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Open AccessArticle

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

1
College of Science, China Jiliang University, Hangzhou 310018, Zhejiang, China
2
Department of Mathematics, Qinghai Normal University, Xining 810008, Qinghai, China
3
Center for Mathematics and Interdisciplinary Sciences of Qinghai Province, Xining 810008, Qinghai, China
4
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
5
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK
*
Authors to whom correspondence should be addressed.
Symmetry 2020, 12(10), 1711; https://doi.org/10.3390/sym12101711
Received: 18 September 2020 / Revised: 11 October 2020 / Accepted: 12 October 2020 / Published: 16 October 2020
(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices)

Abstract

Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk(G) of G is defined by SGutk(G)=SV(G),|S|=kvSdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(G¯) for a connected graph G of order n, m edges, maximum degree Δ and minimum degree δ.
Keywords: distance; Steiner distance; Gutman index; Steiner Gutman k-index distance; Steiner distance; Gutman index; Steiner Gutman k-index

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 ( G ) and E ( G ) represent its sets of vertices and edges, respectively. Let | E ( G ) | = m be the size of G. The complement of G is conventionally denoted by G ¯ . For a vertex v V ( G ) , d e g G ( v ) is the degree of v. The maximum and minimum degrees are, respectively, denoted by Δ and δ . Like degrees, distance is a fundamental concept of graph theory [2]. For two vertices u , v V ( G ) 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 V ( G ) 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 , E ) of G, which is a subtree satisfying S V . If G is connected with order no less than 2 and S V is nonempty, the Steiner distance d ( S ) 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 G with S V ( H ) and | E ( H ) | = d ( S ) , H is a tree. When T is subtree of G, we have d ( S ) = min { | E ( T ) | , S V ( T ) } . For S = { u , v } , d ( S ) = d ( u , v ) reduces to the classical distance between the two vertices u and v. Another basic observation is that if | S | = k , d ( S ) 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 ( G ) of G is defined by
SW k ( G ) = | S | = k S V ( G ) d ( S ) .
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 k n 1 , and the two trivial cases give SW 1 ( G ) = 0 and SW n ( G ) = 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 ( G ) of G is given as
SDD k ( G ) = | S | = k S V ( G ) v S d e g G ( v ) d G ( S ) .
The Gutman index of a connected graph G is defined as
Gut ( G ) = u , v V ( G ) d e g G ( u ) d e g G ( v ) 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 ( G ) of G is defined by
SGut k ( G ) = | S | = k S V ( G ) v S d e g G ( v ) d G ( S ) .
Note that this index is a natural generalization of the classical Gutman index—in particular, for k = 2 , SGut k ( G ) = G u t ( G ) . 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 ( G ) and a positive integer n, the well-known Nordhaus–Gaddum problem is to determine sharp bounds for: ( 1 ) f ( G ) + f ( G ¯ ) and ( 2 ) f ( G ) · f ( G ¯ ) over the class of connected graph G, with order n, m edges, maximum degree Δ and minimum degree δ 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 SGut k of graph G. In Section 3, we obtain sharp upper and lower bounds of SGut k ( G ) + SGut k ( G ¯ ) and SGut k ( G ) · SGut k ( G ¯ ) for a connected graph G in terms of n, m, maximum degree Δ and minimum degree δ .

