Bounds for the largest and the smallest Aα eigenvalues of a graph in terms of vertex degrees

Linear Algebra and its Applications 590 (2020) 210–223

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Bounds for the largest and the smallest Aα eigenvalues of a graph in terms of vertex degrees Sai Wang a,1 , Dein Wong a,∗,2 , Fenglei Tian b a

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China b School of Management, Qufu Normal University, Rizhao, 276826, Shandong, China

a r t i c l e

i n f o

Article history: Received 23 July 2019 Accepted 24 December 2019 Available online 3 January 2020 Submitted by R. Brualdi MSC: 05C50 Keywords: Aα -spectrum of graphs Spectral radius The smallest eigenvalue

a b s t r a c t Let G be a graph with adjacency matrix A(G) and with D(G) the diagonal matrix of its vertex degrees. Nikiforov defined the matrix Aα (G), with α ∈ [0, 1], as Aα (G) = αD(G) + (1 − α)A(G). The largest and the smallest eigenvalues of Aα (G) are respectively denoted by ρ(G) and λn (G). In this paper, we present a tight upper bound for ρ(G) in terms of the vertex degrees of G for α = 1. For any graph G,

ρ(G) ≤ max{

α(du + dw ) +

 α2 (du + dw )2 + 4(1 − 2α)du dw 2

u∼w

}.

If G is connected, for α ∈ [0, 1), the equality holds if and only if G is regular or bipartite semi-regular. As an application, we solve the problem raised by Nikiforov in [15]. For the smallest eigenvalue λn (G) of a bipartite graph G of order n with no isolated vertices, for α ∈ [0, 1), then α(du + dw ) − λn (G) ≥ min { u∼w



α2 (du + dw )2 + 4(1 − 2α)du dw 2

* Corresponding author. 1 2

E-mail addresses: [email protected] (S. Wang), [email protected] (D. Wong). The first author works at Xuhai College, China University of Mining and Technology. Supported by the Fundamental Research Funds for the Central Universities (No. 2018ZDPY06).

https://doi.org/10.1016/j.laa.2019.12.039 0024-3795/© 2020 Elsevier Inc. All rights reserved.

}.

S. Wang et al. / Linear Algebra and its Applications 590 (2020) 210–223

211

If G is connected and α = 1/2, the equality holds if and only if G is regular or semi-regular. © 2020 Elsevier Inc. All rights reserved.

1. Introduction All graphs considered here are simple and undirected. Let G be a graph with vertex set V (G) and edge set E(G). The neighborhood Nu of a vertex u consists of vertices v ∈ V (G) which are adjacent to u, and the number of vertices in Nu is the degree of u, which is written as du . Let G be a graph with adjacency matrix A(G), and with D(G) the diagonal matrix of its vertex degrees. Nikiforov [15] defined the matrix Aα (G), with α ∈ [0, 1], as Aα (G) = αD(G) + (1 − α)A(G). Setting α = 0, 12 or 1, one can obtain A(G) = A0 (G); Q(G) = 2A 12 (G); and D(G) = A1 (G). Since Aα (G) is the linear combinations of A(G) and D(G), it was claimed in [16] that the matrices Aα (G) can underpin a unified theory of A(G) and Q(G). The largest eigenvalue of Aα (G) is also known as Aα spectral radius of G. For trees, Nikiforov et al. [16] gave several interesting results about their Aα spectral radius. Recently, Nikiforov et al. [17] and Xue et al. [19] gave three edge graft transformations on Aα spectral radius independently. As applications, Xue et al. [19] determined the graphs with maximum (resp. minimum) Aα spectral radius among all connected graphs with given diameter (resp. clique number). More results on Aα spectra of graphs, one can see [2], [4], [8], [9], [10], [11], [12], [13], [14], [18]. Further, the cycle, the path, and the star all on n vertices are denoted by Cn , Pn and K1,n−1 . Anderson [1] showed that the signless laplacian spectral radius of a graph G is less than or equal to max{du + dw }, with equality holding if and only if G is regular or u∼w bipartite semi-regular. In this paper, we extend this result to Aα spectra of graphs, obtaining the following result. Theorem 1.1. For any graph G and α ∈ [0, 1). Then  ρ(G) ≤ max{ u∼w

α(du + dw ) +

2

α2 (du + dw ) + 4(1 − 2α)du dw 2

}.

(1)

If G is connected, the equality holds if and only if G is regular or bipartite semi-regular.

