Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.bg.ac.yu
Appl. Anal. Discrete Math. 3 (2009), 379–385.
doi:10.2298/AADM0902379H
A SHARP LOWER BOUND OF THE SPECTRAL
RADIUS OF SIMPLE GRAPHS
Shengbiao Hu
Let G be a simple connected graph with n vertices and let ρ(G) be its spectral
radius. The 2-degree of vertex i is denoted
Pby ti , which is the
P sum of degrees
of the vertices adjacent to i. Let Ni =
tj and Mi =
Nj . We find a
j∼i
j∼i
sharp lower bound of ρ(G), which only contains two parameter Ni and Mi .
Our result extends recent known results.
1. INTRODUCTION
Let G be a simple connected graph with vertex set V = {1, 2, . . . , n}. Let
d(i, j) denote the distance between vertices i and j. For i ∈ V , the degree of i and
the average of the degree of the vertices adjacent to i are denoted by di and mi ,
respectively. The 2-degree of vertex i is denoted by ti , which is the sum of degrees
of the vertices adjacent to i, that is ti = mi di . Let Ni be the sum of the 2-degree
of vertices adjacent to i.
Let A(G) be the adjacency matrix of G. By the Perron-Frobenius theorem [1,
2], the spectral radius ρ(G) is simple and there is a unique positive unit eigenvector.
Since A(G) is a real symmetric matrix, its eigenvalues must be real, and may
ordered as λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G). The sequence of n eigenvalues is called
the spectrum of G, the largest eigenvalue λ1 (G) is often called the spectral radius
of G, denoted by ρ(G) = λ1 (G).
In this paper, we give a sharp lower bound on the spectral radius of simple
graphs. For some recent surveys of the known results about this problem and
related topics, we refer the reader to [3, 4, 7] and references therein.
2000 Mathematics Subject Classification. 05C50.
Keywords and Phrases. Eigenvalues, spectral radius, lower bound, trees.
379
380
Shengbiao Hu
2. MAIN RESULTS
Lemma 1. Let G be a bipartite graph with V = V1 ∪ V2 , V1 = {1, 2, . . . , s} and
V2 = {s+ 1,s + 2, . . . , n}. Let Y1 = (y1 , y2 , . . . , ys )T and Y2 = (ys+1 , ys+2 , . . . , yn )T .
Y1
If Y =
is an eigenvector of A(G) corresponding to ρ(G), then k Y1 k=k Y2 k.
Y2
0 B
Proof. Let A(G) =
, where B is an s × (n − s) matrix. We have
BT 0
Y1
0 B
Y1
=
ρ
A(G)
,
BT 0
Y2
Y2
BY2 = ρ A(G) Y1 ⇒ Y1T BY2 = ρ A(G) Y1T Y1 ,
and
B T Y1 = ρ A(G) Y2 ⇒ Y2T B T Y1 = ρ A(G) Y2T Y2 .
Since (Y1T BY2 )T = Y2T B T Y1 , we have that Y1T Y1 = Y2T Y2 , that is
k Y1 k=k Y2 k .
Theorem 2. Let G be a simple connected graph of order n and ρ(G) be the spectral
radius of G. Then
s
P
. P
n
n
(1)
ρ(G) ≥
Mi 2
Ni 2 ,
i=1
where Ni =
P
tj and Mi =
j∼i
P
i=1
Nj . The equality in (1) holds if and only if
j∼i
M2
Mn
M1
=
= ··· =
N1
N2
Nn
or G is a bipartite graph with V = V1 ∪ V2 , V1 = {1, 2, . . . , s} and V2 = {s + 1, s +
2, . . . , n}, such that
M1
M2
Ms
Ms+1
Ms+2
Mn
=
= ··· =
and
=
= ··· =
.
N1
N2
Ns
Ns+1
Ns+2
Nn
Proof. By Rayleigh quotient, we have
xT A(G)2 x
ρ(G)2 = ρ A(G)2 = max
.
x6=0
xT x
(2) (2)
Let A(G)2 = aij , where aij is the number of (i, j)-walks of length 2 in G.
(2)
(2)
(2)
Clearly, aii = di and aij = aji .
381
A sharp lower bound of the spectral radius of simple graphs
For a fixed (i, j)-walk in G, denote by w(i, j) the length of this walk. Then
(2)
(2)
aij 6= 0 if w(i, j) = 2, and aij = 0 otherwise. If X = (N1 , N2 , . . . , Nn )T , we have
X T A(G)2 X =
n
P
Ni
i=1
=
n
P
P
n
j=1
n
P
i=1
di Ni 2 +
i=1
and X T X =
P
n
(2)
aij Nj =
Ni
P
w(i,j)=2,i6=j
Ni 2 . So
i=1
s
ρ(G) =
s
P
n
=
i=1
P
aij Nj =
w(j,i)=2
(2)
aij Ni Nj
xT A(G)2 x
max
≥
x6=0
xT x
(2)
P
(2)
aij Ni Nj
w(i,j)=2
n P
P
i=1
Mi2
Nj
j∼i
. P
n
i=1
2
=
n
P
i=1
Mi 2
Ni2 .