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 k n . Then
(1) 
S G u t k ( K n ) = n k ( n 1 ) n ( k 1 ) ;
(2) 
S G u t k ( S n ) = ( k n 2 k + 1 ) n 1 k 1 ;
(3) 
S G u t k ( P n ) = 2 k ( k 1 ) n k + 1 .
For connected graph G of order n with m edges, the authors in [14] derived the following upper and lower bounds on SGut k ( G ) .
Lemma 2.
([14]). Let G be a connected graph of order n with m edges, and let k be an integer with 2 k n . Then
( n 1 ) 2 m k k n 1 k 1 k SGut k ( G ) 2 m ( k 1 ) n 1 k 1 if δ 2 ( k 1 ) n k if δ = 1 .
We now give lower and upper bounds for SGut k ( G ) in terms of n, m, maximum degree Δ and minimum degree δ :
Proposition 1.
Let G be a connected graph of order n 3 with m edges and maximum degree Δ, minimum degree δ. Additionally, let k be an integer with 2 k n . Then
2 m ( n 1 ) n 1 k 1 Δ k 1 k SGut k ( G ) 2 m ( k 1 ) n 1 k 1 δ k 1 k if δ 2 k p k + 2 q ( k 1 ) n k p k if δ = 1 ,
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 ( δ 2 ) , or G P n and k = n > 3 ( δ = 1 ) , or G P 3 and k = 2 ( δ = 1 ) .
Proof. 
Upper bound: For any S V ( G ) and | S | = k , we have k 1 d G ( S ) n 1 , and hence
( k 1 ) | S | = k S V ( G ) v S d e g G ( v ) SGut k ( G ) ( n 1 ) | S | = k S V ( G ) v S d e g G ( v ) .
Let
M = | S | = k S V ( G ) v S d e g G ( v ) = { v 1 , v 2 , , v k } V ( G ) d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) .
and
N = { v 1 , v 2 , , v k } V ( G ) [ d e g G ( v 1 ) + d e g G ( v 2 ) + + d e g G ( v k ) ] .
We first prove the upper bound. Without loss of generality, we can assume that d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) . Since
d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) Δ k 1 d e g G ( v 1 )
Δ k 1 k ( d e g G ( v 1 ) + d e g G ( v 2 ) + + d e g G ( v k ) ) ,
it follows that
M = { v 1 , v 2 , , v k } V ( G ) d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) Δ k 1 k { v 1 , v 2 , , v k } V ( G ) [ d e g G ( v 1 ) + d e g G ( v 2 ) + + d e g G ( v k ) ] Δ k 1 k N .
For each v V ( G ) , there are n 1 k 1 k-subsets in G such that each of them contains v. The contribution of vertex v is exactly n 1 k 1 d e g G ( v ) . From the arbitrariness of v, we have
N = n 1 k 1 v V ( G ) d e g G ( v ) = 2 m n 1 k 1 ,
and hence
SGut k ( G ) ( n 1 ) M ( n 1 ) Δ k 1 k N = 2 m ( n 1 ) n 1 k 1 Δ k 1 k .
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 ( S ) = n 1 for any S V ( G ) , | S | = k . Since G is connected, then there exists an S V ( G ) such that | d G ( S ) | = k 1 . If k n 1 , then one can easily see that the upper bound is strict as | d G ( S ) | = k 1 n 2 for some S. Otherwise, k = n . Since G is connected, we have | d G ( S ) | = n 1 for any S V ( G ) . 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 d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) . First we assume that δ 2 . Then
d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) δ k 1 d e g G ( v k ) δ k 1 k ( d e g G ( v 1 ) + d e g G ( v 2 ) + + d e g G ( v k ) ) ,
since d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) . Furthermore, we have
SGut k ( G ) ( k 1 ) { v 1 , v 2 , , v k } V ( G ) d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k )
( k 1 ) δ k 1 k { v 1 , v 2 , , v k } V ( G ) [ d e g G ( v 1 ) + d e g G ( v 2 ) + + d e g G ( v k ) ] = ( k 1 ) δ k 1 k N = 2 m ( k 1 ) n 1 k 1 δ k 1 k .
Next we assume that δ = 1 . If d e g G ( v 1 ) = d e g G ( v 2 ) = = d e g G ( v k ) = 1 , then d G ( S ) k and d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) = 1 . If there exists some v i such that d e g G ( v i ) 2 , then d G ( S ) k 1 and d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k ) 2 max { k p , 1 } = 2 q , where 1 i k . Therefore, we have
SGut k ( G ) k d e g G ( v 1 ) = d e g G ( v 2 ) = = d e g G ( v k ) = 1 { v 1 , v 2 , , v k } V ( G ) , d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k )
+ ( k 1 ) s o m e d e g G ( v i ) 2 { v 1 , v 2 , , v k } V ( G ) , d e g G ( v 1 ) d e g G ( v 2 ) d e g G ( v k )
k p k + 2 q ( k 1 ) n k p k .
Suppose that the right equality holds. Then all the inequalities in the above must be equalities. Suppose that δ 2 . From the equality in (6), d G ( S ) = k 1 for any S V ( G ) and | S | = k , that is, G [ S ] is connected for any S V ( G ) and | S | = k , and hence G is ( n k + 1 ) -connected. From the equality in (7), we have d e g G ( v 1 ) = d e g G ( v 2 ) = = d e g G ( v k ) for any S = { v 1 , v 2 , , v k } V ( G ) , and hence G is a regular graph. Thus, G is a regular ( n k + 1 ) -connected graph of order n.
Next suppose that δ = 1 . From the equality in (9), we obtain d e g G ( v i ) = 1 or d e g G ( v i ) = 2 for any vertex v i V ( G ) . Since G is connected, G P n and p = 2 . If k 3 , then q = k p 1 . In this case d G ( S ) = k 1 for any S V ( G ) and | S | = k . One can easily see that G P n and k = n > 3 (otherwise, d G ( S ) > k 1 for some S V ( G ) as q = k p ). Otherwise, k = p = 2 and hence q = 1 . In this case G 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 ( δ 2 ) , or G P n and k = n > 3 ( δ = 1 ) , or G P 3 and k = 2 ( δ = 1 ) . □
Example 1.
Let G K n with k = n . Then
SGut k ( G ) = ( n 1 ) n + 1 = 2 m ( n 1 ) n 1 k 1 Δ k 1 k .
Let G K n s K 2 ( n = 2 s ) with k = 3 . Then G is a n 2 regular graph of order n. Then
SGut k ( G ) = 2 ( n 2 ) 3 n 3 = 2 m ( k 1 ) n 1 k 1 δ k 1 k .
Let G P n with k = n > 3 . Then
SGut k ( G ) = 2 n 2 ( n 1 ) = k p k + 2 q ( k 1 ) n k p k a s   p = 2 .
Let G P n with k = 2 . Then
SGut k ( G ) = 6 = k p k + 2 q ( k 1 ) n k p k a s   p = 2 .

