Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

Abstract

Building upon the notion of the Gutman index $SGut\left(G\right)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index ${SGut}_k\left(G\right)$ of G is defined by ${SGut}_k\left(G\right)$ $={\sum }_{S\subseteq V\left(G\right),\left|S\right|=k}\left({\prod }_{v\in S}de{g}_G\left(v\right)\right)d_G\left(S\right)$, in which $d_G\left(S\right)$ is the Steiner distance of S and $de{g}_G\left(v\right)$ is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on ${SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter ${SGut}_k$. We obtain sharp upper and lower bounds of ${SGut}_k\left(G\right)+{SGut}_k\left(\overline{G}\right)$ and ${SGut}_k\left(G\right)·{SGut}_k\left(\overline{G}\right)$ for a connected graph G of order n, m edges, maximum degree $\Delta$ and minimum degree $\delta$.

1. Introduction

We consider simple, undirected graphs in this paper. For the standard theoretical graph terminology and notation not defined here, follow [1]. For a graph G, let $V\left(G\right)$ and $E\left(G\right)$ represent its sets of vertices and edges, respectively. Let $\left|E\right(G\left)\right|=m$ be the size of G. The complement of G is conventionally denoted by $\overline{G}$. For a vertex $v\in V\left(G\right)$, $de{g}_G\left(v\right)$ is the degree of v. The maximum and minimum degrees are, respectively, denoted by $\Delta$ and $\delta$. Like degrees, distance is a fundamental concept of graph theory [2]. For two vertices $u,v\in V\left(G\right)$ with connected G, the distance $d(u,v)=d_G(u,v)$ between these two vertices is defined as the length of a shortest path connecting them. An excellent survey paper on this subject can be found in [3].

The above classical graph distance was extended by Chartrand et al. in 1989 to the Steiner distance, which since then has become an essential concept of graph theory. Given a graph $G(V,E)$ and a vertex set $S\subseteq V\left(G\right)$ containing no less than two vertices, an S-Steiner tree (or an S-tree, a Steiner tree connecting S) is defined as a subgraph $T(V^{\prime },E^{\prime })$ of G, which is a subtree satisfying $S\subseteq V^{\prime }$. If G is connected with order no less than 2 and $S\subseteq V$ is nonempty, the Steiner distance $d\left(S\right)$ among the vertices of S (sometimes simply put as the distance of S) is the minimum size of connected subgraph whose vertex sets contain the set S. Clearly, for a connected subgraph $H\subseteq G$ with $S\subseteq V\left(H\right)$ and $\left|E\right(H\left)\right|=d\left(S\right)$, H is a tree. When T is subtree of G, we have $d\left(S\right)=min\left\{\right|E\left(T\right)|,S\subseteq V(T\left)\right\}$. For $S=\{u,v\}$, $d\left(S\right)=d(u,v)$ reduces to the classical distance between the two vertices u and v. Another basic observation is that if $\left|S\right|=k$, $d\left(S\right)\ge k-1$. For more results regarding varied properties of the Steiner distance, we refer to the reader to [3,4,5,6,7,8].

In [9], Li et al. generalized the concept of Wiener index through incorporating the Steiner distance. The Steiner k-Wiener index ${SW}_k\left(G\right)$ of G is defined by

$${SW}_k\left(G\right)=\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}d\left(S\right).$$

For $k=2$, it is easy to see the Steiner Wiener index coincides with the ordinary Wiener index. The interesting range of the Steiner k-Wiener index ${SW}_k$ resides in $2\le k\le n-1$, and the two trivial cases give ${SW}_1\left(G\right)=0$ and ${SW}_n\left(G\right)=n-1$.

Gutman [10] studied the Steiner degree distance, which is a generalization of ordinary degree distance. Formally, the k-center Steiner degree distance ${SDD}_k\left(G\right)$ of G is given as

$${SDD}_k\left(G\right)=\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\sum _{v\in S}de{g}_G\left(v\right)\right)d_G\left(S\right).$$

The Gutman index of a connected graph G is defined as

$$Gut\left(G\right)=\sum _{u,v\in V\left(G\right)}de{g}_G\left(u\right)de{g}_G\left(v\right)d_G(u,v).$$

The Gutman index of graphs attracted attention very recently. For its basic properties and applications, including various lower and upper bounds, see [11,12,13] and the references cited therein. Recently, Mao and Das [14] further extended the concept of the Gutman index by incorporating Steiner distance and considering the weights as multiplications of degrees. The Steiner k-Gutman index ${SGut}_k\left(G\right)$ of G is defined by

$${SGut}_k\left(G\right)=\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_G\left(v\right)\right)d_G\left(S\right).$$

Note that this index is a natural generalization of the classical Gutman index—in particular, for $k=2$, ${SGut}_k\left(G\right)=Gut\left(G\right)$. This is the reason the product of the degrees comes to the definition of Steiner k-Gutman index. The weighting of multiplication of degree or expected degree has also been extensively explored in, for example, the field of random graphs [15,16] and proves to be very prolific. For more results on Steiner Wiener index, Steiner degree distance and Steiner Gutman index, we refer to the reader to [9,10,14,17,18,19].