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For convenience, we denote the right side of (1) with N . Nikiforov in [15] showed, for any graph G ρ(G) ≤ max{αdu + (1 − α)dw },

(2)

u∼w

and raised the problem: Find all cases of α in the equality (2). The following result shows that the upper bound in (1) is better than that in (2) and gives the answer to the problem. Corollary 1.2. For any graph G and 0 ≤ α < 1, we have ρ(G) ≤ N ≤ max{αdu + (1 − α)dw }.

(3)

u∼w

If α = 1/2 and G is connected, the equality (2) holds if and only if G is regular or bipartite semi-regular. If α = 1/2 and G is connected, the equality (2) holds if and only if G is regular. The smallest eigenvalue of A(G), which is second in importance after the spectral radius, has numerous relations with the structure of the graph. To a great extent this is also true for the smallest eigenvalue λmin (Q(G)) of the signless laplacian matrix Q(G) (see, e.g., [5], [6], [7]). It is well known that the smallest eigenvalue of Q(G) is zero if and only if one component of G is bipartite. This is certainly not true for Aα (G) if α = 12 . Therefore, it is interesting to determine the smallest Aα eigenvalue λn (G) of a graph G of order n. In this paper, for α ∈ [0, 1), we give a lower bound for λn (G) when G is a bipartite graph of order n with no isolated vertices. Theorem 1.3. Let G be a bipartite graph with no isolated vertices and α ∈ [0, 1). Then λn (G) ≥ min{ u∼w

α(du + dw ) −

 2 α2 (du + dw ) + 4(1 − 2α)du dw 2

}.

(4)

If G is connected and α = 1/2, the equality holds if and only if G is regular or semiregular. The rest of this paper is organized as follows. In Section 2, we give proofs for Theorem 1.1 and Corollary 1.2. In Section 3, we define the matrix Lα (G) with α ∈ [0, 1] as Lα (G) = αD(G) + (α − 1)A(G), which is analogous as Aα (G). In this section, we study the connection between Lα (G) and Aα (G), proving that a graph G is bipartite if and only if the Aα spectra and the Lα spectra of G are equal. Also, we study the condition for Lα (G) to be positive semidefinite. As an application, we present a proof for Theorem 1.3.

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2. A tight upper bound for the largest Aα eigenvalue ρ(G) We begin with some fundamental properties of Aα (G). For a vector x on vertices of G, we denote by xu the entry of x at u ∈ V (G). If x is an eigenvector of Aα (G) with respect to an eigenvalue λ, then we have λxu = αdu xu + (1 − α)



xw , for u ∈ V (G).

w∼u

The quadratic form Aα (G) x, x can be represented as 

Aα (G)x, x = (2α − 1)

x2u du + (1 − α)

u∈V (G)



2

(xu + xw ) .

uw∈E(G)

The following lemma will be applied latter. Lemma 2.1. (Corollary 12, [15]) Let Δ be the maximal degree of G. If α ∈ [0, 12 ], then ρ(G) ≥ α(Δ + 1). If α ∈ [ 12 , 1), then ρ(G) ≥ αΔ + 1 − α. A graph G is called bipartite semi-regular if it is a bipartite graph and the degrees of vertices on the same chromatic set are constant. The largest and the smallest Aα eigenvalues of a bipartite semi-regular graph have bounds as follows. Lemma 2.2. Let G be a connected bipartite semi-regular graph with V (G) = U ∪ W be the bipartition of V (G) into two chromatic sets. Suppose du = r for u ∈ U and dw = s for w ∈ W . For the largest and the smallest eigenvalue of Aα (G), we have ρ (G) ≥

α (r + s) +

 2 α2 (r + s) + 4 (1 − 2α) rs 2

,

and λn (Aα (G)) ≤

α (r + s) −



2

α2 (r + s) + 4 (1 − 2α) rs 2

.

Proof. If G is bipartite semi-regular, then the vertices of Aα (G) has an equitable partition. The quotient matrix with respect to the equitable partition is 

rα (1 − α)r (1 − α)s sα

 .