3. Nordhaus–Gaddum-Type Results on SGut k ( G )

We are now in a position to give the Nordhaus–Gaddum-type results on SGut k ( G ) .
Theorem 1.
Let G be a connected graph of order n with m edges, maximum degree Δ, minimum degree δ and a connected G ¯ . Additionally, let k be an integer with 2 k n . Then
( 1 )
SGut k ( G ) + SGut k ( G ¯ ) ( n 1 ) 2 n k s 1 k 1
and
SGut k ( G ) · SGut k ( G ¯ ) 2 m ( n 2 n 2 m ) ( n 1 ) 2 n 1 k 1 2 Δ k 1 ( n δ 1 ) k 1 k 2 ,
where s 1 = max { Δ , n δ 1 } . Moreover, the upper bounds are sharp.
( 2 )
SGut k ( G ) + SGut k ( G ¯ ) ( n 1 ) ( k 1 ) n k t 1 k 1 if δ 2 , Δ n 3 2 m ( k 1 ) n 1 k 1 δ k 1 k + k n k if δ 2 , Δ = n 2 k n k + [ n ( n 1 ) 2 m ] ( k 1 ) n 1 k 1 ( n Δ 1 ) k 1 k if δ = 1 , Δ n 3 2 k n k if δ = 1 , Δ = n 2 ,
where t 1 = min { δ , n Δ 1 } .
( 3 )
SGut k ( G ) · SGut k ( G ¯ ) 2 m ( n 2 n 2 m ) ( k 1 ) 2 n 1 k 1 2 δ k 1 ( n Δ 1 ) k 1 k 2 if δ 2 , Δ n 3 2 m ( k 1 ) n k n 1 k 1 δ k 1 if δ 2 , Δ = n 2 [ n ( n 1 ) 2 m ] ( k 1 ) n k n 1 k 1 ( n Δ 1 ) k 1 if δ = 1 , Δ n 3 k 2 n k 2 if δ = 1 , Δ = n 2 .
Proof. 
( 1 ) From Proposition 1, we have
SGut k ( G ) 2 m ( n 1 ) n 1 k 1 Δ k 1 k
and
SGut k ( G ¯ ) [ n ( n 1 ) 2 m ] ( n 1 ) n 1 k 1 ( n δ 1 ) k 1 k ,
and hence
SGut k ( G ) + SGut k ( G ¯ ) ( n 1 ) 2 n k s 1 k 1
and
SGut k ( G ) · SGut k ( G ¯ ) 2 m ( n 2 n 2 m ) ( n 1 ) 2 n 1 k 1 2 Δ k 1 ( n δ 1 ) k 1 k 2 .
( 2 ) From Proposition 1, if δ 2 and Δ n 3 , then
SGut k ( G ) + SGut k ( G ¯ ) 2 m ( k 1 ) n 1 k 1 δ k 1 k + [ n ( n 1 ) 2 m ] ( k 1 ) n 1 k 1 ( n Δ 1 ) k 1 k ( n 1 ) ( k 1 ) n k t 1 k 1 .
If δ ( G ) 2 and Δ = n 2 , then
SGut k ( G ) + SGut k ( G ¯ ) 2 m ( k 1 ) n 1 k 1 δ k 1 k + k p k + 2 q ( k 1 ) n k p k 2 m ( k 1 ) n 1 k 1 δ k 1 k + k p k + 2 ( k 1 ) n k p k 2 m ( k 1 ) n 1 k 1 δ k 1 k + k p k + k n k p k = 2 m ( k 1 ) n 1 k 1 δ k 1 k + k n k ,
where p is the number of pendant vertices in G, and q = max { k p , 1 } .
If δ = 1 and Δ n 3 , then
SGut k ( G ) + SGut k ( G ¯ ) k p k + 2 q ( k 1 ) n k p k + [ n ( n 1 ) 2 m ] ( k 1 ) n 1 k 1 ( n Δ 1 ) k 1 k k n k + [ n ( n 1 ) 2 m ] ( k 1 ) n 1 k 1 ( n Δ 1 ) k 1 k ,
where p is the number of pendant vertices in G ¯ , and q = max { k p , 1 } .
If δ = 1 and Δ = n 2 , then
SGut k ( G ) + SGut k ( G ¯ ) k p k + 2 q ( k 1 ) n k p k + k p k + 2 q ( k 1 ) n k p k k n k + k n k 2 k n k ,
where p , p are the number of pendant vertices in G , G ¯ , respectively, and q = max { k p , 1 } , q = max { k p , 1 } .
