J Comb Optim (2016) 31:182–195DOI 10.1007/s10878-014-9728-y

Maximally edge-connected graphs and Zeroth-ordergeneral Randic index for α ≤ −1

Guifu Su · Liming Xiong ·Xiaofeng Su · Guojun Li

Published online: 11 March 2014© Springer Science+Business Media New York 2014

Abstract Let G be a connected graph with order n, minimum degree δ = δ(G) andedge-connectivity λ = λ(G).AgraphG is maximally edge-connected if λ = δ. In thispaper,we present two sufficient conditions for graphs to bemaximally edge-connected,which generalize two results recently proved by P. Dankelmann, A. Hellwig and L.Volkmann. The extremal graphs are also characterized.

Keywords Degree (of vertex) · Zeroth-order general Randic index ·Edge-connectivity · Maximally edge-connected

1 Introduction

Throughout this paper, we consider finite undirected simple connected graphs. Let Gbe such a graphwith vertex set V = V (G) and edge set E = E(G). Then the order and

G. Su (B) · L. XiongSchool of Mathematics, Beijing Institute of Technology, Beijing 100081,People’s Republic of Chinae-mail: gfs1983@126.com

L. Xionge-mail: lmxiong@bit.edu.cn

G. Su · G. LiComputational Systems Biology Lab, Department of Biochemistry andMolecular Biology, University of Georgia, Athens, GA 30602, USAe-mail: guojunsdu@gmail.com

X. SuCollege of Arts and Science,Shanghai Maritime University, Shanghai 201306, People’s Republic of Chinae-mail: umpire.su@gmail.com

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J Comb Optim (2016) 31:182–195 183

size of G are n = |V | and m = |E |, respectively. The degree of a vertex u ∈ V is thenumber of edges incident with u in G, denoted by deg(u) = degG(u). The minimumvalue of these numbers is the minimum degree of G, denoted by δ = δ(G). Thedistance between two vertices u and v of G is the length of a shortest path connectingthem in G. The maximum value of these numbers is the diameter of G, denoted bydiam(G).

Let u and v be two distinct vertices of a connected graphG,we call a nonempty set Sof edges inG a {u, v}-separator if the removal of S leaves u and v in distinct connectedcomponents of the resulting graph. The set S is aminimum {u, v}-separator if it hasthe smallest cardinality among all {u, v}-separators, and we use λ(u, v) to denote thesize of a minimum {u, v}-separator. We call that a connected graph G with at leasttwo vertices have edge-connectivity λ = λ(G) if the removal of some set of λ edgesdisconnects G, but there is no smaller set of edges whose removal disconnectes G.Formally,

λ(G) = min{λ(u, v) : u, v ∈ V with u �= v}.

A vertex-cut in a graph G is a set X of vertices of G such that G − X is disconnected.The vertex-connectivity or simply the connectivity κ = κ(G) of a graph G is theminimum cardinality of a vertex-cut in G if it is not complete, and κ(G) = n − 1 ifG is the complete graph Kn of order n. Other terminology and notation needed willbe introduced as it naturally occurs in the following and we use (Bondy and Murty1976) for those not defined here.

Whitney in 1932 presented a basic inequality on connectivity, edge-connectivityand minimum degree of a graph.

Theorem 1.1 (Whitney 1932). Let G be a connected graph. Then κ ≤ λ ≤ δ.

Chartrand and Harary (1968) showed that this result is best possible in the sensethat if a, b and c are positive integers such that a ≤ b ≤ c, then there exists a graphG with κ = a, λ = b and δ = c.

Because of Theorem 1.1, it is reasonable to say that a graph G is maximallyconnected if κ = δ and maximally edge-connected if λ = δ.

Sufficient conditions for whether a graph is maximally edge-connected weregiven by several researchers. The following is a classical result for maximally edge-connected graphs due to Chartrand (1966).

Theorem 1.2 (Chartrand 1966). Let G be a connected graph with order n, minimumdegree δ and edge-connectivity λ. If δ ≥ ⌊ n

2

⌋, then λ = δ.