It has eigenvalues α (r + s) ±



2

α2 (r + s) + 4 (1 − 2α) rs 2

,

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which interlace those of Aα (G) (see Corollary 2.4.4, [3]). Thus, we have the required inequalities. 2 Now, we are ready to give a proof for Theorem 1.1. Proof of Theorem 1.1. If G is connected, let x ∈ Rn be a positive unit eigenvector with respect to ρ(G). We have Aα (G)x = ρ(G)x. Suppose xu = max{xv : v ∈ V (G)}, xw = max{xv : v ∼ u}. Considering the entry of Aα (G)x at u, we get ρ(G)xu = αdu xu + (1 − α)



xv

v∼u

≤ αdu xu + (1 − α)du xw , which follows (ρ(G) − αdu )xu ≤ (1 − α)du xw .

(5)

Similarly, by considering the entry of Aα (G)x at w, we get ρ(G)xw = αdw xw + (1 − α)



xv

v∼w

≤ αdw xw + (1 − α)dw xu , and (ρ(G) − αdw )xw ≤ (1 − α)dw xu .

(6)

By Lemma 2.1, both ρ(G) − αdu and ρ(G) − αdw are positive. Combining (5) and (6), we have (ρ(G) − αdu )(ρ(G) − αdw ) ≤ (1 − α)2 du dw .

(7)

From this inequality we have

ρ(G) ≤

α(du + dw ) +

which proves the required inequality.

 2 α2 (du + dw ) + 4(1 − 2α)du dw 2

,

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Next, if ρ(G) attains the upper bound in (1), then all inequalities in above argument turn into equalities. In particular, we have xv = xw , ∀ v ∈ Nu ;

xv = xu , ∀ v ∈ Nw .

Let U = {v : xv = xu } and W = {v : xv = xw }. Then the following claims hold for G. Claim 1. V (G) = U ∪ W . It has been shown that Nw ⊆ U and Nu ⊆ W . A similar discussion shows that xz = xw and xr = xu for z ∈ Nu , r ∈ Nz . Further, we have xs = xw for s ∈ Nr , r ∈ Nz . Proceeding in this way, we have xv = xu or xv = xw for each v ∈ V (G), and thus V (G) = U ∪ W . Claim 2. If G contains an odd cycle, then G must be regular. In this case, one can easily obtain xv = xu for all v ∈ V (G), and thus we have ρ(G)xu = αdv xu + (1 − α)dv xu , ∀ v ∈ V (G), which gives dv = ρ(G), ∀ v ∈ V (G). Hence, G is regular. Claim 3. If G is a bipartite graph, then G is regular or semi-regular. If xu = xw , then a similar discussion as that in Claim 2 shows that G is regular. Suppose xu = xw . For any v ∈ U , since xv = xu and xz = xw for z ∈ Nv , we have ρ(G)xu = αdv xu + (1 − α)dv xw , which leads to dv =

ρ(G)xu . αxu + (1 − α)xw

Similarly, we have dv =

ρ(G)xw , ∀ v ∈ W. αxw + (1 − α)xu

Hence, G is regular or semi-regular. Therefore, if ρ(G) attains the upper bound, then G is either a regular graph or a bipartite semi-regular graph. Conversely, if G is a Δ-regular graph, then  α(du + dw ) +

2

α2 (du + dw ) + 4(1 − 2α)du dw 2

=Δ

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for any adjacent vertices u, w. Also, we have Aα (G)1 = Δ1, where 1 is the all-ones vector, which implies that ρ(G) = Δ and thus ρ(G) attains the upper bound. Suppose G is a bipartite semi-regular graph with V (G) = U ∪W the bipartite partition of V (G), where dv = r for v ∈ U and dv = s for v ∈ W . The upper bound for ρ(G), which has been proved, implies that ρ(G) ≤

α(r + s) +



α2 (r + s)2 + 4(1 − 2α)rs . 2

Combining this inequality with that obtained in Lemma 2.2, we derive the required equality for ρ(G). If G is not connected, then G is consisted of some components and isolate vertices. Suppose G1 , G2 , . . . , Gs are connected components of G, we have ρ(G) = max{ρ(G1 ), · · · , ρ(Gs )}. As proof above, the inequality holds for each component as that in (1). Hence the inequality also holds for G. 2 As an application of Theorem 1.1, we next give the proof of Corollary 1.2. Let M = max{αdu + (1 − α)dw , (1 − α)du + αdw }. u∼w

Clearly M is equal to the right side of (2), and is maximized for uw ∈ E(G). Proof of Corollary 1.2. Without loss of generality, we suppose du ≥ dw for each uw ∈ E(G). If 1/2 < α < 1, then for each uw ∈ E(G), max{αdu + (1 − α)dw , (1 − α)du + αdw } = αdu + (1 − α)dw . Consider the difference αdu + (1 − α)dw −