For a given a graph parameter $f\left(G\right)$ and a positive integer n, the well-known Nordhaus–Gaddum problem is to determine sharp bounds for: $\left(1\right)$ $f\left(G\right)+f\left(\overline{G}\right)$ and $\left(2\right)$ $f\left(G\right)·f\left(\overline{G}\right)$ over the class of connected graph G, with order n, m edges, maximum degree $\Delta$ and minimum degree $\delta$ characterizing the extremal graphs. Many Nordhaus–Gaddum type relations have attracted considerable attention in graph theory. Comprehensive results regarding this topic can be found in e.g., [20,21,22,23,24].

In Section 2, we obtain sharp upper and lower bounds on ${\mathrm{SGut}}_k$ of graph G. In Section 3, we obtain sharp upper and lower bounds of ${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)$ and ${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)$ for a connected graph G in terms of n, m, maximum degree $\Delta$ and minimum degree $\delta$.

2. Sharp Bounds for the Steiner Gutman Index

In [14], the following results have been obtained:

Lemma 1

([14]). Let $K_n$, $S_n$ and $P_n$ be the complete graph, star graph and path graph of order n, respectively, and let k be an integer such that $2\le k\le n$. Then

(1) 

${SGut}_k\left(K_n\right)=\left(\genfrac{}{}{0pt}{}{n}{k}\right){(n-1)}^n(k-1)$;

(2) 

${SGut}_k\left(S_n\right)=(kn-2k+1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)$;

(3) 

${SGut}_k\left(P_n\right)=2^k(k-1)\left(\genfrac{}{}{0pt}{}{n}{k+1}\right)$.

For connected graph G of order n with m edges, the authors in [14] derived the following upper and lower bounds on ${\mathrm{SGut}}_k\left(G\right)$.

Lemma 2.

([14]). Let G be a connected graph of order n with m edges, and let k be an integer with $2\le k\le n$. Then

$$(n-1){\left(\frac{2m}{k}\right)}^k{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^k\ge {\mathrm{SGut}}_k\left(G\right)\ge \left\{\begin{array}{cc}2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\hfill & \mathrm{if}\delta \ge 2\hfill \\ (k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta =1.\hfill \end{array}\right.$$

We now give lower and upper bounds for ${\mathrm{SGut}}_k\left(G\right)$ in terms of n, m, maximum degree $\Delta$ and minimum degree $\delta$:

Proposition 1.

Let G be a connected graph of order $n\ge 3$ with m edges and maximum degree Δ, minimum degree δ. Additionally, let k be an integer with $2\le k\le n$. Then

$$2m(n-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\Delta }^{k-1}}{k}\ge {\mathrm{SGut}}_k\left(G\right)\ge \left\{\begin{array}{cc}2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}\hfill & \mathrm{if}\delta \ge 2\hfill \\ k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]\hfill & \mathrm{if}\delta =1,\hfill \end{array}\right.$$

where p is the number of pendant vertices in G, and $q=max\{k-p,1\}$. The equality of upper bound holds if and only if G is a regular graph with $k=n$. The equality of lower bound holds if and only if G is a regular $(n-k+1)$-connected graph of order n $(\delta \ge 2)$, or $G\cong P_n$ and $k=n>3$ $(\delta =1)$, or $G\cong P_3$ and $k=2$ $(\delta =1)$.

Proof. 

Upper bound: For any $S\subseteq V\left(G\right)$ and $\left|S\right|=k$, we have $k-1\le d_G\left(S\right)\le n-1$, and hence

$$(k-1)\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_G\left(v\right)\right)\le {\mathrm{SGut}}_k\left(G\right)\le (n-1)\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_G\left(v\right)\right).$$

(1)

Let

$$M=\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_G\left(v\right)\right)=\sum _{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)}de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\cdots de{g}_G\left(v_k\right).$$

and

$$N=\sum _{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)}[de{g}_G\left(v_1\right)+de{g}_G\left(v_2\right)+\cdots +de{g}_G\left(v_k\right)].$$

We first prove the upper bound. Without loss of generality, we can assume that $de{g}_G\left(v_1\right)\le de{g}_G\left(v_2\right)\le \dots \le de{g}_G\left(v_k\right)$. Since

$$\begin{array}{ccc}& & de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)\le {\Delta }^{k-1}de{g}_G\left(v_1\right)\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& \le & \frac{{\Delta }^{k-1}}{k}(de{g}_G\left(v_1\right)+de{g}_G\left(v_2\right)+\cdots +de{g}_G\left(v_k\right)),\hfill \end{array}$$

(3)

it follows that

$$\begin{array}{ccc}\hfill M& =& \sum _{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)}de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)\hfill \\ & \le & \frac{{\Delta }^{k-1}}{k}\sum _{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)}[de{g}_G\left(v_1\right)+de{g}_G\left(v_2\right)+\cdots +de{g}_G\left(v_k\right)]\hfill \\ & \le & \frac{{\Delta }^{k-1}}{k}N.\hfill \end{array}$$

For each $v\in V\left(G\right)$, there are $\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)$k-subsets in G such that each of them contains v. The contribution of vertex v is exactly $\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)de{g}_G\left(v\right)$. From the arbitrariness of v, we have

$$N=\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\sum _{v\in V\left(G\right)}de{g}_G\left(v\right)=2m\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right),$$

and hence

$${\mathrm{SGut}}_k\left(G\right)\le (n-1)M\le (n-1)\frac{{\Delta }^{k-1}}{k}N=2m(n-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\Delta }^{k-1}}{k}.$$

