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 SPECTR AL
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 by t
i
, which is the sum of degrees
of the vertices adjacent to i. Let N
i
=
P
j∼i
t
j
and M
i
=
P
j∼i
N
j
. We find a
sharp lower bound of ρ(G), which only contains two parameter N
i
and M
i
.
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 o f i and
the average of the degree of the vertices adjacent to i are denoted by d
i
and m
i
,
respectively. The 2-degree of vertex i is denoted by t
i
, which is the sum of degrees
of the vertices adjacent to i, that is t
i
= m
i
d
i
. Let N
i
be the sum of the 2-degree
of vertices adjacent to i.
Let A(G) be the adjacency matrix o f 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 a s λ
1
(G) ≥ λ
2
(G) ≥ · · · ≥ λ
n
(G). The sequence of n eigenva lues is called
the spectrum of G, the largest eigenvalue λ
1
(G) is often called the spectral radius
of G, denoted by ρ(G) = λ
1
(G).
In this paper, we give a sharp lower bound on the s pectral radius of simple
graphs. For some r e c e nt 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 = V
1
∪ V
2
, V
1
= {1, 2, . . . , s} and
V
2
= {s +1, s+2, . . . , n}. Let Y
1
= (y
1
, y
2
, . . . , y
s
)
T
and Y
2
= (y
s+1
, y
s+2
, . . . , y
n
)
T
.
If Y =
Y
1
Y
2
is an eigenvector of A(G) corresponding to ρ(G), then k Y
1
k=k Y
2
k.
Proof. Let A(G) =
0 B
B
T
0
, where B is an s × (n − s) matrix. We have
0 B
B
T
0
Y
1
Y
2
= ρ
A(G)
Y
1
Y
2
,
BY
2
= ρ
A(G)
Y
1
⇒ Y
T
1
BY
2
= ρ
A(G)
Y
T
1
Y
1
,
and
B
T
Y
1
= ρ
A(G)
Y
2
⇒ Y
T
2
B
T
Y
1
= ρ
A(G)
Y
T
2
Y
2
.
Since (Y
T
1
BY
2
)
T
= Y
T
2
B
T
Y
1
, we have that Y
T
1
Y
1
= Y
T
2
Y
2
, that is
k Y
1
k=k Y
2
k .
Theorem 2. Let G be a simple connected graph of order n and ρ(G) be the spectral
radius of G. Then
(1) ρ(G) ≥
s
n
P
i=1
M
2
i
.
n
P
i=1
N
2
i
,
where N
i
=
P
j∼i
t
j
and M
i
=
P
j∼i
N
j
. The equality in (1) holds if and only if
M
1
N
1
=
M
2
N
2
= · · · =
M
n
N
n
or G is a bipartite graph with V = V
1
∪ V
2
, V
1
= {1, 2, . . . , s} and V
2
= {s + 1, s +
2, . . . , n}, such that
M
1
N
1
=
M
2
N
2
= · · · =
M
s
N
s
and
M
s+1
N
s+1
=
M
s+2
N
s+2
= · · · =
M
n
N
n
.
Proof. By Rayleigh quotient, we have
ρ(G)
2
= ρ
A(G)
2
= max
x6=0
x
T
A(G)
2
x
x
T
x
.
Let A(G)
2
=
a
(2)
ij
, where a
(2)
ij
is the number o f (i, j)-walks of length 2 in G.
Clearly, a
(2)
ii
= d
i
and a
(2)
ij
= a
(2)
ji
.
A sharp lower boun d of the spectral radius of simple graphs 381
For a fixed (i, j)-walk in G, denote by w(i, j) the length of this walk . Then
a
(2)
ij
6= 0 if w(i, j) = 2, and a
(2)
ij
= 0 other w ise. If X = (N
1
, N
2
, . . . , N
n
)
T
, we have
X
T
A(G)
2
X =
n
P
i=1
N
i
n
P
j=1
a
(2)
ij
N
j
=
n
P
i=1
N
i
P
w(j,i)=2
a
(2)
ij
N
j
=
P
w(i,j)=2
a
(2)
ij
N
i
N
j
=
n
P
i=1
d
i
N
2
i
+
P
w(i,j)=2,i6=j
a
(2)
ij
N
i
N
j
=
n
P
i=1
P
j∼i
N
j
2
=
n
P
i=1
M
2
i
and X
T
X =
n
P
i=1
N
2
i
. So
ρ(G) =
s
max
x6=0
x
T
A(G)
2
x
x
T
x
≥
s
n
P
i=1
M
2
i
.
n
P
i=1
N
2
i
.
If the equality holds, then X is a positive eigenvector of A(G)
2
correspond-
ing to ρ(G)
2
. Moreover, if the eigenvalue ρ
A(G)
2
of A(G)
2
has the multiplicity
one, then by the Perron-Frobenius theorem, X is an eigenvector of A(G) corre-
sp onding to ρ(A(G)), therefore A(G)X = ρ(G)X. For all i = 1, 2, . . . , n, we have
A(G)X
i
=
ρ(G)X
i
, that is
P
j∼i
N
j
= ρ(G)N
i
. Since
P
j∼i
N
j
= M
i
, we get
M
i
N
i
= ρ(G) i = 1, 2, . . . , n,
and therefore
M
1
N
1
=
M
2
N
2
= · · · =
M
n
N
n
= ρ(G).
