Skip to main content

On the spectral radius of bipartite graphs which are nearly complete

Abstract

For p,q,r,s,t Z + with rtp and stq, let G=G(p,q;r,s;t) be the bipartite graph with partite sets U={ u 1 ,, u p } and V={ v 1 ,, v q } such that any two edges u i and v j are not adjacent if and only if there exists a positive integer k with 1kt such that (k1)r+1ikr and (k1)s+1jks. Under these circumstances, Chen et al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture:

Assume that pq, k<p, |U|=p, |V|=q and |E(G)|=pqk. Then whether it is true that

λ 1 (G) λ 1 ( G ( p , q ; k , 1 ; 1 ) ) = p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2 .

In this paper, we prove this conjecture for the range min v h V {deg v h } p 1 2 .

MSC:05C05, 05C50.

1 Introduction

Let G be a (simple) graph with the vertex and edge sets given by V(G)={ v 1 , v 2 ,, v n } and E(G)={ v i v j v i  and  v j  are adjacent}, respectively. The adjacency matrix of G on n vertices is an n×n matrix A(G) whose entries a i j are given by

a i j ={ 1 ; if  v i v j E ( G ) , 0 ; otherwise .

Since A(G) is symmetric, all the eigenvalues of A(G) are real. In fact, the eigenvalues of A(G) are called eigenvalues of the graph G. We can list the eigenvalues of the graph G in a non-increasing order as follows:

λ 1 (G) λ 2 (G) λ n 1 (G) λ n (G).

The largest eigenvalue λ 1 (G) is often called the spectral radius of G.

Throughout this paper, we will consider only finite, simple, undirected, bipartite graphs. So, let us suppose that G=(UV,E) is such a bipartite graph, where U={ u 1 , u 2 ,, u p }, V={ v 1 , v 2 ,, v q } are two sets of vertices and E is the set of edges defined as a subset of U×V. As a usual notation, the degrees of vertices u i U and v j V will be denoted by deg u i and deg v j , respectively. For the integers p,q,r,s,t Z + satisfying rtp and stq, let us denote the bipartite graph G by G(p,q;r,s;t) with the above partite sets U and V such that u i U and v j V are not adjacent if and only if there exists a k Z + with 1kt such that (k1)r+1ikr and (k1)s+1jks.

In the literature, upper bounds for the spectral radius in terms of various parameters for unweighted and weighted graphs have been widely investigated [110]. As a special case, in [3], Chen et al. studied the spectral radius of bipartite graphs which are close to a complete bipartite graph. For partite sets U and V having |U|=p, |V|=q and pq, in the same reference, the authors also gave an affirmative answer to the conjecture [[11], Conjecture 1.2] by taking |E(G)|=pq2 into account of a bipartite graph. Furthermore, refining the same conjecture for the number of edges is at least pqp+1, there still exists the following conjecture.

Conjecture 1 [3]

For positive integers p, q and k satisfying pq and k<p, let G be a bipartite graph with partite sets U and V having |U|=p and |V|=q, and |E(G)|=pqk. Then

λ(G)λ ( G ( p , q ; k , 1 ; 1 ) ) = p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2 .

We note that similar conjectures in this topic have been resolved by the first author in the papers [1216]. In here, as the main goal, we present the proof of Conjecture 1 for the range min v h V {deg v h } p 1 2 .

2 Main result

The following lemma will be needed for the proof of our main result.

Lemma 1 [3]

Let λ 1 be the spectral radius of the bipartite graph G(p,q;k,1;1). Then

λ 1 = p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2 .

We now present an upper bound on the spectral radius of the bipartite graph G.

Theorem 1 For positive integers p, q and k satisfying pq and k<p, let G be a bipartite graph with partite sets U and V having |U|=p and |V|=q, and |E(G)|=pqk. If min v h V {deg v h } p 1 2 , then

λ 1 (G) p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2
(1)

with equality if and only if GG(p,q;k,1;1).

Proof Let Z= ( x 1 , x 2 , , x p , y 1 , y 2 , , y q ) T be an eigenvector of A(G) corresponding to an eigenvalue λ 1 (G). For the sets U and V, let x i = max 1 h p x h and y j = max 1 h q y h , respectively. Also, let us suppose that v 1 is the vertex having minimum degree in V. Then we have

p 1 2 min v h V {deg v h }=deg v 1 = d 1 (say).

Now,

A(G)Z= λ 1 (G)Z.
(2)

Considering (2), we get

λ 1 (G) x i (q1) y j + y 1 for  u i U
(3)

and

λ 1 (G) y 1 d 1 x i for  v 1 V.
(4)

However, from (3) and (4), we clearly obtain

λ 1 2 (G) y 1 d 1 [ ( q 1 ) y j + y 1 ] ,

which can be written shortly as

( λ 1 2 ( G ) d 1 ) y 1 (q1) d 1 y j .
(5)

Since v 1 is the vertex with the minimum degree d 1 in V and the total number of edges in bipartite graph G is pqk, we have

h = 1 p λ 1 (G) x h (pqk d 1 ) y j + d 1 y 1 .
(6)

For v j V, from (2) we get

λ 1 (G) y j = u h : u h v j E x h .