From the above argument, we have
SGut k ( G ) + SGut k ( G ¯ ) ( n 1 ) ( k 1 ) n k t 1 k 1 if δ 2 , Δ n 3 2 m ( k 1 ) n 1 k 1 δ k 1 k + k n k if δ 2 , Δ = n 2 k n k + [ n ( n 1 ) 2 m ] ( k 1 ) n 1 k 1 ( n Δ 1 ) k 1 k if δ = 1 , Δ n 3 2 k n k if δ = 1 , Δ = n 2 .
For ( 3 ) , from Proposition 1, if δ 2 and Δ n 3 , then
SGut k ( G ) · SGut k ( G ¯ ) 2 m ( n 2 n 2 m ) ( k 1 ) 2 n 1 k 1 2 δ k 1 ( n Δ 1 ) k 1 k 2 .
If δ 2 and Δ = n 2 , then
SGut k ( G ) · SGut k ( G ¯ ) 2 m ( k 1 ) n 1 k 1 δ k 1 k k p k + 2 q ( k 1 ) n k p k 2 m ( k 1 ) n k n 1 k 1 δ k 1 ,
where p is the number of pendant vertices in G ¯ , and q = max { k p , 1 } .
If δ = 1 and Δ n 3 , then
SGut k ( G ) · SGut k ( G ¯ ) [ n ( n 1 ) 2 m ] ( k 1 ) n 1 k 1 ( n Δ 1 ) k 1 k k p k + 2 q ( k 1 ) n k p k [ n ( n 1 ) 2 m ] ( k 1 ) n k n 1 k 1 ( n Δ 1 ) k 1 ,
where p is the number of pendant vertices in G, and q = max { k p , 1 } .
If δ ( G ) = 1 and Δ = n 2 , then
SGut k ( G ) · SGut k ( G ¯ ) k p k + 2 q ( k 1 ) n k p k k p k + 2 q ( k 1 ) n k p k k 2 n k 2 ,
where p , p are the number of pendant vertices in G and G ¯ , respectively, and q = max { k p , 1 } , q = max { k p , 1 } .
From the above argument, we have
SGut k ( G ) · SGut k ( G ¯ ) 2 m ( n 2 n 2 m ) ( k 1 ) 2 n 1 k 1 2 δ k 1 ( n Δ 1 ) k 1 k 2 if δ ( G ) 2 , Δ n 3 2 m ( k 1 ) n k n 1 k 1 δ k 1 if δ ( G ) 2 , Δ = n 2 [ n ( n 1 ) 2 m ] ( k 1 ) n k n 1 k 1 ( n Δ 1 ) k 1 if δ ( G ) = 1 , Δ n 3 k 2 n k 2 if δ ( G ) = 1 , Δ = n 2 .
To show the sharpness of the upper bound and the lower bound for δ ( G ) 2 , Δ n 3 , we let G and G ¯ be two n 1 2 -regular graphs of order n, where n is odd. If k = n , then SGut k ( G ) = ( n 1 ) ( n 1 2 ) n , SGut k ( G ¯ ) = ( n 1 ) ( n 1 2 ) n , s 1 = max { Δ , n δ 1 } = n 1 2 , Δ ( n δ 1 ) = ( n 1 2 ) 2 , t 1 = min { δ , n Δ 1 } = n 1 2 and δ ( n Δ 1 ) = ( n 1 2 ) 2 . Furthermore, we have SGut k ( G ) + SGut k ( G ¯ ) = 2 ( n 1 ) ( n 1 2 ) n = ( n 1 ) 2 n k s 1 k 1 , SGut k ( G ) · SGut k ( G ¯ ) = ( n 1 ) 2 ( n 1 2 ) 2 n = 2 m ( n 2 n 2 m ) ( n 1 ) 2 n 1 k 1 2 Δ k 1 ( n δ 1 ) k 1 k 2 , SGut k ( G ) + SGut k ( G ¯ ) = 2 ( n 1 ) ( n 1 2 ) n = ( n 1 ) ( k 1 ) n k t 1 k 1 and SGut k ( G ) · SGut k ( G ¯ ) = ( n 1 ) 2 ( n 1 2 ) 2 n = 2 m ( n 2 n 2 m ) ( k 1 ) 2 n 1 k 1 2 δ k 1 ( n Δ 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 4 with maximum degree Δ and minimum degree δ. Then
( 1 )