(4)

Suppose that the left equality holds. Then all the inequalities in the above must be equalities. From the equality in (3), one can easily see that G is a regular graph. From the equality in (4), we have $d\left(S\right)=n-1$ for any $S\subseteq V\left(G\right)$, $\left|S\right|=k$. Since G is connected, then there exists an $S\subseteq V\left(G\right)$ such that $|d_G\left(S\right)|=k-1$. If $k\le n-1$, then one can easily see that the upper bound is strict as $|d_G\left(S\right)|=k-1\le n-2$ for some S. Otherwise, $k=n$. Since G is connected, we have $|d_G\left(S\right)|=n-1$ for any $S\subseteq V\left(G\right)$. Hence G is a regular graph with $k=n$.

Conversely, one can see easily that the left equality holds for regular graph with $k=n$.

Lower bound: Without loss of generality, we can assume that $de{g}_G\left(v_1\right)\le de{g}_G\left(v_2\right)\le \dots \le de{g}_G\left(v_k\right)$. First we assume that $\delta \ge 2$. Then

$$\begin{array}{ccc}& & de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\cdots de{g}_G\left(v_k\right)\ge {\delta }^{k-1}de{g}_G\left(v_k\right)\hfill \\ & & \ge \frac{{\delta }^{k-1}}{k}(de{g}_G\left(v_1\right)+de{g}_G\left(v_2\right)+\cdots +de{g}_G\left(v_k\right)),\hfill \end{array}$$

(5)

since $de{g}_G\left(v_1\right)\le de{g}_G\left(v_2\right)\le \cdots \le de{g}_G\left(v_k\right)$. Furthermore, we have

$$\begin{array}{ccc}\hfill {\mathrm{SGut}}_k\left(G\right)& \ge & (k-1)\sum _{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)}de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& \ge & (k-1)\frac{{\delta }^{k-1}}{k}\sum _{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)}[de{g}_G\left(v_1\right)+de{g}_G\left(v_2\right)+\cdots +de{g}_G\left(v_k\right)]\hfill \\ & =& (k-1)\frac{{\delta }^{k-1}}{k}N\hfill \\ & =& 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}.\hfill \end{array}$$

(7)

Next we assume that $\delta =1$. If $de{g}_G\left(v_1\right)=de{g}_G\left(v_2\right)=\cdots =de{g}_G\left(v_k\right)=1$, then $d_G\left(S\right)\ge k$ and $de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)=1$. If there exists some $v_i$ such that $de{g}_G\left(v_i\right)\ge 2$, then $d_G\left(S\right)\ge k-1$ and $de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)\ge 2^{max\{k-p,1\}}=2^q$, where $1\le i\le k$. Therefore, we have

$$\begin{array}{ccc}\hfill {\mathrm{SGut}}_k\left(G\right)& \ge & k\sum _{\stackrel{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right),}{de{g}_G\left(v_1\right)=de{g}_G\left(v_2\right)=\cdots =de{g}_G\left(v_k\right)=1}}de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +(k-1)\sum _{\stackrel{\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right),}{somede{g}_G\left(v_i\right)\ge 2}}de{g}_G\left(v_1\right)de{g}_G\left(v_2\right)\dots de{g}_G\left(v_k\right)\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& \ge & k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right].\hfill \end{array}$$

(9)

Suppose that the right equality holds. Then all the inequalities in the above must be equalities. Suppose that $\delta \ge 2$. From the equality in (6), $d_G\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$ and $\left|S\right|=k$, that is, $G\left[S\right]$ is connected for any $S\subseteq V\left(G\right)$ and $\left|S\right|=k$, and hence G is $(n-k+1)$-connected. From the equality in (7), we have $de{g}_G\left(v_1\right)=de{g}_G\left(v_2\right)=\cdots =de{g}_G\left(v_k\right)$ for any $S=\{v_1,v_2,\dots ,v_k\}\subseteq V\left(G\right)$, and hence G is a regular graph. Thus, G is a regular $(n-k+1)$-connected graph of order n.

Next suppose that $\delta =1$. From the equality in (9), we obtain $de{g}_G\left(v_i\right)=1$ or $de{g}_G\left(v_i\right)=2$ for any vertex $v_i\in V\left(G\right)$. Since G is connected, $G\cong P_n$ and $p=2$. If $k\ge 3$, then $q=k-p\ge 1$. In this case $d_G\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$ and $\left|S\right|=k$. One can easily see that $G\cong P_n$ and $k=n>3$ (otherwise, $d_G\left(S\right)>k-1$ for some $S\subseteq V\left(G\right)$ as $q=k-p$). Otherwise, $k=p=2$ and hence $q=1$. In this case $G\cong P_3$ and $k=2$.

Conversely, one can see easily that the equality holds on lower bound for a regular $(n-k+1)$-connected graph of order n$(\delta \ge 2)$, or $G\cong P_n$ and $k=n>3$ $(\delta =1)$, or $G\cong P_3$ and $k=2$ $(\delta =1)$. □

Example 1.