α(du + dw ) +

 α2 (du + dw )2 + 4(1 − 2α)du dw . 2

Namely αdu + (2 − 3α)dw −

 α2 (du + dw )2 + 4(1 − 2α)du dw . 2

Note that αdu + (2 − 3α)dw ≥ αdw + (2 − 3α)dw = 2(1 − α)dw ≥ 0, by (αdu + (2 − 3α)dw )2 − α2 (du + dw )2 − 4(1 − 2α)du dw = 4(1 − α)(2α − 1)dw (du − dw ) ≥ 0,

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we obtain α(du + dw ) +

 α2 (du + dw )2 + 4(1 − 2α)du dw ≤ αdu + (1 − α)dw , 2

for each uw ∈ E (G). By the definition of M , we get ρ(G) ≤ N ≤ M . If 0 ≤ α < 1/2, then for each uw ∈ E (G), max{αdu + (1 − α)dw , (1 − α)du + αdw } = (1 − α)du + αdw . By analogous discussion as above, we have ((2 − 3α)du + αdw )2 − α2 (du + dw )2 − 4(1 − 2α)du dw = 4(1 − α)(1 − 2α)du (du − dw ) ≥ 0, and α(du + dw ) +



α2 (du + d2w + 4(1 − 2α)du dw ≤ (1 − α)du + αdw , 2

for each uw ∈ E (G). We also have ρ(G) ≤ N ≤ M . Hence the inequality (3) holds for α ∈ [0, 1). If G is connected, from Theorem 1.1, the first equality holds in (1) if and only if G is regular or bipartite semi-regular. The equality (2) holds if and only if both equalities in (3) hold. So the second equality in (3) would hold only if G is regular or bipartite semi-regular. If G is bipartite semi-regular and α = 1/2, the second equality in (3) does not hold. Hence G must be regular for α ∈ [0, 1) and α = 1/2. If α = 1/2, the equality (2) holds if and only if G is regular or bipartite semi-regular by Anderson [1]. 2 3. A lower bound for the smallest Aα eigenvalue of G In this section, we study the smallest eigenvalue of Aα (G). To achieve the goal of Theorem 1.3, we introduce a matrix Lα (G) for G, which is analogous to the definition of Aα (G). For α ∈ [0, 1], define Lα (G) as Lα (G) = αD(G) + (α − 1)A(G). It is clear that Lα (G) = −A(G) if α = 0, and Lα (G) is essentially equivalent to the Laplacian matrix L(G) if α = 12 , i.e., L 12 (G) = 12 L(G). We write the eigenvalues of Lα (G) in decreasing order as μ1 (Lα (G))  μ2 (Lα (G))  · · ·  μn (Lα (G)).

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The largest eigenvalue μ1 (Lα (G)) is called the Lα -spectral radius of G. If x ∈ Rn is an eigenvector of Lα (G) with respect to an eigenvalue μ, then the eigenequations of Lα (G) can be written as μxu = αdu xu + (α − 1)



xw , ∀ u ∈ V (G).

(8)

w∼u

The quadratic form Lα (G)x, x can be represented as 

Lα (G)x, x = (2α − 1)



x2u du + (1 − α)

u∈V (G)

2

(xu − xw ) .

uw∈E(G)

Since Lα (G) is a real symmetric matrix, by Rayleigh principle, it follows that μ(G) := μ1 (Lα (G)) = max Lα (G), x; x=1

μn (G) := μn (Lα (G)) = min Lα (G), x. x=1

The following lemma reveals the relation between Lα (G) and Aα (G), which generalizes a result about the relation between L(G) and Q(G). Lemma 3.1. If G is a connected graph, α ∈ [0, 1), then μ(G) ≤ ρ(G), the equality holds if and only if G is a bipartite graph. Proof. Suppose the vertex set of G is {v1 , v2 , . . . , vn }. Let x = (x1 , x2 , . . . , xn )T ∈ Rn be a unit eigenvector of Lα (G) with respect to μ(G) and y = (y1 , y2 , . . . , yn )T ∈ Rn be a unit eigenvector of Aα (G) with respect to ρ(G). Thus 

μ(G) = (2α − 1)



2

(xi − xj ) ,

vi ∼vj ,i
vi ∈V (G)

ρ(G) = (2α − 1)



x2i dvi + (1 − α)



yi2 dvi + (1 − α)

2

(yi + yj ) .

