\alpha -(n-1)^\alpha ]+2[(n+1)^\alpha -n^\alpha ] +[(n+2)^\alpha -6^\alpha ]>0. \end{aligned}$$

Hence we obtain our desired result. $\square$

Remark 3

Let G is a graph with base $\widehat{T_1^4}$ (or $\widehat{T_2^4}$), if there are two vertices u, v in its base with degrees no less 3 and at least one pendent edge attaching at each one, its general sum-connectivity index is less than $T_1^4$ (or $T_2^4$).

By repeated translations I–IV and Lemmas 2.1, 2.3, 2.5, 2.6, 3.4 and Remark 3, we have

Theorem 3.5

Let $G\in {\fancyscript{G}}_{n,n+2}^4$ and $G\ne T_1^4,T_2^4$, then $\chi _{\alpha }(G)<\chi _{\alpha }(T_1^4)<\chi _{\alpha }(T_2^4)$.

Lemma 3.6

Let $T_1^6, T_2^6, T_3^6$ be the graphs as shown in Fig. 9, then $\chi _{\alpha }(T_1^6)>\chi _{\alpha }(T_2^6)>\chi _{\alpha }(T_3^6)$.

Fig. 9

The graphs $T_1^6, T_2^6, T_3^6$

Proof

By direct calculation, we have

$$\begin{aligned} \chi _{\alpha }(T_1^6)= & {} (n-5)n^\alpha +3(n+1)^\alpha +(n+3) ^\alpha +3\cdot 6^\alpha ,\\ \chi _{\alpha }(T_2^6)= & {} (n-5)n^\alpha +2(n+1)^\alpha +2(n+2) ^\alpha +2\cdot 5^\alpha +6^\alpha ,\\ \chi _{\alpha }(T_3^6)= & {} (n-6)(n-2)^\alpha +2n^\alpha +(n-1) ^\alpha +3\cdot 5^\alpha +2\cdot 6^\alpha .\\ \end{aligned}$$

Then

$$\begin{aligned} \chi _{\alpha }(T_1^6)-\chi _{\alpha }(T_2^6)= & {} [(n+1)^\alpha -(n+2)^\alpha ]+[(n+3)^\alpha -(n+2) ^\alpha ]+2[6^\alpha -5^\alpha ]\\= & {} [(n+1)^\alpha -(n+2)^\alpha ]+[(n+3)^\alpha -(n+2) ^\alpha ]+2[6^\alpha -5^\alpha ]. \end{aligned}$$

Let

$$\begin{aligned} f_{\alpha ,1}(n+2)= & {} (n+3)^\alpha -(n+2)^\alpha ,\\ f_{\alpha ,2}(n+1)= & {} (n+2)^\alpha -(n+1)^\alpha , \end{aligned}$$

then by Lemma 2.2, we have $f_{\alpha ,1}(n+2)>f_{\alpha ,2}(n+1).$ So $\chi _{\alpha }(T_1^6)>\chi _{\alpha }(T_2^6)$. Further,

$$\begin{aligned}&\chi _{\alpha }(T_2^6)-\chi _{\alpha }(T_3^6)\\&\quad =(n-6)[n^\alpha -(n-2)^\alpha ]+2(n+1)^\alpha +2(n+2) ^\alpha -n^\alpha -(n-1)^\alpha -5^\alpha -6^\alpha >0. \end{aligned}$$

Hence we have our desirable result. $\square$

Remark 4

Let G is a graph with base $\widehat{T_1^6}$ ($\widehat{T_2^6}$ or $\widehat{T_3^6}$), if there are two vertices u, v in its base with degrees no less 3 and at least one pendent edge attaching at each one, its general sum-connectivity index is less than $T_1^6$ ($T_2^6$ or $T_3^6$).

By repeated translations I–IV and Lemmas 2.1, 2.3, 2.5, 2.6, 3.6 and Remark 4, we have

Theorem 3.7

Let $G\in {\fancyscript{G}}_{n,n+2}^6$ and $G\ne T_1^6, T_2^6, T_3^6$, $\chi _{\alpha }(G)<\chi _{\alpha }(T_3^6)<\chi _{\alpha }(T_2^6) <\chi _{\alpha }(T_1^6)$.

Lemma 3.8

Let $T_a^7, T_b^7$ be the graphs as shown in Fig. 10, where $s\ge t\ge 1$ and $T_b^7=T_a^7-vy_t+ux_{s+1}.$ Then $\chi _\alpha (T_a^7)<\chi _\alpha (T_b^7)$.