Let $G\cong K_n$ with $k=n$. Then

$${\mathrm{SGut}}_k\left(G\right)={(n-1)}^{n+1}=2m(n-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\Delta }^{k-1}}{k}.$$

Let $G\cong K_n\setminus s{K}_2$ $(n=2s)$ with $k=3$. Then G is a $n-2$ regular graph of order n. Then

$${\mathrm{SGut}}_k\left(G\right)=2{(n-2)}^3\left(\genfrac{}{}{0pt}{}{n}{3}\right)=2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}.$$

Let $G\cong P_n$ with $k=n>3$. Then

$${\mathrm{SGut}}_k\left(G\right)=2^{n-2}(n-1)=k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]as p=2.$$

Let $G\cong P_n$ with $k=2$. Then

$${\mathrm{SGut}}_k\left(G\right)=6=k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]as p=2.$$

3. Nordhaus–Gaddum-Type Results on ${\mathbf{SGut}}_{\mathit{k}}\mathbf{\left(}\mathit{G}\mathbf{\right)}$

We are now in a position to give the Nordhaus–Gaddum-type results on ${\mathrm{SGut}}_k\left(G\right)$.

Theorem 1.

Let G be a connected graph of order n with m edges, maximum degree Δ, minimum degree δ and a connected $\overline{G}$. Additionally, let k be an integer with $2\le k\le n$. Then

$\left(1\right)$

$${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\le {(n-1)}^2\left(\genfrac{}{}{0pt}{}{n}{k}\right)s_1^{k-1}$$

and

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\le 2m(n^2-n-2m){(n-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\Delta }^{k-1}{(n-\delta -1)}^{k-1}}{k^2},$$

where $s_1=max\{\Delta ,n-\delta -1\}$. Moreover, the upper bounds are sharp.

$\left(2\right)$

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left\{\begin{array}{cc}(n-1)(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)t_1^{k-1}\hfill & \mathrm{if}\delta \ge 2,\Delta \le n-3\hfill \\ 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta \ge 2,\Delta =n-2\hfill \\ k\left(\genfrac{}{}{0pt}{}{n}{k}\right)+[n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^{k-1}}{k}\hfill & \mathrm{if}\delta =1,\Delta \le n-3\hfill \\ 2k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta =1,\Delta =n-2,\hfill \end{array}\right.\hfill \end{array}$$

where $t_1=min\{\delta ,n-\Delta -1\}$.

$\left(3\right)$

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left\{\begin{array}{cc}2m(n^2-n-2m){(k-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\delta }^{k-1}{(n-\Delta -1)}^{k-1}}{k^2}\hfill & \mathrm{if}\delta \ge 2,\Delta \le n-3\hfill \\ 2m(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){\delta }^{k-1}\hfill & \mathrm{if}\delta \ge 2,\Delta =n-2\hfill \\ [n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){(n-\Delta -1)}^{k-1}\hfill & \mathrm{if}\delta =1,\Delta \le n-3\hfill \\ k^2{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\delta =1,\Delta =n-2.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

$\left(1\right)$ From Proposition 1, we have

$${\mathrm{SGut}}_k\left(G\right)\le 2m(n-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\Delta }^{k-1}}{k}$$

and

$${\mathrm{SGut}}_k\left(\overline{G}\right)\le [n(n-1)-2m](n-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\delta -1)}^{k-1}}{k},$$

and hence

$${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\le {(n-1)}^2\left(\genfrac{}{}{0pt}{}{n}{k}\right)s_1^{k-1}$$

and

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\le 2m(n^2-n-2m){(n-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\Delta }^{k-1}{(n-\delta -1)}^{k-1}}{k^2}.$$

$\left(2\right)$ From Proposition 1, if $\delta \ge 2$ and $\Delta \le n-3$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+[n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^{k-1}}{k}\hfill \\ & \ge & (n-1)(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)t_1^{k-1}.\hfill \end{array}$$

If $\delta \left(G\right)\ge 2$ and $\Delta =n-2$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+k\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)+2^{q^{\prime }}(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)\right]\hfill \\ & \ge & 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+k\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)+2(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)\right]\hfill \\ & \ge & 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+k\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)+k\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)\right]\hfill \\ & =& 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+k\left(\genfrac{}{}{0pt}{}{n}{k}\right),\hfill \end{array}$$

where $p^{\prime }$ is the number of pendant vertices in G, and $q^{\prime }=max\{k-p^{\prime },1\}$.

If $\delta =1$ and $\Delta \le n-3$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]+[n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^{k-1}}{k}\hfill \\ & \ge & k\left(\genfrac{}{}{0pt}{}{n}{k}\right)+[n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^{k-1}}{k},\hfill \end{array}$$

where p is the number of pendant vertices in $\overline{G}$, and $q=max\{k-p,1\}$.

If $\delta =1$ and $\Delta =n-2$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]+k\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)+2^{q^{\prime }}(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)\right]\hfill \\ & \ge & k\left(\genfrac{}{}{0pt}{}{n}{k}\right)+k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\ge 2k\left(\genfrac{}{}{0pt}{}{n}{k}\right),\hfill \end{array}$$

where $p,p^{\prime }$ are the number of pendant vertices in $G,\overline{G}$, respectively, and $q=max\{k-p,1\}$, $q^{\prime }=max\{k-p^{\prime },1\}$.