If the equality holds, then X is a positive eigenvector
of A(G)2 correspond
2
2
2
ing to ρ(G) . Moreover, if the eigenvalue ρ A(G) of A(G) has the multiplicity
one, then by the Perron-Frobenius theorem, X is an eigenvector of A(G) corresponding to ρ(A(G)), therefore A(G)X
= ρ(G)X. For all
P
Pi = 1, 2, . . . , n, we have
A(G)X i = ρ(G)X i , that is
Nj = ρ(G)Ni . Since
Nj = Mi , we get
j∼i
j∼i
Mi
= ρ(G)
Ni
i = 1, 2, . . . , n,
and therefore
M1
M2
Mn
=
= ··· =
= ρ(G).
N1
N2
Nn
If the eigenvalue
ρ A(G)2 of A(G)2 has the multiplicity two, it is well known
that −ρ A(G) is an eigenvalue of A(G). Hence G is a bipartite graph. Without
loss of generality, we assume that
0 B
A=
,
BT 0
hence
2
A =
BB T 0
0 BT B
.
Let X1 = (N1 , N2 , . . . , Ns )T and X2 = (Ns+1 , Ns+2 , . . . , Nn )T , we have
X1
BB T 0
X1
2
=
ρ
A(G)
,
0 BT B
X2
X2
BB T X1 = ρ(A(G)2 )X1 and B T BX2 = ρ A(G)2 X2 .
Let Y = (y1 , y2 , . . . , yn )T be a positive eigenvector of A(G) corresponding to ρ(G).
Let Y1 = (y1 , y2 , . . . , ys )T and Y2 = (ys+1 , ys+2 , . . . , yn )T . Thus
BB T Y1 = ρ A(G)2 Y1 and B T BY2 = ρ A(G)2 Y2 .
382
Shengbiao Hu
Since BB T and B T B have the same nonzero eigenvalues, BB T and B T B have
eigenvalues ρ(A(G)2 ) with multiplicity one, respectively. Hence by the PerronFrobenius theorem, we have Y1 = aX1 (a 6= 0) and Y2 = bX2 (b 6= 0). Now, it
follows from A(G)Y = ρ(G)Y that
P
bNj = ρ(G)aNi , i = 1, 2, . . . , s
j∼i
and
P
aNj = ρ(G)bNi ,
i = s + 1, s + 2, . . . , n.
j∼i
Since
P
Nj = M i ,
j∼i
thus we have
Mi
a
= ρ(G)
Ni
b
and
Mi
b
= ρ(G)
Ni
a
i = 1, 2, . . . , s
i = s + 1, s + 2, . . . , n.
Therefore,
M1
M2
Ms
a
=
= ··· =
= ρ(G)
N1
N2
Ns
b
i = 1, 2, . . . s
and
Ms+1
Ms+2
Mn
b
=
= ··· =
= ρ(G)
Ns+1
Ns+2
Nn
a
i = s + 1, s + 2, . . . , n.
In addition, by Lemma 1 we have
2
a2 (N12 + · · · + Ns2 ) = b2 (Ns+1
+ · · · + Nn2 ).
(2)
Conversely, we have:
(i) If
M1
M
M
= 2 = · · · = n , then
N1
N2
Nn
s
P
n
Mi 2
i=1
. P
n
i=1
Ni 2 = ρ(G).
M1
M
M
a
= 2 = · · · = s = ρ(G) i =
N1
N2
Ns
b
Mn
b
= ··· =
= ρ(G) i = s + 1, s + 2, . . . , n. Then
Nn
a
(ii) If G is a bipartite graph with
1, 2, . . . , s and
Ms+1
M
= s+2
Ns+1
Ns+2
by (2) we have
v
v
uP
2
u
2
u n M2
u ρ(G)2 a N 2 + . . . + N 2 ) + b N 2 + · · · + N 2 u
i
t
s
n
1
s+1
2
2
u i=1
b
a
u n
=
= ρ(G),
tP 2
N12 + N22 + · · · + Nn2
Ni
i=1
383
A sharp lower bound of the spectral radius of simple graphs
and the proof follows.
We now show that our bound improves the bound of Hong and Zhang [6].