Theorem 1.2 was the starting point for many other sufficient conditions for max-imally edge-connected of graphs. Several years later, Lesniak (1974) proved that agraph is maximally edge-connected if deg(u) + deg(v) ≥ n − 1 for all pairs u, v

of nonadjacent vertices, which obviously weakened the condition in Theorem 1.2.Plesník (1975) showed that every graph of diameter two is maximally edge-connectedand thus generalizedLesniak’s result.We encourage the readers to consult the improve-ments of Plesnik’s result in Daneklmann and Volkmann (1995) and Plesník and Znám

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184 J Comb Optim (2016) 31:182–195

(1989). In Goldsmith and White (1978) presented that if the vertex set of G can bepartitioned into � n

2 � pairs of vertices ui and vi such that deg(ui ) + deg(vi ) ≥ n fori = 1, 2, · · · , � n

2 �, then it is maximally edge-connected. Later, (Xu 1994) gave ananalogous sufficient condition for directed graphs, which also includes the results byGoldsmith and White. Dankelmann and Volkmann (1997) presented simple degreesequence conditions for the equality of edge-connectivity and minimum degree of adigraph. In addition, some other sufficient conditions, depending on paraments notdirectly related to the vertex degree, for graphs to be maximally edge-connected wereshowed by several authors, we encourage the readers to consult the literature Fiol(1993), (Febrega and Fiol 1989,1996), Soneoka et al. (1987).

Let α be a given real number, the Zeroth-order general Randic index R0α(G) was

defined by in Li and Zheng (2005):

R0α(G) =

∑

u∈Vdegα(u).

Li and Zhao (2004) characterized the trees with the first three largest and smallestZeroth-order general Randic index with α being equal to m,−m, 1

m ,− 1m for some

integer m ≥ 2. In Hu (2005), investigated the molecular graphs having the smallestand largest Zeroth-order general Randic index. Hua and Deng (2006) gave sharp lowerand upper bounds for this index among all unicyclic graphs.

Let α = −1,we getR(G) = R0−1(G) = ∑u∈V 1

deg(u),which is the known inverse

degree of graphs. The inverse degree first attracted attention through conjectures ofthe computer programm Graffiti (Fajtlowicz 1987). Later, Erdös et al. (1988) andDankelmann et al. (2008) independently investigated the inverse degree of graphs.In Dankelmann et al. (2009), gave sufficient conditions for graphs to be maximallyedge-connected in terms of the inverse degree, the minimum degree and the order ofa graph.

Motivated by the results of Dankelmann et al. (2009), in this paper, we give suffi-cient conditions for arbitrary graphs and triangle-free graphs to be maximally edge-connected in terms of the Zeroth-order general Randic, minimum degree and theorder.

2 Preliminary lemmas

In this section, we will list or prove some lemmas which will be used in our laterproofs.

Lemma 2.1 (Lin et al. 2009).Let x1, x2 ∈ N and α ∈ R. If x1 − 2 ≥ x2 ≥ 1, then

(i) (x1 − 1)α + (x2 + 1)α < xα1 + xα

2 if α < 0 or α > 1;(ii) (x1 − 1)α + (x2 + 1)α > xα

1 + xα2 if 0 < α < 1.

We will repeatedly make use of Jenson’s Inequality, which states that: Let C be aconvex subset of a real vector space X, let xi ∈ C and σi ≥ 0 (i = 1, 2, · · · , n) with

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J Comb Optim (2016) 31:182–195 185

∑ni=1 σi = 1, then

(i) �

(k∑

i=1

σi xi

)

≤k∑

i=1

σi�(xi )if�(x) : C → Ris a convex function;

(ii) �

(k∑

i=1

σi xi

)

≥k∑

i=1

σi�(xi )if�(x) : C → Ris a concave function.

We also use the following inequalities.