From the above argument, we have

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left\{\begin{array}{cc}(n-1)(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)t_1^{k-1}\hfill & \mathrm{if}\delta \ge 2,\Delta \le n-3\hfill \\ 2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}+k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta \ge 2,\Delta =n-2\hfill \\ k\left(\genfrac{}{}{0pt}{}{n}{k}\right)+[n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^{k-1}}{k}\hfill & \mathrm{if}\delta =1,\Delta \le n-3\hfill \\ 2k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta =1,\Delta =n-2.\hfill \end{array}\right.\hfill \end{array}$$

For $\left(3\right)$, from Proposition 1, if $\delta \ge 2$ and $\Delta \le n-3$, then

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\ge 2m(n^2-n-2m){(k-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\delta }^{k-1}{(n-\Delta -1)}^{k-1}}{k^2}.$$

If $\delta \ge 2$ and $\Delta =n-2$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left[2m(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^{k-1}}{k}\right]\left[k\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)+2^{q^{\prime }}(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)\right]\right]\hfill \\ & \ge & 2m(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){\delta }^{k-1},\hfill \end{array}$$

where $p^{\prime }$ is the number of pendant vertices in $\overline{G}$, and $q^{\prime }=max\{k-p^{\prime },1\}$.

If $\delta =1$ and $\Delta \le n-3$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left[[n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^{k-1}}{k}\right]\left[k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]\right]\hfill \\ & \ge & [n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){(n-\Delta -1)}^{k-1},\hfill \end{array}$$

where p is the number of pendant vertices in G, and $q=max\{k-p,1\}$.

If $\delta \left(G\right)=1$ and $\Delta =n-2$, then

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left[k\left(\genfrac{}{}{0pt}{}{p}{k}\right)+2^q(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p}{k}\right)\right]\right]\left[k\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)+2^{q^{\prime }}(k-1)\left[\left(\genfrac{}{}{0pt}{}{n}{k}\right)-\left(\genfrac{}{}{0pt}{}{p^{\prime }}{k}\right)\right]\right]\hfill \\ & \ge & k^2{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2,\hfill \end{array}$$

where $p,p^{\prime }$ are the number of pendant vertices in G and $\overline{G}$, respectively, and $q=max\{k-p,1\}$, $q^{\prime }=max\{k-p^{\prime },1\}$.

From the above argument, we have

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left\{\begin{array}{cc}2m(n^2-n-2m){(k-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\delta }^{k-1}{(n-\Delta -1)}^{k-1}}{k^2}\hfill & \mathrm{if}\delta \left(G\right)\ge 2,\Delta \le n-3\hfill \\ 2m(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){\delta }^{k-1}\hfill & \mathrm{if}\delta \left(G\right)\ge 2,\Delta =n-2\hfill \\ [n(n-1)-2m](k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){(n-\Delta -1)}^{k-1}\hfill & \mathrm{if}\delta \left(G\right)=1,\Delta \le n-3\hfill \\ k^2{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\delta \left(G\right)=1,\Delta =n-2.\hfill \end{array}\right.\hfill \end{array}$$

To show the sharpness of the upper bound and the lower bound for $\delta \left(G\right)\ge 2,\Delta \le n-3$, we let G and $\overline{G}$ be two $\frac{n-1}{2}$-regular graphs of order n, where n is odd. If $k=n$, then ${\mathrm{SGut}}_k\left(G\right)=(n-1){\left(\frac{n-1}{2}\right)}^n$, ${\mathrm{SGut}}_k\left(\overline{G}\right)=(n-1){\left(\frac{n-1}{2}\right)}^n$, $s_1=max\{\Delta ,n-\delta -1\}=\frac{n-1}{2}$, $\Delta (n-\delta -1)={\left(\frac{n-1}{2}\right)}^2$, $t_1=min\{\delta ,n-\Delta -1\}=\frac{n-1}{2}$ and $\delta (n-\Delta -1)={\left(\frac{n-1}{2}\right)}^2$. Furthermore, we have ${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)=2(n-1){\left(\frac{n-1}{2}\right)}^n={(n-1)}^2\left(\genfrac{}{}{0pt}{}{n}{k}\right)s_1^{k-1}$, ${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)={(n-1)}^2{\left(\frac{n-1}{2}\right)}^{2n}=2m(n^2-n-2m){(n-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\Delta }^{k-1}{(n-\delta -1)}^{k-1}}{k^2}$, ${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)=2(n-1){\left(\frac{n-1}{2}\right)}^n=(n-1)(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)t_1^{k-1}$ and ${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)={(n-1)}^2{\left(\frac{n-1}{2}\right)}^{2n}=2m(n^2-n-2m){(k-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\delta }^{k-1}{(n-\Delta -1)}^{k-1}}{k^2}$. □

The following corollary is immediate from the above theorem.

Corollary 1.