Corollary 3 (Hong and Zhang [6]). Let G be a simple connected graph of order
n, then
s
P
. P
n
n
(3)
ρ(G) ≥
Ni 2
ti2 ,
i=1
i=1
with equality if and only if
N1
N2
Nn
=
= ··· =
t1
t2
tn
or G a bipartite graph with V = V1 ∪ V2 , V1 = {1, 2, . . . , s} and V2 = {s + 1, s +
2, . . . , n} such that
N1
N2
Ns
Ns+1
Ns+2
Nn
=
= ··· =
and
=
= ··· =
.
t1
t2
ts
ts+1
ts+2
tn
Proof. By Cauchy-Schwartz inequality, we have
P
n
Mi 2
i=1
P
n
n
P
2 P
n
n
P 2
P P P 2
ti2 ≥
ti M i =
ti N j =
ti
tk
i=1
i=1
i=1
=
P
n
j∼i
i=1 j∼i
P
ti
tk
i=1 w(k,i)=2
=
P
n
di ti2 +
i=1
=
2
=
P
k∼j
P
ti tj
w(i,j)=2
ti tj
w(i,j)=2,i6=j
2
2
P
2
n P 2 2
n
P
tj
=
Ni 2 .
j∼i
i=1
i=1
The equality holds if and only if
M1
M2
Mn
=
= ··· =
.
N1
N2
Nn
Hence
n
P
i=1
n
P
i=1
Mi 2
≥
Ni
2
n
P
i=1
n
P
Ni 2
i=1
.
ti2
Therefore it follows from Theorem 1 that the result holds.
384
Shengbiao Hu
A graph is called pseudo-semiregular if its vertices have the same average
degree. A bipartite graph is called pseudo-semiregular if all vertices in the same
part of a bipartition have the same average degree.
Corollary 4 (Yu et al. [8]). Let G be a simple connected graph of order n. Then
s
P
. P
n
n
(4)
ρ(G) ≥
ti2
di2 ,
i=1
i=1
with equality if and only if G is either a pseudo-regular graph or a pseudo-semiregular
graph.
Proof. This corollary follows from Corollary 3 (See [3]).
Corollary 5 (Hofmeister [5]). Let G be a simple connected graph with degree
sequence d1 , d2 , . . . , dn . Then
s
n
1 P
(5)
ρ(G) ≥
di2 .
n
i=1
Proof. This corollary follows from Corollary 4 (See [6].
The lower bound (1) is complicated though it is better than (3). But if G is
a tree, then the Ni and the Mi have a simple expression
P
P P P
Ni =
tj =
dk = di2 +
dj
j∼i
j∼i
k∼j
d(j,i)=2
and
Mi =
P
Nj =
j∼i
Hence we have
P
j∼i
dj2 +
P
d(k,j)=2
P
P
dk =
dj2 + (di − 1) dj +
j∼i
j∼i
P
dj .
d(j,i)=3
Corollary 6. Let T be a tree of order n and ρ(T ) be the spectral radius of T. Then
v
uP
2
P
P
u n P 2
dj + (di − 1)
dj +
dj
u
u i=1 j∼i
j∼i
d(j,i)=3
(6)
ρ(T ) ≥ u
.
2
u
n P
P
t
di2 +
dj
i=1
d(j,i)=2
The equality holds if and only if
M1
M2
Mn
=
= ··· =
N1
N2
Nn
or for the bipartite graph T with V = V1 ∪ V2 , V1 = {1, 2, . . . , s} and V2 = {s +
1, s + 2, . . . , n} such that
M1
M2
Ms
Ms+1
Ms+2
Mn
=
= ··· =
and
=
= ··· =
,
N1
N2
Ns
Ns+1
Ns+2
Nn
A sharp lower bound of the spectral radius of simple graphs
385
where
Mi =
P
j∼i
and
dj2 + (di − 1)
P
dj +
P
dj .
j∼i
Ni = di2 +
P
dj
d(j,i)=3
d(j,i)=2
Proof. This corollary follows from Theorem 2.
Acknowledgements. The author wish to thanks the referee for the comments,
corrections and suggestions. This work was supported by NNSF of China (No.
10861009).
REFERENCES
1. A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences.
Academic Press, New York, 1979. SIAM, Philadelphia, PA, 1994.
2. N. Biggs: Algebraic Graph Theory, second ed. Cambridge University Prees, Cambridge, 1995.
3. D. Cvetković, M. Doob, H. Sachs: Spectra of Graphs - Theory and Application.
Academic Press, New York, 1980.
4. D. Cvetković, P. Rowlinson: The largest eigenvalue of a graph : A survey. Linear
and Multilinear Algebra, 28 (1990), 3–33.
5. M. Hofmeister: Spectral radius and degree sequence. Math. Nachr., 139 (1988),
37–44.
6. Yuan Hong, Xiao-Dong Zhang: Sharp upper and lower bounds for the Laplacian
matrices of trees. Discrete Mathematics, 296 (2005), 187–197.
7. V. Nikiforov: Some inequalities for the largest eigenvalue of a graph. Combinatorics,
Probability and Computing, 11 (2002), 179–189.
8. A. M. Yu, M. Lu, F. Tian: On the spectral radius of graphs. Linear Algebra Appl.,
387 (2004), 41–49.
Department of Mathematics,
Qinghai Nationalities University,
Xining, Qinghai 810007,
P. R. China
E–mail: [email protected]
(Received February 15, 2009)
(Revised May 28, 2009)