Lemma 2.2 Let x1, · · · , xp and A be positive reals with∑p

i=1 xi ≤ A. For any realnumber α < 0, we have

(i)∑p

i=1 xαi ≥ p1−αAα;

(ii) If, in addition x1, · · · , xp, A are positive integers, and a, b are integers withA = ap + b and 0 ≤ b < p, then

∑pi=1 x

αi ≥ (p − b)aα + b(a + 1)α .

Proof (i) This follows by applying Jenson’s Inequality to the function σ1(x) = xα,

which is convex for α < 0.(ii) Let x1, · · · , xp and A be positive integers with

∑pi=1 xi ≤ A. We assume that xi

are chosen such that∑p

i=1 xαi is minimum. If no two of xi differ by more than 1,

then p − b of the xi equal a and the remaining b of the xi equal a + 1, and theinequality follows. So assume that two of the xi , say x1 and x2, differ by morethan 1.Assume that x1 > x2. Let y1 = x1 −1, y2 = x2 +1 and yi = xi for i ≥ 3.Then by Lemma 2.1 (i), we have

p∑

i=1

yαi −

p∑

i=1

xαi = (x1 − 1)α + (x2 + 1)α − xα

1 − xα2 < 0,

which contradicts to the choice of the xi . This implies the desired result. The following results can be obtained from the definition of the convex and concave

function, respectively, so we omit the proofs here.

Lemma 2.3 Let �(x) be a continuous function on interval [L , R] and l + r = L + Rfor l, r ∈ [L , R]. Then(i) �(L) + �(R) ≥ �(l) + �(r) if �(x) is convex;(ii) �(L) + �(R) ≤ �(l) + �(r) if �(x) is concave.

We call that a graph is triangle-free if it does not contain a triangle as a subgraph.

Lemma 2.4 (Daneklmann and Volkmann 1995). Let G be a triangle-free graph withorder n ≤ 4δ − 1. Then λ = δ.

A complete r -multipartite graph Kn1,··· ,nr is a simple graph whose vertices can bepartitioned into r(≥ 2) sets so that each pair of vertices are connected by an edge ifand only if they belong to different sets of the partition.

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186 J Comb Optim (2016) 31:182–195

The Turán graph Tn,r is the complete r -partite graph with b partite sets of sizea + 1 and r − b partite sets of size a, where a = � n

r � and b = n − ra.The following is a famous result due to Turán (1941).

Lemma 2.5 (Turán 1941).

(i) Among all the n-vertex simple graphswith no (r+1)-clique, Tn,r has themaximumnumber of edges.

(ii) |E(Tn,r )| ≤⌊(1 − 1

r

) · n22

⌋.

For two subsets X and Y of V (G), let [X,Y ] be the set of edges with one endpoint inX and the other one in Y, and |[X,Y ]| denotes the cardinality of [X,Y ].

The following lemma was proved by in Dankelmann and Volkmann (2000).

Lemma 2.6 (Dankelmann and Volkmann 2000).Let G be a connected graph. If thereexist two disjoint, nonempty sets X,Y ⊂ V (G)with X ∪Y = V (G) and |[X,Y ]| < δ,

then |X | ≥ δ + 1 and |Y | ≥ δ + 1.

The result above also can be found in other literature, e.g. (Daneklmann andVolkmann1995) and (Plesník and Znám 1989).

Lemma 2.7 Let x be a real number and α < 0. Then xα + (x + 12 )

α ≤ (x + 1)α +(x − 1

2 )α.

Proof Let f (t) = tα − (t − 12 )

α. Easily to find that f (t) is increasing, so we getf (x + 1) ≥ f (x), i.e., (x + 1)α − (x + 1

2 ) ≥ xα − (x − 12 ), as desired.

3 Main results

Theorem 3.1 Let G be a connected graph with order n, minimum degree δ and edge-connectivity λ. If for α ≤ −1 we have

R0α(G) < 2δα − δα+1 + (δ − 1)(δ + 1)α + (δ − 1)(n − δ − 1)α

+(2n − 3δ − 2)(n − δ − 2)α,

then λ = δ.