Let G be a connected graph of order $n\ge 4$ with maximum degree Δ and minimum degree δ. Then

$\left(1\right)$

$$\begin{array}{ccc}& & {(n-1)}^2\left(\genfrac{}{}{0pt}{}{n}{k}\right)s_1^{k-1}\ge {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \ge & \left\{\begin{array}{cc}(n-1)(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)t_1^{k-1}\hfill & \mathrm{if}\delta \ge 2,\Delta \le n-3\hfill \\ n(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{\delta }^k}{k}+k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta \ge 2,\Delta =n-2\hfill \\ k\left(\genfrac{}{}{0pt}{}{n}{k}\right)+n(k-1)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)\frac{{(n-\Delta -1)}^k}{k}\hfill & \mathrm{if}\delta =1,\Delta \le n-3\hfill \\ 2k\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\delta =1,\Delta =n-2,\hfill \end{array}\right.\hfill \end{array}$$

where $s_1=min\{\Delta ,n-\delta -1\}$, $t_1=min\{\delta ,n-\Delta -1\}$;

$\left(2\right)$

$$\begin{array}{c}\hfill n^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\Delta }^{k-1}{(n-\delta -1)}^{k-1}{(n-1)}^4}{4k^2}\ge {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\\ \hfill \ge \left\{\begin{array}{cc}{n}^2{(k-1)}^2{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}^2\frac{{\delta }^k{(n-\Delta -1)}^k}{k^2}\hfill & \mathrm{if}\delta \ge 2,\Delta \le n-3\hfill \\ n(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){\delta }^k\hfill & \mathrm{if}\delta \ge 2,\Delta =n-2\hfill \\ n(k-1)\left(\genfrac{}{}{0pt}{}{n}{k}\right)\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right){(n-\Delta -1)}^k\hfill & \mathrm{if}\delta =1,\Delta \le n-3\hfill \\ k^2{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\delta =1,\Delta =n-2.\hfill \end{array}\right.\end{array}$$

The following is the famous inequality by Pólya and Szegö:

Lemma 3.

(Pólya–Szegö inequality) [25] Let $(a_1,a_2,\dots ,a_r)$ and $(b_1,b_2,\dots ,b_r)$ be two positive r-tuples such that there exist positive numbers $M_1$, $m_1$, $M_2$, $m_2$ satisfying:

$$0<m_1\le a_i\le M_1,0<m_2\le b_i\le M_2,1\le i\le r.$$

Then

$$\frac{{\displaystyle \sum _{i=1}^r}{a}_i^2{\displaystyle \sum _{i=1}^r}{b}_i^2}{{\left({\displaystyle \sum _{i=1}^r}{a}_i{b}_i\right)}^2}\le \frac{1}{4}{\left(\sqrt{\frac{M_1{M}_2}{m_1{m}_2}}+\sqrt{\frac{m_1{m}_2}{M_1{M}_2}}\right)}^2.$$

(10)

We now give more lower and upper bounds for ${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)$ in terms of n, $\Delta$ and $\delta$.

Theorem 2.

Let G be a connected graph of order n with maximum degree Δ, minimum degree δ and a connected $\overline{G}$. Additionally, let k be an integer with $2\le k\le n$. Then

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\ge \left\{\begin{array}{cc}{(k-1)}^2{\delta }^k{(n-\delta -1)}^k{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\Delta +\delta \le n-1,\hfill \\ {(k-1)}^2{\Delta }^k{(n-\Delta -1)}^k{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\Delta +\delta \ge n-1\hfill \end{array}\right.$$

(11)

with equality holding if and only if G is a regular graph with $d_G\left(S\right)=d_{\overline{G}}\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$, $\left|S\right|=k$, and

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\le \frac{{(n-1)}^{2k+2}}{2^{2k+2}}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\left[{\left(\frac{\Delta (n-\delta -1)}{\delta (n-\Delta -1)}\right)}^k+{\left(\frac{\delta (n-\Delta -1)}{\Delta (n-\delta -1)}\right)}^k+2\right],$$

Moreover, the equality holds if and only if G is a $\left(\frac{n-1}{2}\right)$-regular graph with $k=n$, n is odd.

Proof. 

Lower bound: By Cauchy–Schwarz inequality with (1), we have

$$\begin{array}{ccc}\hfill {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)& \ge & {(k-1)}^2\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_G\left(v\right)\right)\sum _{\stackrel{S\subseteq V\left(\overline{G}\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& \ge & {(k-1)}^2{\left(\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}{\left(\prod _{v\in S}de{g}_G\left(v\right)\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)}^{1/2}\right)}^2\hfill \\ & \ge & {(k-1)}^2{\left(\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}{\left(\prod _{v\in S}de{g}_G\left(v\right)(n-1-de{g}_G\left(v\right))\right)}^{1/2}\right)}^2.\hfill \end{array}$$

(13)

Since $\delta \le de{g}_G\left(v\right)\le \Delta$, one can easily see that

$$\begin{array}{ccc}\hfill de{g}_G\left(v\right)(n-1-de{g}_G\left(v\right))& \ge & \left\{\begin{array}{cc}\delta (n-\delta -1)\hfill & \mathrm{if}\Delta +\delta \le n-1,\hfill \\ \Delta (n-\Delta -1)\hfill & \mathrm{if}\Delta +\delta \ge n-1.\hfill \end{array}\right.\hfill \end{array}$$

(14)

From the above results, we have

$$\begin{array}{ccc}\hfill {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)& \ge & \left\{\begin{array}{cc}{(k-1)}^2{\delta }^k{(n-\delta -1)}^k{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\Delta +\delta \le n-1,\hfill \\ {(k-1)}^2{\Delta }^k{(n-\Delta -1)}^k{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\hfill & \mathrm{if}\Delta +\delta \ge n-1.\hfill \end{array}\right.\hfill \end{array}$$