( n 1 ) 2 n k s 1 k 1 SGut k ( G ) + SGut k ( G ¯ ) ( n 1 ) ( k 1 ) n k t 1 k 1 if δ 2 , Δ n 3 n ( k 1 ) n 1 k 1 δ k k + k n k if δ 2 , Δ = n 2 k n k + n ( k 1 ) n 1 k 1 ( n Δ 1 ) k k if δ = 1 , Δ n 3 2 k n k if δ = 1 , Δ = n 2 ,
where s 1 = min { Δ , n δ 1 } , t 1 = min { δ , n Δ 1 } ;
( 2 )
n 2 n 1 k 1 2 Δ k 1 ( n δ 1 ) k 1 ( n 1 ) 4 4 k 2 SGut k ( G ) · SGut k ( G ¯ ) n 2 ( k 1 ) 2 n 1 k 1 2 δ k ( n Δ 1 ) k k 2 if δ 2 , Δ n 3 n ( k 1 ) n k n 1 k 1 δ k if δ 2 , Δ = n 2 n ( k 1 ) n k n 1 k 1 ( n Δ 1 ) k if δ = 1 , Δ n 3 k 2 n k 2 if δ = 1 , Δ = n 2 .
The following is the famous inequality by Pólya and Szegö:
Lemma 3.
(Pólya–Szegö inequality) [25] Let ( a 1 , a 2 , , a r ) and ( b 1 , b 2 , , 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 a i M 1 , 0 < m 2 b i M 2 , 1 i r .
Then
i = 1 r a i 2 i = 1 r b i 2 i = 1 r a i b i 2 1 4 M 1 M 2 m 1 m 2 + m 1 m 2 M 1 M 2 2 .
We now give more lower and upper bounds for SGut k ( G ) · SGut k ( G ¯ ) in terms of n, Δ and δ .
Theorem 2.
Let G be a connected graph of order n with maximum degree Δ, minimum degree δ and a connected G ¯ . Additionally, let k be an integer with 2 k n . Then
SGut k ( G ) · SGut k ( G ¯ ) ( k 1 ) 2 δ k ( n δ 1 ) k n k 2 if Δ + δ n 1 , ( k 1 ) 2 Δ k ( n Δ 1 ) k n k 2 if Δ + δ n 1
with equality holding if and only if G is a regular graph with d G ( S ) = d G ¯ ( S ) = k 1 for any S V ( G ) , | S | = k , and
SGut k ( G ) · SGut k ( G ¯ ) ( n 1 ) 2 k + 2 2 2 k + 2 n k 2 Δ ( n δ 1 ) δ ( n Δ 1 ) k + δ ( n Δ 1 ) Δ ( n δ 1 ) k + 2 ,
Moreover, the equality holds if and only if G is a n 1 2 -regular graph with k = n , n is odd.
Proof. 
Lower bound: By Cauchy–Schwarz inequality with (1), we have
SGut k ( G ) · SGut k ( G ¯ ) ( k 1 ) 2 | S | = k S V ( G ) v S d e g G ( v ) | S | = k S V ( G ¯ ) v S d e g G ¯ ( v )
( k 1 ) 2 | S | = k S V ( G ) v S d e g G ( v ) v S d e g G ¯ ( v ) 1 / 2 2 ( k 1 ) 2 | S | = k S V ( G ) v S d e g G ( v ) ( n 1 d e g G ( v ) ) 1 / 2 2 .
Since δ d e g G ( v ) Δ , one can easily see that
d e g G ( v ) ( n 1 d e g G ( v ) ) δ ( n δ 1 ) if Δ + δ n 1 , Δ ( n Δ 1 ) if Δ + δ n 1 .
From the above results, we have
SGut k ( G ) · SGut k ( G ¯ ) ( k 1 ) 2 δ k ( n δ 1 ) k n k 2 if Δ + δ n 1 , ( k 1 ) 2 Δ k ( n Δ 1 ) k n k 2 if Δ + δ n 1 .
The equality holds in (12) if and only if d G ( S ) = d G ¯ ( S ) = k 1 for any S V ( G ) with | S | = k . By the Cauchy–Schwarz inequality, the equality holds in (13) if and only if
