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A new upper bound on the largest normalized Laplacian eigenvalue

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A new upper bound on the largest normalized Laplacian eigenvalue
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     P   R   E   P   R   I   N   T OperatorsandMatrices Preprint A NEW UPPER BOUND ON THE LARGESTNORMALIZED LAPLACIAN EIGENVALUE O SCAR  R OJO AND  R ICARDO  L. S OTO( Communicated by Richard A. Brualdi )  Abstract.  Let  G   be a simple undirected connected graph on  n  vertices. Suppose that the verticesof   G   are labelled 1 , 2 ,... , n .  Let  d  i  be the degree of the vertex  i .  The Randi´c matrix of   G   ,denoted by  R ,  is the  n × n  matrix whose  ( i ,  j ) − entry is  1 √  d  i d   j if the vertices  i  and  j  areadjacent and 0 otherwise. The normalized Laplacian matrix of   G   is  L    =  I  −  R ,  where  I   is the n × n  identity matrix. In this paper, by using an upper bound on the maximum modulus of thesubdominant Randi´c eigenvalues of   G   , we obtain an upper bound on the largest eigenvalue of  L   .  We also obtain an upper bound on the largest modulus of the negative Randi´c eigenvaluesand, from this bound, we improve the previous upper bound on the largest eigenvalue of   L   . 1. Introduction Let  G   = ( V  ,  E  )  be a simple undirected graph on  n  vertices. Some matrices on  G  are the adjacency matrix  A ,  the Laplacian matrix  L  =  D −  A  and the signless Laplacianmatrix  Q  =  D +  L ,  where  D  is the diagonal matrix of vertex degrees. It is well knownthat  L  and  Q  are positive semidefinite matrices and that  ( 0 , 1 )  is an eigenpair of   L where  1  is the all ones vector. Fiedler [16] proved that  G   is a connected graph if andonly if the second smallest eigenvalue of   L  is positive. This eigenvalue is called thealgebraic connectivity of   G   . The signless Laplacian matrix has recently attracted theattention of several researchers. Recent papers on this matrix are [5, 6, 7, 8, 9] andsome of its basic properties [6] are:1. For a connected graph, the smallest eigenvalue of   Q  is equal to 0 if and onlyif the graph is bipartite. In this case, 0 is a simple eigenvalue. Then, for aconnected graph, the smallest eigenvalue of   Q  is positive if and only if the graphis not bipartite.2. If   G   is a bipartite graph then  Q  and  L  have the same characteristic polynomial.  Mathematics subject classification  (2010): 05C50, 15A48. Keywords and phrases : normalized Laplacian matrix, Randi´c matrix, upper bound, largest eigenvalue,subdominant eigenvalue. This research was supported by Project Fondecyt 1100072, Chile. It was finished when the first author was a visitorat the Department of Mathematics, Universidade de Aveiro, Aveiro, Portugal.c      , ZagrebPaper OaM-0594 1     P   R   E   P   R   I   N   T 2  O. R OJO AND  R. L. S OTO Othermatricesonthegraph  G   arethenormalizedLaplacianmatrixandtheRandi´cmatrixof   G  .  Supposethattheverticesof   G   arelabelled 1 , 2 ,..., n .  Let  d  i  bethedegreeof the vertex  i .  Let  D − 12  be the diagonal matrix whose diagonal entries are1 √  d  1 , 1 √  d  2 ,... , 1 √  d  n whenever  d  i  = 0 .  If   d  i  = 0 for some  i  then the correspondingdiagonalentry of   D − 1   2  isdefinedtobe 0 .  ThenormalizedLaplacianmatrixof   G   , denotedby  L    , was introducedby F. Chung [15] as L    =  D − 12  LD − 12 =  I  −  D − 12  AD − 12 .  (1)The eigenvalues of   L    are called the normalized Laplacian eigenvalues of   G  . From  ( 1 ) , we have  D 12 L    D 12 =  D −  A =  L and thus  D 12 L    D 12 1  =  L 1  =  0 1 . Hence 0 is an eigenvalue of   L    with eigenvector  D 12 1 . We recall the following results on  L    [15] :1. The eigenvalues of   L    lie in the interval  [ 0 , 2 ] . 2. 0 is a simple eigenvalue of   L    if and only if   G   is connected.3. 2 is an eigenvalue of   L    if and only if a connected component of   G   is bipartiteand nontrivial.Among papers on  L    , we mention [10, 11, 13, 14] and [17].From now on, we assume that  G   is connected graph. Then  d  i  >  0 for all  i . Thenotation  i ∼  j  means that the vertices  i  and  j  are adjacent. The matrix  R =  D − 12  AD − 12 in  ( 1 )  is the Randi´c matrix of   G   in which the  ( i ,  j ) -entry is  1 √  d  i d   j if   i  ∼  j  and 0otherwise. Moreover  I  − L    =  R .The eigenvalues of   R  are called the Randi´c eigenvalues of   G   . Clearly  L    and  R  arebothreal symmetricmatrices. The Randi´cmatrixwas earlierstudied in connectionwiththe Randi´c index [1, 2, 18] and [19]. Two recent papers on the Randi´c matrix are [3]and [4].Throughout this paper0  =  λ  n  λ  n − 1  ...  λ  1 and ρ n  ρ n − 1  ...  ρ 1     P   R   E   P   R   I   N   T B OUND ON THE LARGEST NORMALIZED  L APLACIAN EIGENVALUE  3are the normalized Laplacian eigenvalues and the Randi´c eigenvalues of   G   , respec-tively. It follows that λ  i  =  1 − ρ n − i + 1  ( 1  i  n ) . If   M   is a nonnegative matrix then, by the Perron-Frobenius Theorem,  M   hasan eigenvalue equal to its spectral radius, called the Perron root of   M  .  In addition,if   M   is irreducible then the Perron root of   M   is a simple eigenvalue with a corre-sponding positive eigenvector, called the Perron vector of   M  .  Since  G   is a connectedgraph, Randi´c matrix of   G   is a irreducible nonnegative matrix. Let  v  =  D 12 1 .  Then v  =  √  d  1 , √  d  2 ,... , √  d  n  T  .  An easy computation shows that  R v  = v . Hence, 1 and  v  are the Perron root and the Perron vector of   R ,  respectively.Let  ∆  and  δ   be the largest and smallest vertex degrees of   G  ,  respectively, and let q n  be the smallest eigenvalue of   Q .A recent result involving the largest eigenvalue of   L    and the smallest eigenvalueof   Q  isT HEOREM  1. [17]  Let   G   be a connected graph. Then 2 − q n δ    λ  1  2 − q n ∆  .  (2)We may consider 2 − q n ∆  as an upper bound on  λ  1 .  Observe that 2 − q n ∆  = 2 if andonly if   G   is a bipartite graph.In this paper, we search for a new upper bound on  λ  1  not exceeding the trivialupper bound 2 . 2. Searching for an upper bound on  λ  1 Since  ∑ ni = 1 ρ i  = tr  (  R ) =  0 ,  it follows that  ρ n  <  0 .  We have λ  1  =  1 − ρ n  =  1 + | ρ n | . In order to find an upper bound on  λ  1  not exceeding 2 ,  we look for an upper bound on | ρ n |  not exceeding 1 . An eigenvalue of a nonnegative matrix  M   which is different from the Perron rootis called a subdominant eigenvalue of   M  .  Let  ξ  (  M  )  be the maximum modulus of the subdominant eigenvalues of   M  .  Special attention has been devoted to find upperbounds on  ξ  (  M  ) . In [20], we can find a unified presentation of results concerningupper bounds on  ξ  (  M  ) .  These upper bounds are important because  ξ  (  M  )  plays amajor role in convergence properties of powers of   M  .  Since λ  1  1 + ξ (  R ) ,  (3)we focus our attention on upper bounds on  ξ  (  R ) .  