Proof Suppose to contrary that λ < δ. Let S be a minimum {u, v}-separator for anytwo distinct vertices u, v inG and let X and Y denote the vertex set of two componentsof G − S. This implies that λ(u, v) = λ(G), i.e., S is a minimum edge-cut, then byLemma 2.6, we have |X | ∈ [δ + 1, n − δ − 1] and |Y | ∈ [δ + 1, n − δ − 1] (also see,for example, Dankelmann and Volkmann 1997).

Clearly, each vertex in X is adjacent to at most |X | − 1 vertices of X, and exactlyλ edges join a vertex of X to that of Y. Hence

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J Comb Optim (2016) 31:182–195 187

∑

x∈Xdeg(x) ≤ |X |(|X | − 1) + λ.

By Lemma 2.2 (ii), we have

∑

x∈Xdegα(x) ≥(|X | − λ)(|X | − 1)α + λ|X |α

=[(|X | − 1) + (1 − λ)](|X | − 1)α + λ|X |α=(|X | − 1)α+1 + (|X | − 1)α − λ[(|X | − 1)α − |X |α].

Since there must be a vertex v with degree δ, we may assume that v ∈ Y. Then itfollows

∑

y∈Y−{v}deg(y) ≤ (|Y | − 1)2 + λ.

By Lemma 2.2 (ii), we have

∑

y∈Ydegα(y) ≥δα + (|Y | − 1 − λ)(|Y | − 1)α + λ|Y |α

=δα + (|Y | − 1)α+1 − λ[(|Y | − 1)α − |Y |α].

Now the Zeroth-order general Randic index can be presented as

R0α(G) =

∑

x∈Xdegα(x) +

∑

y∈Ydegα(y)

≥ (|X | − 1)α+1 + (|X | − 1)α + δα + (|Y | − 1)α+1

− λ[(|X | − 1)α − |X |α + (|Y | − 1)α − |Y |α].

To minimize the right side of the last inequality, we consider the function σ2(t) =(t − 1)α − tα. It is easy to verify that σ ′′

2 (t) > 0 for α ≤ −1 and t > 1, so σ2(t) isconvex. By Lemma 2.3 (i), we get:

(|X | − 1)α − |X |α + (|Y | − 1)α − |Y |α ≤ δα − (δ + 1)α

+(n − δ − 2)α − (n − δ − 1)α.

Noting that σ1(t) = tα is a decreasing function for α ≤ −1. Thus, in conjunctionwith λ ≤ δ − 1, we have

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188 J Comb Optim (2016) 31:182–195

R0α(G) ≥ (n − δ − 2)α+1 + (n − δ − 2)α + δα − (n − δ − 2)α+1

+ (δ − 1)[(δ + 1)α − δα + (n − δ − 1)α − (n − δ − 2)α]= 2δα − δα+1 + (δ − 1)(δ + 1)α + (δ − 1)(n − δ − 1)α

+ (2n − 3δ − 2)(n − δ − 2)α,

a contradiction to the hypothesis. Hence λ = δ. The following result is an immediate consequence of Theorem 3.1 for α = −1.

Corollary 3.2 (Dankelmann et al. 2009). Let G be a connected graph with order n,

minimum degree δ and edge-connectivity λ. If

R(G) < 2 + 2

δ(δ + 1)+ n − 2δ

(n − δ − 2)(n − δ − 1)

then λ = δ.

Now we pay our attention to the maximally edge-connected triangle-free graphs.

Theorem 3.3 Let G be a connected triangle-free graphwith order n,minimum degreeδ and edge-connectivity λ. If for α ≤ −1 we have

R0α(G) < δα − δα+1 + (δ − 1)(δ + 1)α + (δ − 1)(n − 2δ + 2)α

2α

+ (2n − 5δ + 1)(n − 2δ)α

2α,

then λ = δ.

