Proceedings of the Ninth International Conference
on Matrix Theory and its Applications
2010, pp. xxx-xx
( Will be set by the pu blisher )
The real eigenvalues of signless P-Laplacian matrix for star graph
Gui-hai Yu, Qing-wen Wang*
Department of Mathematics, Shanghai Universtiy
99 Shangda Road, Shanghai 200444
Abstract. Let G  (V , E ) be a simple connected undirected graph with
The
signless
Qp (G) f (v) 
P-Laplacian

Q p (G )
[ p 1]
( f (v)  f (u ))
of
a
function
n vertices and m edges.
on V is given by
f
, where the symbol t [ p ] denotes a power function that
uV ,u v
preserves the sign of t , i.e. t [ p ]  sign(t )  t
p

. A real number
Q p (G ) if there exists a function f  0 on V such that  f (v)
p 1
is called an eigenvalues of
  ( f (u )  f (v)) p 1 . And
u v
this function f is called the corresponding eigenfunction to  . In this paper we investigate the
real eigenvalues of star graph and get the only real eigenvalues of Q p (G ) are 0, 1 and (1  n
1
p 1 p 1
)
.
Keywords: Signless P-Laplacian eigenvalues; Star graph; Eigenfunction
1. Introduction
Let G  (V , E ) be the simple connected undirected graph with vertex set V  v1 , v2 ,
edge set E . d v1 , d v2 ,
, d vn denotes the degree sequence of vertices V  v1 , v2 ,
, vn  and
, vn  . The signless P-
Laplacian Q p (G ) of a function f on V is given by
Qp (G) f (v) 

[ p 1]
( f (v)  f (u ))
,
uV ,u v
where 1  p   and the symbol t [ p ] denotes a power function that preserves the sign of t , i.e.
t [ p ]  sign(t )  t . Notice that for p  2 , Q2 (G) is the well-known signless Laplacian of a graph.[1]
p
1
1
This research was supported by the Natural Science Foundation of China (No. 10471085), the Natural Science Foundation of
Shanghai, the Development Foundation of Shanghai Educational Committee, and the Special Founds for Major Specialities of
Shanghai Education Committee.
*Corresponding author:
Email address: [email protected]
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Recently there is an increasing interest signless P-Laplacian , see e.g. [2], [3]. The related eigenvalue
problems have been occasionally occurred in fields like network analysis [4, 5], pattern recongnition [6],
or image processing [7].

A real number
is called an eigenvalues of Q p (G ) if there exists a function f  0 on
 f (v) p 1   ( f (u )  f (v)) p 1
V such that
(1)
u v
And this function f is called the corresponding eigenfunction to  .
In this paper we only consider the star graph Sn 1 with n  1 vertices. We shall investigate the
signless P-Laplacian Q p (G ) and characterize the real eigenvalues of Q p (G ) .
2. Result and proof
Theorem 1: Let Sn 1 be the star graph with n  1 vertices. Then the only eigenvalues of Q p (G ) are
0, 1 and (1  n
1
p 1 p 1
)
.
Proof: Let V  v, v1 , v2 ,
, vn  be vertices of Sn 1 and dv  n . Let f be an eigenfunction
associated to the eigenvalue  .
By the definition we have
 f (v )
p 1
d
  ( f (v)  f (vi ))
p 1
(2)
i 1
 f (vi ) p 1  ( f (vi )  f (v)) p 1 , i  1, 2,
,n.
(3)
By (2) and (3), we can conclude that 0 is an eigenvalue of Q p (G ) and the corresponding eigenfunction
f satisfies f (v)   f (v1 )   f (v2 ) 
  f (vn )  a for some real umber a  0 .
From (3), we can get
(
1
p 1
 1) f (vi )  f (v), i  1, 2,
,n.
(4)
If   1 , then f (v )  0 .
Hence   1 is an eigenvalue of Q p (G ) and the corresponding eigenfunction f satisfies f (v )  0
n
and
 ( f (v ))
i 1
i
p 1
0.
Assume that   0,1 . By (3) we have
2
1
f (vi )  (1   p 1 )1 f (v), i  1, 2,
,n .
(5)
From above equation, we obtain
 ( f (vi ))
p 1
  (
1
p 1
 1)1 p  ( f (v)) p 1 .
Hence
1
( f (v)  f (vi )) p 1   ( p 1  1)1 p  ( f (v)) p 1 .
(6)
Combining (1) and (6), we have
1
 ( f (v)) p 1  n ( p 1  1)1 p  ( f (v)) p 1 .
We claim that f (v )  0 if   0,1 . Otherwise, f (vi )  0, i  1, 2,
, n which is impossible.
So we have
1  n(
1
1
p 1
 1)1 p ,
1
Which implies  p 1  1  n p 1 . We complete the proof.
Theorem 2: If G is a finite connected graph and  is an real eigenvalue of Q p (G ) , then   0 .
Proof: Let f be an function of the P-Laplacian Q p (G ) corresponding to  . Since G is finite, f
can attains a positive maximum at a vertex v0 .
Therefore
 ( f (v0 )) p 1 

( f (v0 )  f (v))
(1 
f (v )
)
f (v0 )
vV , v v0
p 1
,
which implies


vV ,v v0
p 1
 0.
Corollary 1: Let G is a finite connected graph and  is the smallest eigenvalue of Q p (G ) . Then
  0 associated to the eigenfunction f such that f (v0 )  a( 0) for some vertex v0 and
f (v)   f (v0 )  a for other vertex.
Proof： By means of (1) and the proof of theorem 2, we can get the result.
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References
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[2] T. Bityikoglu, etc, “Largest Eigenvalues of the Discrete P-Laplacian of Trees with Degree Sequences”,
Electronic Journal of Linear Algebra, 18(2009), pp.202-210.
[3] S. Amghibech, “Eigenvalues of the Discrete P-Laplacian for Graphs ”, Ars Comb., 67(2003), pp. 283-302.
[4] T. Kayano, etc., “Boundary Limit of Discrete Dirichlet Potentials”, Hiroshima Math. J., 14(1984), pp. 401-406.
[5] Maretsugu Yamasaki, “Ideal Boundary Limit of Discrete Dirichlet Functions”, Hiroshima Math. J. 16(1986),
PP. 353-360.
[6] D. Zhou, etc., “Regularization on Discrete Spaces”, Proceeding of the 27th DAGM Sysposium, 3663(2005),
pp.361-368.
[7] O. Lezoray, etc., “Graph Regularization for Color Image Processing”, Computer Vision and Image
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