In other words, by (6),

λ 1 2 (G) y j = u h : u h v j E λ 1 (G) x h h = 1 p λ 1 (G) x h (pqk d 1 ) y j + d 1 y 1 ,

that is,

( λ 1 2 ( G ) p q + k + d 1 ) y j d 1 y 1 .
(7)

From (5) and (7), we get

λ 1 4 (G)(pqk) λ 1 2 (G)+ d 1 (pqkq d 1 )0,

that is,

λ 1 (G) p q k + p 2 q 2 2 p q k + k 2 4 p q d 1 + 4 k d 1 + 4 q d 1 2 2 .
(8)

Let us consider a function

f(x)=4q x 2 +4kx4pqx,where x p 1 2 .

Then

f (x)=4q ( p k q 2 x ) <0,as x p 1 2  and k<pq.

Thus f(x) is a decreasing function on 1x p 1 2 . Since pk d 1 p 1 2 , from (8), we get the required result (1).

Suppose now that equality holds in (1). Then all inequalities in the above argument must become equalities. Thus we have d 1 =pk. From the equality in (3), we get

From the equality in (4), we get

From the equality in (7), we get

Hence we conclude that GG(p,q;k,1;1).

Conversely, by Lemma 1, one can easily see that the equality holds in (1) for the graph G(p,q;k,1;1). □

Remark 1 In Theorem 1, we proved Conjecture 1 for the range min v h V {deg v h } p 1 2 . However, this conjecture is still open for the range p 1 2 < min v h V {deg v h }<p.

References

  1. 1.

    Berman A, Zhang XD: On the spectral radius of graphs with cut vertices. J. Comb. Theory, Ser. B 2001, 83: 233–240. 10.1006/jctb.2001.2052

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brualdi RA, Hoffman AJ:On the spectral radius of a (0,1) matrix. Linear Algebra Appl. 1985, 65: 133–146.

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chen YF, Fu HL, Kim IJ, Stehr E, Watts B: On the largest eigenvalues of bipartite graphs which are nearly complete. Linear Algebra Appl. 2010, 432: 606–614. 10.1016/j.laa.2009.09.008

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cvetković D, Doob M, Sachs H: Spectra of Graphs. Academic Press, New York; 1980.

    Google Scholar 

  5. 5.

    Cvetković D, Rowlinson P: The largest eigenvalue of a graph: a survey. Linear Multilinear Algebra 1990, 28: 3–33. 10.1080/03081089008818026

    Article  Google Scholar 

  6. 6.

    Das KC, Kumar P: Bounds on the greatest eigenvalue of graphs. Indian J. Pure Appl. Math. 2003, 34(6):917–925.

    MathSciNet  Google Scholar 

  7. 7.

    Das KC, Kumar P: Some new bounds on the spectral radius of graphs. Discrete Math. 2004, 281: 149–161. 10.1016/j.disc.2003.08.005

    MathSciNet  Article  Google Scholar 

  8. 8.

    Das KC, Bapat RB: A sharp upper bound on the spectral radius of weighted graphs. Discrete Math. 2008, 308: 3180–3186. 10.1016/j.disc.2007.06.020

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hong Y: Bounds of eigenvalues of graphs. Discrete Math. 1993, 123: 65–74. 10.1016/0012-365X(93)90007-G

    MathSciNet  Article  Google Scholar 

  10. 10.

    Stanley RP: A bound on the spectral radius of graphs with e edges. Linear Algebra Appl. 1987, 67: 267–269.

    Article  Google Scholar 

  11. 11.

    Bhattacharya A, Friedland S, Peled UN: On the first eigenvalue of bipartite graphs. Electron. J. Comb. 2008., 15: Article ID #R144

    Google Scholar 

  12. 12.

    Das KC: On conjectures involving second largest signless Laplacian eigenvalue of graphs. Linear Algebra Appl. 2010, 432: 3018–3029. 10.1016/j.laa.2010.01.005

    MathSciNet  Article  Google Scholar 

  13. 13.

    Das KC: Conjectures on index and algebraic connectivity of graphs. Linear Algebra Appl. 2010, 433: 1666–1673. 10.1016/j.laa.2010.06.012

    MathSciNet  Article  Google Scholar 

  14. 14.

    Das KC: Proofs of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs. Linear Algebra Appl. 2011, 435: 2420–2424. 10.1016/j.laa.2010.12.018

    MathSciNet  Article  Google Scholar 

  15. 15.

    Das KC: Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs. Discrete Math. 2012, 312: 992–998. 10.1016/j.disc.2011.10.030

    MathSciNet  Article  Google Scholar 

  16. 16.

    Das KC: Proof of conjectures on adjacency eigenvalues of graphs. Discrete Math. 2013, 313(1):19–25. 10.1016/j.disc.2012.09.017

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The first author is supported by BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea, and the other authors are partially supported by Research Project Offices of Uludag (2012-15 and 2012-19) and Selcuk Universities.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ismail Naci Cangul.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. Moreover, all authors read and approved the final manuscript.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Das, K.C., Cangul, I.N., Maden, A.D. et al. On the spectral radius of bipartite graphs which are nearly complete. J Inequal Appl 2013, 121 (2013). https://doi.org/10.1186/1029-242X-2013-121

Download citation

Keywords

  • bipartite graph
  • adjacency matrix
  • spectral radius