The equality holds in (12) if and only if $d_G\left(S\right)=d_{\overline{G}}\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$ with $\left|S\right|=k$. By the Cauchy–Schwarz inequality, the equality holds in (13) if and only if

$$\frac{{\prod }_{v\in S_1}de{g}_G\left(v\right)}{{\prod }_{v\in S_1}de{g}_{\overline{G}}\left(v\right)}=\frac{{\prod }_{v\in S_2}de{g}_G\left(v\right)}{{\prod }_{v\in S_2}de{g}_{\overline{G}}\left(v\right)}\mathrm{for} \mathrm{any}{S}_1,S_2\in V\left(G\right)\mathrm{with} |S_1|=|S_2|=k,$$

that is, if and only if $de{g}_G\left(u\right)=de{g}_G\left(v\right)$ for any $u,v\in V\left(G\right)$, that is, if and only if G is a regular graph. Hence the equality holds in (11) if and only if G is a regular graph with $d_G\left(S\right)=d_{\overline{G}}\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$, $\left|S\right|=k$.

Upper bound: Let $\overline{\Delta }$ and $\overline{\delta }$ be the maximum degree and the minimum degree of graph $\overline{G}$, respectively. Then $\overline{\Delta }=n-\delta -1$ and $\overline{\delta }=n-\Delta -1$. By (1) and (10), we have

$$\begin{array}{ccc}& & {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\hfill \\ & \le & {(n-1)}^2\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_G\left(v\right)\right)\sum _{\stackrel{S\subseteq V\left(\overline{G}\right)}{\left|S\right|=k}}\left(\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)\hfill \\ & \le & {(n-1)}^2{\left(\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}{\left(\prod _{v\in S}de{g}_G\left(v\right)\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)}^{1/2}\right)}^2\frac{1}{4}{\left({\left(\frac{\Delta \overline{\Delta }}{\delta \overline{\delta }}\right)}^{k/2}+{\left(\frac{\delta \overline{\delta }}{\Delta \overline{\Delta }}\right)}^{k/2}\right)}^2\hfill \\ & \le & \frac{{(n-1)}^2}{4}{\left(\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}{\left(\prod _{v\in S}de{g}_G\left(v\right)(n-1-de{g}_G\left(v\right))\right)}^{1/2}\right)}^2{\left({\left(\frac{\Delta \overline{\Delta }}{\delta \overline{\delta }}\right)}^{k/2}+{\left(\frac{\delta \overline{\delta }}{\Delta \overline{\Delta }}\right)}^{k/2}\right)}^2.\hfill \end{array}$$

One can easily see that

$$de{g}_G\left(v\right)(n-1-de{g}_G\left(v\right))\le \frac{{(n-1)}^2}{4}\mathrm{for} \mathrm{any} v\in V\left(G\right).$$

Using this result in the above with $\overline{\Delta }=n-\delta -1$ and $\overline{\delta }=n-\Delta -1$, we get

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)\le \frac{{(n-1)}^{2k+2}}{2^{2k+2}}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\left[{\left(\frac{\Delta (n-\delta -1)}{\delta (n-\Delta -1)}\right)}^k+{\left(\frac{\delta (n-\Delta -1)}{\Delta (n-\delta -1)}\right)}^k+2\right].$$

Moreover, the above equality holds if and only if G is a $\left(\frac{n-1}{2}\right)$-regular graph with $k=n$, n is odd (very similar proof of the Proposition 1). □

Example 2.

Let $G\cong C_n$ with $k=n$. Then $\delta =2$ and hence

$${\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)={(n-1)}^2{(n-3)}^n{2}^n={(k-1)}^2{\delta }^k{(n-\delta -1)}^k{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2.$$

Let G be a $\left(\frac{n-1}{2}\right)$-regular graph of order n with $k=n$ and odd n. Then $\Delta =\delta =\frac{n-1}{2}$ and hence

$$\begin{array}{cc}\hfill {\mathrm{SGut}}_k\left(G\right)·{\mathrm{SGut}}_k\left(\overline{G}\right)& =\frac{{(n-1)}^{2n+2}}{2^{2n}}\hfill \\ \hfill & =\frac{{(n-1)}^{2k+2}}{2^{2k+2}}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2\left[{\left(\frac{\Delta (n-\delta -1)}{\delta (n-\Delta -1)}\right)}^k+{\left(\frac{\delta (n-\Delta -1)}{\Delta (n-\delta -1)}\right)}^k+2\right].\hfill \end{array}$$

We now give more lower and upper bounds of ${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)$ in terms of n, $\Delta$ and $\delta$.

Theorem 3.

Let G be a connected graph of order n with maximum degree Δ, minimum degree δ and a connected $\overline{G}$. Additionally, let k be an integer with $2\le k\le n$. Then

$${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\ge \left\{\begin{array}{cc}2(k-1){\delta }^{k/2}{(n-\delta -1)}^{k/2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\Delta +\delta \le n-1,\hfill \\ 2(k-1){\Delta }^{k/2}{(n-\Delta -1)}^{k/2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\Delta +\delta \ge n-1\hfill \end{array}\right.$$

(15)

with equality holding if and only if G is a $\left(\frac{n-1}{2}\right)$-regular graph with odd n and $d_G\left(S\right)=d_{\overline{G}}\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$, $\left|S\right|=k$, and

$${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)\le (n-1)\left[{\Delta }^k+{(n-\delta -1)}^k\right]\left(\genfrac{}{}{0pt}{}{n}{k}\right)$$

(16)

with equality holding if and only if G is a regular graph with $k=n$.