Proof Suppose to contrary that λ < δ. By Lemma 2.4, we have δ ≤ n4 . Let S be

a minimum {u, v}-separator for any two distinct vertices u, v in G and let X and Ydenote the vertex sets of the two components of G− S. The assumption λ < δ and thefact n > 4δ imply that |X | ∈ [2δ, n−2δ] and |Y | ∈ [2δ, n−2δ] (also see, for example,

Dankelmann and Volkmann 1997). By Lemma 2.5, we have |[X, X ]| ≤⌊ |X |2

4

⌋and

therefore

∑

x∈Xdeg(x) ≤ 2

⌊ |X |24

⌋+ λ ≤

{ |X |22 + δ − 1, if |X | is even;

(|X |−1)2

2 + |X | + δ − 2, if |X | is odd. (1)

We first consider the case |X | is even. By Lemma 2.2 (ii),∑

x∈X degα(x) is mini-mized subject to (1) if |X | − δ + 1 of the degrees equal |X |

2 and δ − 1 degrees equal|X |2 + 1. Hence

∑

x∈Xdegα(x) ≥ (|X | − δ + 1)

( |X |2

)α

+ (δ − 1)

( |X |2

+ 1

)

= |X |α+1

2α− (δ − 1)

[( |X |2

)α

−( |X |

2+ 1

)α]. (2)

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J Comb Optim (2016) 31:182–195 189

Nowwe assume |X | is odd. ByLemma 2.2 (ii),∑

x∈X degα(x) isminimized subjectto (1) if |X |+1

2 − δ + 1 of the degrees equal |X |−12 and |X |−1

2 + δ − 1 degrees equal|X |+1

2 . Hence

∑

x∈Xdegα(x)≥

( |X | + 1

2− δ + 1

) ( |X | − 1

2

)α

+( |X | − 1

2+ δ − 1

)( |X | + 1

2

)α

=[ |X | − 1

2− (δ − 2)

] ( |X | − 1

2

)α

+[ |X | + 1

2+ (δ − 2)

] ( |X | + 1

2

)α

=( |X | − 1

2

)α+1+

( |X |+1

2

)α+1− (δ − 2)

[( |X | − 1

2

)α

−( |X | + 1

2

)α].

(3)

For convenience, we distinguish the following cases depending on the parities oforder n and |X |.Case 1. Both n and |X | are even.

In this case |Y | = n − |X | is also even. According to (2), we have

R0α(G) =

∑

x∈Xdegα(x) +

∑

y∈Ydegα(y)

≥ |X |α+1

2α− (δ − 1)

[( |X |2

)α

−( |X |

2+ 1

)α]

+ (n − |X |)α+1

2α− (δ − 1)

[(n − |X |

2

)α

−(n − |X |

2+ 1

)α]

= |X |α+1

2α+ (n − |X |)α+1

2α− (δ − 1)σ (|X |),

where σ(|X |) =[( |X |

2

)α −( |X |

2 + 1)α +

(n−|X |

2

)α −(n−|X |

2 + 1)α]

.

Let σ3(t) = ( t2

)α − ( t2 + 1

)α. It is easy to verify that σ ′′

3 (t) > 0 for t > 0 andα ≤ −1, and hence the function σ3(t) is convex. By Lemma 2.3 (i) and |X |, |Y | ∈[2δ, n − 2δ], we get

σ3(|X |) + σ3(n − |X |) ≤ σ3(2δ) + σ3(n − 2δ).

Note that σ4(t) = tα+1 is a decreasing function for α ≤ −1 and t > 0, so|X |α+1 ≥ (n − 2δ)α+1. Hence

R0α(G)≥ (n − 2δ)α+1

2α−1 −(δ − 1)

[δα − (δ + 1)α+

(n − 2δ

2

)α

−(n − 2δ + 2

2

)α]

= δα − δα+1 + (δ − 1)(δ + 1)α + (δ − 1)(n − 2δ + 2)α

2α

+ (2n − 5δ + 1)(n − 2δ)α

2α= β1(α, δ),

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190 J Comb Optim (2016) 31:182–195

a contradiction to the hypothesis.Case 2. n is even and |X | is odd.