vi ∼vj ,i
ui ∈V (G)

Let |x| = (|x1 |, |x2 |, . . . , |xn |)T . Then μ(G) = (2α − 1)

 vi ∈V (G)

≤ (2α − 1)



vi ∈V (G)



x2i dvi + (1 − α)

(xi − xj )

2

vi ∼vj ,i
|xi |2 dvi + (1 − α)



2

(|xi | + |xj |)

vi ∼vj ,i
= Aα (G), |x| ≤ ρ(G). This proves the required inequality. If μ(G) = ρ(G), all inequalities involved turn into equalities. By Perron-Frobenius theorem, |x| is the Perron vector y of Aα (G) and thus xi = 0 for all i. From

S. Wang et al. / Linear Algebra and its Applications 590 (2020) 210–223

(2α − 1)



x2i di + (1 − α)

=(2α − 1)

2

(xi − xj )

vi ∼vj ,i
vi ∈V (G)





219

|xi |2 di + (1 − α)



2

(|xi | + |xj |) ,

vi ∼vj ,i
vi ∈V (G)

it follows (xi − xj )2 = (|xi | + |xj |)2 for vi ∼ vj . Therefore xi xj < 0 for adjacent vertices vi and vj . Let U = {vi : xi > 0};

W = {vj : xj < 0}.

Thus, each edge e ∈ E(G) has one endpoint in U and another endpoint in W . This implies that G is a bipartite graph. If G is a bipartite graph with two chromatic sets U and W , let D = diag(−Ir , Is ) be a diagonal matrix with r = |U |, s = |W |. Then Lα (G) = DAα (G)D−1 and thus Lα (G) and Aα (G) have the same spectrum. In this case, we have μ(G) = ρ(G). 2 Corollary 3.2. A graph G is bipartite if and only if Lα (G) and Aα (G) have the same spectrum. Nikiforov and Rojo [18] studied the positive semidefiniteness of Aα (G) and obtained the smallest α for which Aα (G) is positive semidefinite. When G is connected, we shall show Lα (G) is not positive semidefinite for any α ∈ [0, 12 ). Two well known results will be applied in our proof. Lemma 3.3. (Theorem 2.6.1, [3]) Let A and B be Hermitian matrices of order n, and let their eigenvalues be indexed in decreasing order. If 1 ≤ k ≤ n, then λk (A) + λn (B) ≤ λk (A + B) ≤ λk (A) + λ1 (B). Lemma 3.4. (Proposition 1.4.4, [3]) Let G be a connected graph of order n. Then the smallest signless laplacian eigenvalue of G is 0 if and only if G is bipartite. Theorem 3.5. Let G be a connected graph of order n, and let α0 (G) be the smallest α such that Lα (G) is positive semidefinite. Then α0 (G) = 12 . Proof. Let x ∈ Rn be a nonzero vector. If α > 12 , then for any edge uw ∈ E(G), we see that 2

Lα (G) x, x ≥ (1 − α) (xu − xw ) + (2α − 1) x2u + (2α − 1) x2w ≥ 0

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Hence Lα (G) is positive semidefinite. Select a vertex u with xu = 0 and let uw ∈ E(G). Then the above inequality is strict and thus Lα (G) is positive definite. If G is bipartite, by Corollary 3.2 in this paper and Corollary 7 in [18], we have α0 (G) = 12 . When G is not bipartite, let 1 ≥ α > β ≥ 0. By the definition of Lα (G), we have Lα (G) − Lβ (G) = (α − β) Q (G) . By Lemma 3.3, μn (Lα (G)) − μn (Lβ (G)) ≥ (α − β) λmin Q (G) . Lemma 3.4 implies that λmin (Q(G)) > 0. Thus we have μn (Lα (G)) − μn (Lβ (G)) > 0. Setting α =

1 2

> β ≥ 0, we have   μn L 12 (G) − μn (Lβ (G)) > 0.

