On the spectral radius of bipartite graphs which are nearly complete

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R E S E A R C H

Open Access

On the spectral radius of bipartite graphs

which are nearly complete

Kinkar Chandra Das

1

_{, Ismail Naci Cangul}

2*

_{, Ayse Dilek Maden}

3

_{and Ahmet Sinan Cevik}

3

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey

Full list of author information is available at the end of the article

Abstract

Forp,q,r,s,t∈Z+_with_rt_≤_p_and_st_≤_q_{, let}_G₌_G₍_p_,_q_;_r_,_s_;_t_{) be the bipartite graph} with partite setsU={u1,. . .,up}andV={v1,. . .,vq}such that any two edgesuiandvj are not adjacent if and only if there exists a positive integerkwith 1≤k≤tsuch that (k– 1)r+ 1≤i≤krand (k– 1)s+ 1≤j≤ks. Under these circumstances, Chenet al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture:

Assume thatp≤q,k<p,|U|=p,|V|=qand|E(G)|=pq–k. Then whether it is true that

λ

1(G)≤

λ

1

(

G(p,q;k, 1; 1)

)

=

pq–k+p2_q2_{– 6}_pqk_{+ 4}_pk_{+ 4}_qk2_{– 3}_k2

2 .

In this paper, we prove this conjecture for the range minvh∈V{degvh} ≤

p–1 2 .

MSC: 05C05; 05C50

Keywords: bipartite graph; adjacency matrix; spectral radius

1 Introduction

LetGbe a (simple) graph with the vertex and edge sets given byV(G) ={v,v, . . . ,vn}and

E(G) ={vivj|viandvjare adjacent}, respectively. Theadjacency matrixofGonnvertices

is ann×nmatrixA(G) whose entriesaijare given by

aij=

⎧ ⎨ ⎩

; ifvivj∈E(G), ; otherwise.

SinceA(G) is symmetric, all the eigenvalues ofA(G) are real. In fact, the eigenvalues of

A(G) are calledeigenvalues of the graph G. We can list the eigenvalues of the graphGin a non-increasing order as follows:

λ(G)≥λ(G)≥ · · · ≥λn–(G)≥λn(G).

The largest eigenvalueλ(G) is often called thespectral radiusofG.

Throughout this paper, we will consider only ﬁnite, simple, undirected, bipartite graphs. So, let us suppose thatG= (U∪V,E) is such a bipartite graph, whereU={u,u, . . . ,up},

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U×V. As a usual notation, thedegreesof verticesui∈Uandvj∈V will be denoted by

deguianddegvj, respectively. For the integersp,q,r,s,t∈Z+_satisfying_rt_≤_p_and_st_≤_q_,

let us denote the bipartite graphGbyG(p,q;r,s;t) with the above partite setsUandVsuch thatui∈Uandvj∈V arenot adjacentif and only if there exists ak∈Z+with ≤k≤t

such that (k– )r+ ≤i≤krand (k– )s+ ≤j≤ks.

In the literature, upper bounds for the spectral radius in terms of various parameters for unweighted and weighted graphs have been widely investigated [–]. As a special case, in [], Chenet al.studied the spectral radius of bipartite graphs which are close to a complete bipartite graph. For partite setsUandV having|U|=p,|V|=qandp≤q, in the same reference, the authors also gave an aﬃrmative answer to the conjecture [, Conjecture .] by taking|E(G)|=pq–  into account of a bipartite graph. Furthermore, reﬁning the same conjecture for the number of edges is at leastpq–p+ , there still exists the following conjecture.

Conjecture [] For positive integers p,q and k satisfying p≤q and k<p,let G be a bipartite graph with partite sets U and V having|U|=p and|V|=q,and|E(G)|=pq–k.

Then

λ(G)≤λG(p,q;k, ; )=

pq–k+p_q_{– }_pqk_{+ }_pk_{+ }_qk_{– }_k

 .

We note that similar conjectures in this topic have been resolved by the ﬁrst author in the papers [–]. In here, as the main goal, we present the proof of Conjecture  for the rangemin_v_h_∈_V{degvh} ≤ p–_ .

2 Main result

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

Lemma [] Letλbe the spectral radius of the bipartite graph G(p,q;k, ; ).Then

λ=

pq–k+p_q_{– }_pqk_{+ }_pk_{+ }_qk_{– }_k

 .

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

Theorem  For positive integers p,q and k satisfying p≤q and k<p,let G be a bipar-tite graph with parbipar-tite sets U and V having|U|=p and|V|=q,and|E(G)|=pq–k.If

minvh∈V{degvh} ≤

p–  ,then

λ(G)≤

pq–k+p_q_{– }_pqk_{+ }_pk_{+ }_qk_{– }_k

 ()

with equality if and only if G∼=G(p,q;k, ; ).

Proof LetZ= (x,x, . . . ,xp,y,y, . . . ,yq)Tbe an eigenvector ofA(G) corresponding to an

eigenvalueλ(G). For the setsUandV, letxi=max≤h≤pxhandyj=max≤h≤qyh,

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have

p–  

≥min

vh∈V{

degvh}=degv=d (say).

Now,

A(G)Z=λ(G)Z. ()

Considering (), we get

λ(G)xi≤(q– )yj+y forui∈U ()

and

λ(G)y≤dxi forv∈V. ()

However, from () and (), we clearly obtain

λ_(G)y≤d

(q– )yj+y

, which can be written shortly as

λ_(G) –d

y≤(q– )dyj. ()

Sincevis the vertex with the minimum degreedinV and the total number of edges

in bipartite graphGispq–k, we have

p

h=

λ(G)xh≤(pq–k–d)yj+dy. ()

Forvj∈V, from () we get

λ(G)yj=

uh:uhvj∈E

xh.

In other words, by (),

λ_(G)yj=

uh:uhvj∈E

λ(G)xh≤

p

h=

λ(G)xh≤(pq–k–d)yj+dy,

that is,

λ_(G) –pq+k+d

yj≤dy. ()

From () and (), we get

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that is,

λ(G)≤

pq–k+p_q_{– }_pqk₊_k_{– }_pqd

+ kd+ qd

 . ()

Let us consider a function

f(x) = qx+ kx– pqx, wherex≤ p– 



.

Then

f (x) = –q

p–k

q– x

< , asx≤ p– 



andk<p≤q.

Thusf(x) is a decreasing function on ≤x≤ p–_ . Sincep–k≤d≤ p–_ , from (), we

get the required result ().

Suppose now that equality holds in (). Then all inequalities in the above argument must become equalities. Thus we haved=p–k. From the equality in (), we get

yh=yj, h= , , . . . ,q and

uivh∈E, h= , , . . . ,q. From the equality in (), we get

xh=xi, h=p–d+ ,p–d+ , . . . ,p and

uhv∈E, h=p–d+ ,p–d+ , . . . ,p.

From the equality in (), we get

yh=yj, h= , , . . . ,q and

uhvj∈E, h= , , . . . ,p,j= , , . . . ,q.

Hence we conclude thatG∼=G(p,q;k, ; ).

Conversely, by Lemma , one can easily see that the equality holds in () for the graph

G(p,q;k, ; ).

Remark  In Theorem , we proved Conjecture  for the rangeminvh∈V{degvh} ≤

p–  .

However, this conjecture is still open for the rangep–_ <minvh∈V{degvh}<p.

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.

Author details

1_{Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea.}2_{Department of}

Mathematics, Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey.3_{Department of}

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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.

Received: 19 December 2012 Accepted: 24 February 2013 Published: 21 March 2013 References

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