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 g e nerality, we assume that
A =
0 B
B
T
0
,
hence
A
2
=
BB
T
0
0 B
T
B
.
Let X
1
= (N
1
, N
2
, . . . , N
s
)
T
and X
2
= (N
s+1
, N
s+2
, . . . , N
n
)
T
, we have
BB
T
0
0 B
T
B
X
1
X
2
= ρ
A(G)
2
X
1
X
2
,
BB
T
X
1
= ρ(A(G)
2
)X
1
and B
T
BX
2
= ρ
A(G)
2
X
2
.
Let Y = (y
1
, y
2
, . . . , y
n
)
T
be a positive eigenvector of A(G) corr e sponding to ρ(G).
Let Y
1
= (y
1
, y
2
, . . . , y
s
)
T
and Y
2
= (y
s+1
, y
s+2
, . . . , y
n
)
T
. Thus
BB
T
Y
1
= ρ
A(G)
2
Y
1
and B
T
BY
2
= ρ
A(G)
2
Y
2
.
382 Shengbiao Hu
Since BB
T
and B
T
B have the same nonzero eige nvalues, BB
T
and B
T
B have
eigenvalues ρ(A(G)
2
) with multiplicity one, respectively. Hence by the Perron-
Frobenius theorem, we have Y
1
= aX
1
(a 6= 0) and Y
2
= bX
2
(b 6= 0). Now, it
follows from A(G)Y = ρ(G)Y that
P
j∼i
bN
j
= ρ(G)aN
i
, i = 1, 2, . . . , s
and
P
j∼i
aN
j
= ρ(G)bN
i
, i = s + 1, s + 2, . . . , n.
Since
P
j∼i
N
j
= M
i
,
thus we have
M
i
N
i
=
a
b
ρ(G) i = 1, 2, . . . , s
and
M
i
N
i
=
b
a
ρ(G) i = s + 1, s + 2, . . . , n.
Therefore,
M
1
N
1
=
M
2
N
2
= · · · =
M
s
N
s
=
a
b
ρ(G) i = 1, 2, . . . s
and
M
s+1
N
s+1
=
M
s+2
N
s+2
= · · · =
M
n
N
n
=
b
a
ρ(G) i = s + 1, s + 2, . . . , n.
In addition, by L e mma 1 we have
(2) a
2
(N
2
1
+ · · · + N
2
s
) = b
2
(N
2
s+1
+ · · · + N
2
n
).
Conversely, we have:
(i) If
M
1
N
1
=
M
2
N
2
= · · · =
M
n
N
n
, then
s
n
P
i=1
M
2
i
.
n
P
i=1
N
2
i
= ρ(G).
(ii) If G is a bipartite graph with
M
1
N
1
=
M
2
N
2
= · · · =
M
s
N
s
=
a
b
ρ(G) i =
1, 2, . . . , s and
M
s+1
N
s+1
=
M
s+2
N
s+2
= · · · =
M
n
N
n
=
b
a
ρ(G) i = s + 1, s + 2, . . . , n. Then
by (2) we have
v
u
u
u
u
u
t
n
P
i=1
M
2
i
n
P
i=1
N
2
i
=
v
u
u
t
ρ(G)
2
a
2
b
2
N
2
1
+ . . . + N
2
s
) +
b
2
a
2
N
2
s+1
+ · · · + N
2
n
N
2
1
+ N
2
2
+ · · · + N
2
n
= ρ(G),
A sharp lower boun d of the spectral radius of simple graphs 383
and the proof follows.
We now show that our bound improves the bound of Hong and Zha n g [6].
Corollary 3 (Hong and Zhang [6]). Let G be a simple connected graph of order
n, then
(3) ρ(G) ≥
s
n
P
i=1
N
2
i
.
n
P
i=1
t
2
i
,
with equ ality if and only if
N
1
t
1
=
N
2
t
2
= · · · =
N
n
t
n
or G a bipartite graph with V = V
1
∪ V
2
, V
1
= {1, 2, . . . , s} and V
2
= { s + 1 , s +
2, . . . , n} such that
N
1
t
1
=
N
2
t
2
= · · · =
N
s
t
s
and
N
s+1
t
s+1
=
N
s+2
t
s+2
= · · · =
N
n
t
n
.
Proof. By Cauchy-Schwartz inequality, we have
n
P
i=1
M
2
i
n
P
i=1
t
2
i
≥
n
P
i=1
t
i
M
i
2
=
n
P
i=1
t
i
P
j∼i
N
j
2
=
n
P
i=1
t
i
P
j∼i
P
k∼j
t
k
2
=
n
P
i=1
t
i
P
w(k,i)=2
t
k
2
=
P
w(i,j)=2
t
i
t
j
2
=
n
P
i=1
d
i
t
2
i
+
P
w(i,j)=2,i6=j
t
i
t
j
2
=
n
P
i=1
P
j∼i
t
j
2
2
=
n
P
i=1
N
2
i
2
.
The equality holds if and only if
M
1
N
1
=
M
2
N
2
= · · · =
M
n
N
n
.
Hence
n
P
i=1
M
2
i
n
P
i=1
N
2
i
≥
n
P
i=1
N
2
i
n
P
i=1
t
2
i
.
Therefore it follows from Theorem 1 that the result holds.