In this case |Y | = n − |X | is also odd and we have |X |, |Y | ≥ 2δ + 1. Hence, bythe inequality (3), we get

R0α(G) =

∑

x∈Xdegα(x) +

∑

y∈Ydegα(y)

≥( |X | − 1

2

)α+1+

( |X | + 1

2

)α+1− (δ − 2)

[( |X | − 1

2

)α

−( |X | + 1

2

α)]

+(n − |X | − 1

2

)α+1+

(n − |X | + 1

2

)α+1− (δ − 2)

[(n − |X | − 1

2

)α

−(n − |X | + 1

2

α)]

=( |X | − 1

2

)α+1+

( |X | + 1

2

)α+1+

(n − |X | − 1

2

)α+1+

(n − |X | + 1

2

)α+1

−(δ − 2)

[( |X | − 1

2

)α

−( |X | + 1

2

)α

+(n − |X | − 1

2

)α

−(n − |X | + 1

2

)α].

Note that σ5(t) = ( t−12

)α − ( t+12

)αis a convex function for α ≤ −1 and t > 0. By

Lemma 2.3 (i) and the fact |X |, |Y | ≥ 2δ + 1, we get

σ5(|X |) + σ5(n − |X |) ≤ σ5(2δ + 1) + σ5(n − 2δ − 1).

Hence

R0α(G) ≥

(n − 2δ − 2

2

)α

+(n − 2δ

2

)α

+(n − 2δ − 2

2

)α

+(n − 2δ

2

)α

−(δ − 2)

[δα − (δ + 1)α +

(n − 2δ − 2

2

)α

−(n − 2δ

2

)α]

= 2δα − δα+1 + (δ − 2)(δ + 1)α + (n − δ − 2)(n − 2δ)α

2α

+ (n − 3δ)(n − 2δ − 2)α

2α= β2(α, δ) > β1(α, δ).

It remains to verify that β2(α, δ) > β1(α, δ). We begin with the following claim.

Claim 1 (n − 3δ + 2)(n − 2δ − 1)α < (n − 3δ)(n − 2δ − 2)α .

Proof of Claim 1 It sufficient to show that(n−2δ−2n−2δ−1

)α

> n−3δ+2n−3δ , i.e.,

(1− 1

n−2δ−1

)α

>1 + 2n−3δ . Note that for n>2δ+2, we have

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J Comb Optim (2016) 31:182–195 191

(1 − 1

n − 2δ − 1

)α

= 1 + α

(− 1

n − 2δ − 1

)+ · · ·

+α(α − 1) · · · (α − n + 1)

n!(

− 1

n − 2δ − 1

)n

+ · · ·

> 1 + α

(− 1

n − 2δ − 1

).

Hence, we need prove that 1 − αn−2δ−1 > 1 + 2

n−3δ , i.e.,2

n−3δ + αn−2δ−1 < 0, i.e.,

2(n−2δ−1)+α(n−3δ)(n−3δ)(n−2δ−1) < 0, i.e., 2(n − 2δ − 1) + α(n − 3δ) < 0, i.e., α < 4δ−2n+2

n−3δ . Note

that α ≤ −1, it holds if 4δ−2n+2n−3δ ≤ −1, i.e., 4δ−2n+2+n−3δ

n−3δ ≤ 0, i.e., n ≥ δ + 2. Thisconfirms Claim 1.

Thus the difference between β2(α, δ) and β1(α, δ) can be expressed as

β2(α, δ) − β1(α, δ)

= δα − (δ + 1)α + (n − 3δ)(n − 2δ − 2)α

2α− (δ − 1)(n − 2δ + 2)α

2α

− (n − 4δ + 3)(n − 2δ)α

2α> δα − (δ + 1)α + (n − 3δ + 2)(n − 2δ − 1)α

2α

− (δ − 1)(n − 2δ + 2)α

2α− (n − 4δ + 3)(n − 2δ)α

2α

= δα − (δ + 1)α + (δ − 1)

[(n − 2δ − 1)α

2α− (n − 2δ + 2)α

2α

]

+(n − 4δ + 3)

[(n − 2δ − 1)α

2α− (n − 2δ)α

2α

]> 0.