v S 1 d e g G ( v ) v S 1 d e g G ¯ ( v ) = v S 2 d e g G ( v ) v S 2 d e g G ¯ ( v ) for   any S 1 , S 2 V ( G ) with   | S 1 | = | S 2 | = k ,
that is, if and only if d e g G ( u ) = d e g G ( v ) for any u , v V ( G ) , 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 ( S ) = d G ¯ ( S ) = k 1 for any S V ( G ) , | S | = k .
Upper bound: Let Δ ¯ and δ ¯ be the maximum degree and the minimum degree of graph G ¯ , respectively. Then Δ ¯ = n δ 1 and δ ¯ = n Δ 1 . By (1) and (10), we have
SGut k ( G ) · SGut k ( G ¯ ) ( n 1 ) 2 | S | = k S V ( G ) v S d e g G ( v ) | S | = k S V ( G ¯ ) v S d e g G ¯ ( v ) ( n 1 ) 2 | S | = k S V ( G ) v S d e g G ( v ) v S d e g G ¯ ( v ) 1 / 2 2 1 4 Δ Δ ¯ δ δ ¯ k / 2 + δ δ ¯ Δ Δ ¯ k / 2 2 ( n 1 ) 2 4 | S | = k S V ( G ) v S d e g G ( v ) ( n 1 d e g G ( v ) ) 1 / 2 2 Δ Δ ¯ δ δ ¯ k / 2 + δ δ ¯ Δ Δ ¯ k / 2 2 .
One can easily see that
d e g G ( v ) ( n 1 d e g G ( v ) ) ( n 1 ) 2 4 for   any   v V ( G ) .
Using this result in the above with Δ ¯ = n δ 1 and δ ¯ = n Δ 1 , we get
SGut k ( G ) · SGut k ( G ¯ ) ( n 1 ) 2 k + 2 2 2 k + 2 n k 2 Δ ( n δ 1 ) δ ( n Δ 1 ) k + δ ( n Δ 1 ) Δ ( n δ 1 ) k + 2 .
Moreover, the above equality holds if and only if G is a n 1 2 -regular graph with k = n , n is odd (very similar proof of the Proposition 1). □
Example 2.
Let G C n with k = n . Then δ = 2 and hence
SGut k ( G ) · SGut k ( G ¯ ) = ( n 1 ) 2 ( n 3 ) n 2 n = ( k 1 ) 2 δ k ( n δ 1 ) k n k 2 .
Let G be a n 1 2 -regular graph of order n with k = n and odd n. Then Δ = δ = n 1 2 and hence
SGut k ( G ) · SGut k ( G ¯ ) = ( n 1 ) 2 n + 2 2 2 n = ( n 1 ) 2 k + 2 2 2 k + 2 n k 2 Δ ( n δ 1 ) δ ( n Δ 1 ) k + δ ( n Δ 1 ) Δ ( n δ 1 ) k + 2 .
We now give more lower and upper bounds of SGut k ( G ) + SGut k ( G ¯ ) in terms of n, Δ and δ .
Theorem 3.
Let G be a connected graph of order n with maximum degree Δ, minimum degree δ and a connected G ¯ . Additionally, let k be an integer with 2 k n . Then
SGut k ( G ) + SGut k ( G ¯ ) 2 ( k 1 ) δ k / 2 ( n δ 1 ) k / 2 n k if Δ + δ n 1 , 2 ( k 1 ) Δ k / 2 ( n Δ 1 ) k / 2 n k if Δ + δ n 1
with equality holding if and only if G is a n 1 2 -regular graph with odd n and d G ( S ) = d G ¯ ( S ) = k 1 for any S V ( G ) , | S | = k , and
SGut k ( G ) + SGut k ( G ¯ ) ( n 1 ) Δ k + ( n δ 1 ) k n k
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 0 , that is, a 2 + b 2 2 a b with equality holding if and only if a = b . Therefore we have