We recall the result [12, p. 295] :     P   R   E   P   R   I   N   T 4  O. R OJO AND  R. L. S OTO T HEOREM  2.  If M   = ( m i ,  j )  0  of order n × n has a positive eigenvector  w  = [ w 1 , w 2 ,... , w n ] T  corresponding to the spectral radius  ρ (  M  )  of M then ξ  (  M  )   12 max i <  jn ∑ k  = 1 w k   m i , k  w i − m  j , k  w  j  . where the maximum is taken over all pairs  ( i ,  j )  ,  1  i  <  j  n . In order to apply Theorem 2, it is convenient to observe that the Randi´c matrix of  G   is diagonally similar to the row stochastic matrix S   =  D − 12  RD 12 .  (4)The following lemma gives some immediate properties of   S  . L EMMA  1.  1. The  ( i ,  j ) − entry of S is  1 d  i if j ∼ i and   0  otherwise.2. S  1 = 1  where  1  is the all ones vector.3.  u  is an eigenvector for R corresponding to the eigenvalue  α   if and only if  D − 12 u  is an eigenvector for S corresponding to the eigenvalue  α  . 4. If   G   is an r  − regular graph then S   =  R . Let  N  i  be the set of neighbours of the vertex  v i  and let  |  N  i |  be the cardinality of   N  i . T HEOREM  3.  Let   G   be a simple undirected connected graph. If   λ  1  is the largest eigenvalue of   L    then | λ  1 |  2 − min i <  j    N  i ∩  N   j  max  d  i , d   j   (5) where the minimum is taken over all pairs  ( i ,  j )  ,  1  i  <  j  n . Proof.  We knowthattheRandi´cmatrixof   G   is similartotherowstochasticmatrix S   defined in  ( 4 ) .  Then  ξ  (  R ) =  ξ  ( S  ) .  The eigenvector corresponding to the spectralof   S   is  w = 1 . Applying Theorem 2 to  S   = ( s i ,  j ) ,  we have ξ  ( S  )   12 max i <  jn ∑ k  = 1  s i , k  − s  j , k   =  12 max i <  j   ∑ k  ∈  N  i −  N   j 1 d  i +  ∑ k  ∈  N   j −  N  i 1 d   j +  ∑ k  ∈  N  i ∩  N   j  1 d  i −  1 d   j  =  12 max i <  j   N  i −  N   j  d  i +   N   j −  N  i  d   j +  ∑ k  ∈  N  i ∩  N   j  1 d  i −  1 d   j  =  12 max i <  j  2 −   N  i ∩  N   j  d  i −   N   j ∩  N  i  d   j +  ∑ k  ∈  N  i ∩  N   j  1 d  i −  1 d   j  .     P   R   E   P   R   I   N   T B OUND ON THE LARGEST NORMALIZED  L APLACIAN EIGENVALUE  5Suppose  d  i  =  max  d  i , d   j  .  In this case2 −   N  i ∩  N   j  d  i −   N   j ∩  N  i  d   j +  ∑ k  ∈  N  i ∩  N   j  1 d  i −  1 d   j  =  2 −   N  i ∩  N   j  d  i −   N   j ∩  N  i  d   j +   1 d   j −  1 d  i   N  i ∩  N   j =  2 − 2   N  i ∩  N   j  d  i . Similarly, if   d   j  =  max  d  i , d   j   then2 −   N  i ∩  N   j  d  i −   N   j ∩  N  i  d   j +  ∑ k  ∈  N  i ∩  N   j  1 d  i −  1 d   j  =  2 − 2   N   j ∩  N  i  d   j . Hence ξ  ( S  )   12 max i <  jn ∑ k  = 1  s i , k  − s  j , k   =  12 max i <  j  2 −  2   N   j ∩  N  i  max  d  i , d   j  =  1 − min i <  j    N  i ∩  N   j  max  d  i , d   j  Since  λ  1  1 + ξ (  R ) =  1 + ξ ( S  ) , the upper bound in  ( 5 )  follows.   R EMARK  1. If   G   is a bipartite graph then   N  i ∩  N   j  =  0 ,  for some  i  <  j ,  andconsequently the upper bound in  ( 5 )  is equal to 2 .  This is sufficient condition but it isnot a necessary condition. In fact, there are other instances in which  N  i ∩  N   j  =  0 forsome  i  <  j .  One of them is given by a nonbipartite graph having a bridge. However, if min i <  j   N  i ∩  N   j   1 and  q n  <  1 then2 − min i <  j    N  i ∩  N   j  max  d  i , d   j  <  2 − q n ∆  .  (6)In fact q n  <  1    N  i ∩  N   j   for  i  <  j and q n ∆    1max  d  i , d   j   for  i  <  j .
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