Fig. 10

The graphs $T_a^7, T_b^7$

Proof

By direct calculation, we have

$$\begin{aligned}&\chi _{\alpha }(T_b^7)-\chi _{\alpha }(T_a^7)\\&\quad =(t-1)[(t+3)^\alpha -(t+1)^\alpha ]+[(s+5)^\alpha -(t+4)^\alpha ]\\&\qquad +s[(s+5)^\alpha -(s+4)^\alpha ]+2[(s+7)^\alpha -(s+6) ^\alpha ]+2[(t+5)^\alpha -(t+6)^\alpha ]. \end{aligned}$$

Let

$$\begin{aligned} f_{\alpha ,1}(s+6)= & {} (s+7)^\alpha -(s+6)^\alpha ,\\ f_{\alpha ,1}(t+5)= & {} (t+6)^\alpha -(t+5)^\alpha . \end{aligned}$$

By Lemma 2.2, we have $f_{\alpha ,1}(s+6)>f_{\alpha ,1}(t+5)$ since $s\ge t.$ Hence $\chi _\alpha (T_a^7)<\chi _\alpha (T_b^7)$. $\square$

By Lemma 3.8, we have the following lemma.

Lemma 3.9

Let $T_i^7, i=1,2$ be the graphs as shown in Fig. 11, $\chi _\alpha (T_2^7)<\chi _\alpha (T_1^7)$.

Fig. 11

The graphs $T_i^7,i = 1, 2$

By repeated translations I–IV and Lemmas 2.1, 2.3, 2.5, 2.6 and  3.9, we have

Theorem 3.10

Let $G\in {\fancyscript{G}}_{n,n+2}^7$ and $G\ne T_i^7(i=1,2)$, $\chi _{\alpha }(G)<\chi _{\alpha }(T_2^7)<\chi _{\alpha }(T_1^7)$.

Lemma 3.11

For $n\ge 6$, $\chi _\alpha (T_2^7)<\chi _\alpha (T_2^6)<\min \{\chi _\alpha (T_1^6),\chi _{\alpha }(T_1^7)\}$ and $\chi _\alpha (T_2^3)<\chi _\alpha (T_2^4)<\chi _\alpha (T_2^6)<\min \{\chi _\alpha (T_1^6),\chi _{\alpha }(T_1^7)\}$.

Proof

Let $f_{\alpha ,1}(n+2)=(n+3)^\alpha -(n+2)^\alpha , f_{\alpha ,1}(n+1)=(n+2)^\alpha -(n+1)^\alpha , f_{\alpha ,1}(n)=(n+1)^\alpha -n^\alpha$. Then by Lemma 2.2, we have

$$\begin{aligned} \chi _\alpha (T_2^4)-\chi _\alpha (T_2^3)= & {} [n^\alpha -(n+1)^\alpha ]+[(n+2)^\alpha -(n+1)^\alpha ]+ 2(5^\alpha -4^\alpha )>0,\\ \chi _\alpha (T_2^6)-\chi _\alpha (T_2^4)= & {} [n^\alpha -(n+1)^\alpha ]+[(n+2)^\alpha -(n+1)^\alpha ] +(6^\alpha -4^\alpha )>0,\\ \chi _{\alpha }(T_2^6)-\chi _{\alpha }(T_2^7)= & {} (n-5)[n^\alpha -(n-1) ^\alpha ]+(n+2)^\alpha ]+5^\alpha -2\cdot 7^\alpha \\\ge & {} [n^\alpha -(n-1)^\alpha ]+(n+2)^\alpha ] +5^\alpha -2\cdot 7^\alpha \\\ge & {} (8^\alpha -7^\alpha )+(6^\alpha -7^\alpha )>0,\\ \chi _{\alpha }(T_1^7)-\chi _{\alpha }(T_2^6)= & {} [(n+2) ^\alpha -(n+1)^\alpha ]+[n^\alpha -(n+1)^\alpha ]+2(6^\alpha -5^\alpha )>0. \end{aligned}$$

Further by Lemmas 3.6 and 3.9, we have our desired results. $\square$

Note that

$$\begin{aligned}&\chi _\alpha (T_1^7)-\chi _\alpha (T_1^6)\\&\quad =[(n-4)n^\alpha +3(n+2)^\alpha +3\cdot 6^\alpha ]-[(n-5) n^\alpha +3(n+1)^\alpha +(n+3)^\alpha +3\cdot 6^\alpha ]\\&\quad =[n^\alpha -(n+3)^\alpha ]+3[(n+2)^\alpha -(n+1)^\alpha ]. \end{aligned}$$

It is not easy to confirm the sign of the difference $\chi _\alpha (T_1^7)-\chi _\alpha (T_1^6)$, but there exists a natural number $n_0(\alpha )$ such that $\chi _{\alpha }(T_1^{7})>\chi _{\alpha }(T_1^{6})$ for every $n\ge n_0(\alpha )$. Using mathematical software it can be seen that $n_0(\alpha )=\alpha -2$ for every $\alpha \in N, 8\le \alpha \le 20$.

Theorems 3.3, 3.5, 3.7, 3.10 imply

Theorem 3.12

Let $G\in {\fancyscript{G}}_{n,n+2}(n\ge 6)$ and $G\ne T_2^6, T_1^6, T_1^7$, then $\chi _{\alpha }(G)<\chi _\alpha (T_2^6)<\min \{\chi _\alpha (T_1^6),\chi _{\alpha }(T_1^7)\}< \max (\chi _{\alpha }(T_1^{6}),\chi _{\alpha }(T_1^{7}))$ for $\alpha \ge 1$.