v S d e g G ( v ) + v S d e g G ¯ ( v ) 2 v S d e g G ( v ) v S d e g G ¯ ( v ) 1 / 2 = 2 v S d e g G ( v ) d e g G ¯ ( v ) 1 / 2 = 2 v S d e g G ( v ) ( n d e g G ( v ) 1 ) 1 / 2 .
From the above result with (14), we get
v S d e g G ( v ) + v S d e g G ¯ ( v ) 2 δ k / 2 ( n δ 1 ) k / 2 if Δ + δ n 1 , 2 Δ k / 2 ( n Δ 1 ) k / 2 if Δ + δ n 1 .
Now,
SGut k ( G ) + SGut k ( G ¯ ) = | S | = k S V ( G ) v S d e g G ( v ) d G ( S ) + v S d e g G ¯ ( v ) d G ¯ ( S ) ( k 1 ) | S | = k S V ( G ) v S d e g G ( v ) + v S d e g G ¯ ( v ) 2 ( k 1 ) δ k / 2 ( n δ 1 ) k / 2 n k if Δ + δ n 1 , 2 ( k 1 ) Δ k / 2 ( n Δ 1 ) k / 2 n k if Δ + δ n 1 .
From the above, one can easily see that the equality holds in (15) if and only if G is a n 1 2 -regular graph with odd n and d G ( S ) = d G ¯ ( S ) = k 1 for any S V ( G ) , | S | = k .
Upper bound: By arithmetic-geometric mean inequality, we have
SGut k ( G ) + SGut k ( G ¯ ) = | S | = k S V ( G ) v S d e g G ( v ) d G ( S ) + v S d e g G ¯ ( v ) d G ¯ ( S ) ( n 1 ) | S | = k S V ( G ) v S d e g G ( v ) + v S d e g G ¯ ( v ) ( n 1 ) | S | = k S V ( G ) v S d e g G ( v ) k k + v S d e g G ¯ ( v ) k k = ( n 1 ) k k | S | = k S V ( G ) v S d e g G ( v ) k + v S ( n d e g G ( v ) 1 ) k = ( n 1 ) k k | S | = k S V ( G ) v S d e g G ( v ) k + k ( n 1 ) v S d e g G ( v ) k ( n 1 ) k k | S | = k S V ( G ) ( k Δ ) k + k ( n 1 ) k δ k = ( n 1 ) Δ k + ( n δ 1 ) k n k .
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 n 1 2 -regular graph with odd n and k = n . Then δ = n 1 2 and hence
SGut k ( G ) + SGut k ( G ¯ ) = ( n 1 ) n + 1 2 n 1 = 2 ( k 1 ) δ k / 2 ( n δ 1 ) k / 2 n k
Let G C n with k = n . Then Δ = δ = 2 , Δ ¯ = δ ¯ = 2 and hence
SGut k ( G ) + SGut k ( G ¯ ) = ( n 1 ) 2 n + ( n 3 ) n = ( n 1 ) Δ k + ( n δ 1 ) k n k .

Author Contributions

Conceptualization, Z.W., Y.M., K.C.D. and Y.S.; writing—original draft preparation, Z.W.; methodology, Y.M., K.C.D. and Y.S.; writing—review and editing, Y.M., K.C.D. and Y.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Science Foundation of China grant numbers 12061059, 11601254, 11661068, 11551001, 11161037 and 11461054, and the UoA Flexible Fund from Northumbria University grant number 201920A1001.

Acknowledgments

The authors are very grateful to three anonymous referees for their valuable comments on the paper, which have considerably improved the presentation of this paper.

Conflicts of Interest

The author declares no conflict of interest.

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