Since μn (L 12 (G)) = 12 (λmin (L(G)) = 0, we have μn (Lβ (G)) < 0 for β ∈ [0, 12 ). In this case, we also have α0 (G) = 12 . 2 Corollary 3.6. Let G be a connected graph. Then μn (G) < 0 if α ∈ [0, 12 ), and μn (G) > 0 if α ∈ ( 12 , 1). Lemma 3.7. Let G be a graph of order n with minimum degree δ. Then μn (G) < αδ for α ∈ [0, 1). Proof. Let V (G) = {v1 , v2 , . . . , vn } and let x = (x1 , x2 , . . . , xn )T be a nonzero vector in Rn corresponding to the vertices of G. We have

μn (G) ≤

Lα (G), x = xxT

(2α − 1)

vi ∈V (G)

x2i dvi + (1 − α) n

vi ∼vj ,i
2

(xi − xj )

2 i=1 xi

Putting x = (0, 0, . . . , 1)T , we get μn (G) ≤ αδ. Since x = (0, 0, . . . , 1)T is not an eigenvector with respect to μn (G), the inequality is strict. 2 Lemma 3.8. Let G be a connected bipartite graph of order n, and let x be a unit eigenvector of Lα (G), where α ∈ [0, 1), with respect to μn (G). Then xu xw > 0 for any u, w ∈ V (G).

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Proof. Let V (G) = U ∪ W be the bipartition of V (G) into two chromatic sets. Let |x| be the unit vector such that |x|v = |xv | for all v ∈ V (G). We assert that xu xw ≥ 0 for each edge uw ∈ E(G). Assume to the contrary that G contains an edge uw such that xu xw < 0. We have (|x|u − |x|w )2 < (xu − xw )2 , which is a contradiction, as μn (G) ≤ Lα (G) , |x| < Lα (G) , x = μn (G) . If xu = 0 for a vertex u, then (α − 1)



xw = 0 and thus xw = 0 for all w adjacent

w∼u

to u (noting that xw xu ≥ 0 if w ∼ u), which leads to x = 0 (as G is connected), a contradiction. This proves that xu =  0 for all vertices of G. Finally, we have xu xw > 0 for all u, w ∈ V (G). 2 With the above lemmas in hand, we are ready to give a proof for Theorem 1.3. Proof of Theorem 1.3. Suppose G is connected. Since G is bipartite, by Corollary 3.2, it suffices to give a proof for μn (G). By Lemma 3.8, we can choose a positive unit eigenvector x with respect to μn (G). Suppose xu = max{xv : v ∈ V (G)};

xw = max{xv : v ∼ u}.

Considering the entry of (μn (G) − Lα (G))x at u, we have (μn (G) − αdu )xu = (α − 1)



xv ≥ (α − 1)du xw .

(9)

v∼u

Considering the entry of (μn (G) − Lα (G))x at w, we have (μn (G) − αdw )xw = (α − 1)



xv ≥ (α − 1)dw xu .

(10)

v∼w

In view of Lemma 3.7, both μn (G) − αdu and μn (G) − αdw are negative. Multiplying (9) and (10), we have (μn (G) − αdu )(μn (G) − αdw ) ≤ (1 − α)2 du dw . From this inequality, we obtain the required inequality for μn (G). Now, for α = 1/2, suppose λn (G) = μn (G) attains the lower bound. Then all inequalities in the above argument turn into equalities. In particular, from (9) and (10) we have xv = xw for v ∈ Nu and xv = xu for v ∈ Nw . Let

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U = {v : xv = xu }; W = {v : xv = xw }. By analogous discussion as that in the proof of Theorem 1.1, we have V (G) = U ∪ W . If xu = xw , we have μn (G)xu = αdv xu + (α − 1)dv xu , ∀ v ∈ V (G), which gives dv =

μn (G) , ∀ v ∈ V (G). 2α − 1

Hence, G is regular. If xu = xw , we have μn (G)xu = αdv xu + (α − 1)dv xw , ∀ v ∈ U, which gives dv =

μn (G)xu , ∀ v ∈ U. αxu + (α − 1)xw

dv =

μn (G)xw , ∀ v ∈ W. αxw + (α − 1)xu

Similarly,

Hence, G is semi-regular. Conversely, λn (G) obviously attains the lower bound if G is regular. Now, suppose G is semi-regular with bipartition V (G) = U ∪ W , where dv = r for v ∈ U and dv = s for v ∈ W . The lower bound for λn (G) implies

λn (G) ≥

α (r + s) −

 2 α2 (r + s) + 4 (1 − 2α) rs 2

.

Combining this inequality with one inequality in Lemma 2.2 we have

λn (G) =

α (r + s) −

 2 α2 (r + s) + 4 (1 − 2α) rs 2

.

Thus, the equality holds for a connected bipartite semi-regular graph. If G is not connected, suppose G1 , G2 , . . . , Gs are connected components of G, we have λn (G) = min{λ(G1 ), · · · , λ(Gs )}.

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