Proof. 

For any two real numbers $a,b$, we have ${(a-b)}^2\ge 0$, that is, $a^2+b^2\ge 2ab$ with equality holding if and only if $a=b$. Therefore we have

$$\begin{array}{ccc}\hfill \prod _{v\in S}de{g}_G\left(v\right)+\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)& \ge & 2{\left(\prod _{v\in S}de{g}_G\left(v\right)\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)}^{1/2}\hfill \\ & =& 2{\left(\prod _{v\in S}de{g}_G\left(v\right)de{g}_{\overline{G}}\left(v\right)\right)}^{1/2}\hfill \\ & =& 2{\left(\prod _{v\in S}de{g}_G\left(v\right)(n-de{g}_G\left(v\right)-1)\right)}^{1/2}.\hfill \end{array}$$

From the above result with (14), we get

$$\begin{array}{ccc}\hfill \prod _{v\in S}de{g}_G\left(v\right)+\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)& \ge & \left\{\begin{array}{cc}2{\delta }^{k/2}{(n-\delta -1)}^{k/2}\hfill & \mathrm{if}\Delta +\delta \le n-1,\hfill \\ 2{\Delta }^{k/2}{(n-\Delta -1)}^{k/2}\hfill & \mathrm{if}\Delta +\delta \ge n-1.\hfill \end{array}\right.\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)& =& \sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[\left(\prod _{v\in S}de{g}_G\left(v\right)\right)d_G\left(S\right)+\left(\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)d_{\overline{G}}\left(S\right)\right]\hfill \\ & \ge & (k-1)\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[\prod _{v\in S}de{g}_G\left(v\right)+\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right]\hfill \\ & \ge & \left\{\begin{array}{cc}2(k-1){\delta }^{k/2}{(n-\delta -1)}^{k/2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\Delta +\delta \le n-1,\hfill \\ 2(k-1){\Delta }^{k/2}{(n-\Delta -1)}^{k/2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)\hfill & \mathrm{if}\Delta +\delta \ge n-1.\hfill \end{array}\right.\hfill \end{array}$$

From the above, one can easily see that the equality holds in (15) if and only if G is a $\left(\frac{n-1}{2}\right)$-regular graph with odd n and $d_G\left(S\right)=d_{\overline{G}}\left(S\right)=k-1$ for any $S\subseteq V\left(G\right)$, $\left|S\right|=k$.

Upper bound: By arithmetic-geometric mean inequality, we have

$$\begin{array}{ccc}\hfill {\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)& =& \sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[\left(\prod _{v\in S}de{g}_G\left(v\right)\right)d_G\left(S\right)+\left(\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right)d_{\overline{G}}\left(S\right)\right]\hfill \\ & \le & (n-1)\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[\prod _{v\in S}de{g}_G\left(v\right)+\prod _{v\in S}de{g}_{\overline{G}}\left(v\right)\right]\hfill \\ & \le & (n-1)\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[{\left(\frac{\sum _{v\in S}de{g}_G\left(v\right)}{k}\right)}^k+{\left(\frac{\sum _{v\in S}de{g}_{\overline{G}}\left(v\right)}{k}\right)}^k\right]\hfill \\ & =& \frac{(n-1)}{k^k}\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[{\left(\sum _{v\in S}de{g}_G\left(v\right)\right)}^k+{\left(\sum _{v\in S}(n-de{g}_G\left(v\right)-1)\right)}^k\right]\hfill \\ & =& \frac{(n-1)}{k^k}\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[{\left(\sum _{v\in S}de{g}_G\left(v\right)\right)}^k+{\left(k(n-1)-\sum _{v\in S}de{g}_G\left(v\right)\right)}^k\right]\hfill \\ & \le & \frac{(n-1)}{k^k}\sum _{\stackrel{S\subseteq V\left(G\right)}{\left|S\right|=k}}\left[{(k\Delta )}^k+{\left(k(n-1)-k\delta \right)}^k\right]\hfill \\ & =& (n-1)\left[{\Delta }^k+{(n-\delta -1)}^k\right]\left(\genfrac{}{}{0pt}{}{n}{k}\right).\hfill \end{array}$$

From the above, one can easily see that the equality holds in (16) if and only if G is a regular graph with $k=n$ (very similar proof of the Proposition 1). □

Example 3.

Let G be a $\left(\frac{n-1}{2}\right)$-regular graph with odd n and $k=n$. Then $\delta =\frac{n-1}{2}$ and hence

$${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)=\frac{{(n-1)}^{n+1}}{2^{n-1}}=2(k-1){\delta }^{k/2}{(n-\delta -1)}^{k/2}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$$

Let $G\cong C_n$ with $k=n$. Then $\Delta =\delta =2$, $\overline{\Delta }=\overline{\delta }=2$ and hence

$${\mathrm{SGut}}_k\left(G\right)+{\mathrm{SGut}}_k\left(\overline{G}\right)=(n-1)\left[2^n+{(n-3)}^n\right]=(n-1)\left[{\Delta }^k+{(n-\delta -1)}^k\right]\left(\genfrac{}{}{0pt}{}{n}{k}\right).$$