This confirms that β1(α, δ) < β2(α, δ), which implies that R0α(G) ≥ β1(α, δ), a

contradiction to the hypothesis.

Case 3. Both n and |X | are odd ( or n is odd and |X | is even).In both cases above |X | and |Y | have different parities. Assume without loss of

generality that |X | is odd and |Y | is even. Then, for |X | ≥ 2δ + 1, we get

R0α(G) =

∑

x∈Xdegα(x) +

∑

y∈Ydegα(y)

≥( |X | − 1

2

)α+1+

( |X | + 1

2

)α+1− (δ − 2)

[( |X | − 1

2

)α

−( |X | + 1

2

)α]

+ (n − |X |)α+1

2α− (δ − 1)

[(n − |X |

2

)α

−(n − |X |

2+ 1

)α]

=( |X |−1

2

)α+1+

( |X |+1

2

)α+1+ (n − |X |)α+1

2α+

[( |X |−1

2

)α

−( |X |+1

2

)α]

−(δ − 1)

[( |X |−1

2

)α

−( |X |−1

2+1

)α

+(n − |X |

2

)α

−(n − |X |

2+1

)α].

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192 J Comb Optim (2016) 31:182–195

By Lemma reflema.2.7, we get

( |X | − 1

2

)α

−( |X | − 1

2+ 1

)α

+(n − |X |

2

)α

−(n − |X |

2+ 1

)α

≤( |X | − 1

2

)α

−( |X | − 1

2+ 1

)α

+(n − |X | − 1

2

)α

−(n − |X | − 1

2+ 1

)α

.

Noticing σ5(t) = ( t−12

)α − ( t+12

)αis a convex function, then by Lemma 2.3 (i) we

get

σ5(|X |) + σ5(n − |X |) ≤ σ5(2δ + 1) + σ5(n − 2δ − 1).

On the other hand,( |X |−1

2

)α+1 ≥ ( n−2δ−22

)α+1,

( |X |+12

)α+1 ≥ ( n−2δ2

)α+1and

(n−|X |)α+1

2α ≥ (n−2δ−1)α+1

2α . Hence

R0α(G) ≥

(n − 2δ − 2

2

)α+1

+(n − 2δ

2

)α+1

+ (n − 2δ − 1)α+1

2α

+[(n − 2δ − 2)α

2α− (n − 2δ)α

2α

]

−(δ − 1)

[δα − (δ + 1)α + (n − 2δ − 2)α

2α− (n − 2δ)α

2α

]

= δα − δα+1 + (δ − 1)(δ + 1)α + (n − 4δ + 2)(n − 2δ − 2)α

2α+1

+ (n − 4)(n − 2δ)α

2α+1

+ (n − 2δ − 1)(n − 2δ − 1)α

2α= β3(α, δ) > β1(α, δ),

a contradiction to the hypothesis.It remains to show that β3(α, δ) > β1(α, δ). We begin with the following claim.

Claim 2 (n − 2δ + 1)(n − 2δ)α < (n − 2δ − 1)(n − 2δ − 1)α .

Proof of Claim 2 It sufficient to show that(n−2δ−1n−2δ

)α

> n−2δ+1n−2δ−1 , i.e.,

(1 − 1

n−2δ

)α

>

1 + 2n−2δ−1 . Note that for n > 2δ + 2, we have

(1 − 1

n − 2δ

)α

= 1 + α

(− 1

n − 2δ

)+ · · ·

+α(α − 1) · · · (α − n + 1)

n!(

− 1

n − 2δ

)n

+ · · ·

> 1 + α

(− 1

n − 2δ

).

123

J Comb Optim (2016) 31:182–195 193

Hence, we only show that 1 − αn−2δ > 1 + 2

n−2δ−1 , i.e.,2

n−2δ−1 + αn−2δ < 0, i.e.,

2(n − 2δ) + α(n − 2δ − 1) < 0. Note that α ≤ −1, then it holds if 4δ−2nn−2δ−1 ≤ −1,

i.e., 4δ−2nn−2δ−1 + 1 ≤ 0, i.e., n ≥ 2δ − 1. As desired we complete the proof of Claim 2.

Hence, the difference between β3(α, δ) and β1(α, δ) can be expressed as

β3(α, δ) − β1(α, δ) = (n − 4δ + 2)(n − 2δ − 2)α

2α+1 + (2n − 4δ − 2)(n − 2δ − 1)α

2α+1

− (2δ − 2)(n − 2δ + 2)α

2α+1 − (3n − 10δ + 6)(n − 2δ)α

2α+1

>(n − 4δ + 2)(n − 2δ − 2)α

2α+1 + (2n − 4δ + 2)(n − 2δ)α

2α+1

− (2δ − 2)(n − 2δ + 2)α

2α+1 − (3n − 10δ + 6)(n − 2δ)α

2α+1 > 0.

This confirms that β3(α, δ) > β1(α, δ), which implies thatR0α(G) ≥ β1(α, δ), again

a contradiction to the hypothesis.This completes the proof of Theorem 3.3. Let α = −1 in Theorem 3.3, we get

Corollary 3.4 (Dankelmann et al. 2009). Let G be a connected triangle-free graphwith order n, minimum degree δ and edge-connectivity λ. If

R(G) < 4 − 4(δ − 1)

[1

2δ(2δ + 2)+ 1

(n − 2δ)(n − 2δ + 2)

],

then λ = δ.

4 Sharpness

An independent set of a graph is a subset of pairwise nonadjacent vertices of thegraph. The size of an independent set is the number of vertices it contains.

A matching in a graph is a set of edges no two of which share a vertex. It may bethe case that the graph itself is a matching.

• We present a class of graphs to show that the condition in Theorem 3.1 cannotbe improved. For an arbitrary integer δ and n = 2δ + 2 ≥ 8, let 1 ∼= Kδ+1with vertex set V (1) = {x1, · · · , xδ+1} and let 2 ∼= Kn−δ−1 with vertex setV (2) = {y1, · · · , yn−δ−1}. We define G ′ to be the union of 1 and 2 togetherwith δ − 1 edges x1y1, x2y2, · · · , xδ−1yδ−1. Then n(G ′) = n, δ(G ′) = δ and

R0α(G ′) = 2δα − δα+1 + (δ − 1)(δ + 1)α + (δ − 1)(n − δ − 1)α

+(2n − 3δ − 2)(n − δ − 2)α.

But it is easy to see that λ(G ′) ≤ δ(G ′) − 1.

123

194 J Comb Optim (2016) 31:182–195

• Wealso present a class of graphs to show that the condition in Theorem3.3 cannot beimproved. For arbitrary integer δ and positive integer n = 4δ, letG ′′ be the bipartitegraph obtained from the disjoint union of Kδ,δ and Kn/2−δ,n/2−δ by choosing anindependent set of size δ − 1 in each of the two graphs and adding a matchingbetween the vertices of these sets. Easily to know that G ′′ is a triangle-free graphwith n(G ′′) = n and minimum degree δ(G ′′) = δ. Furthermore, the Zeroth-orderRandic index

R0α(G ′′) = δα − δα+1 + (δ − 1)(δ + 1)α + (δ − 1)(n − 2δ + 2)α

2α

+ (2n − 5δ + 1)(n − 2δ)α

2α.

But we get λ(G ′′) ≤ δ(G ′′) − 1.

Acknowledgments This work is supported by the Natural Science Funds of China (No. 11071016 andNo: 11171129) and by the Beijing Natural Science Foundation (No. 1102015).

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