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A sharp lower bound of the spectral radius of simple graphs

01 Jan 2009-Applicable Analysis and Discrete Mathematics (National Library of Serbia)-Vol. 3, Iss: 2, pp 379-385

AbstractLet G be a simple connected graph with n vertices and let p(G) be its spectral radius. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i. Let Ni = Σj~i tj and Mi = Σj~i Nj. We find a sharp lower bound of p(G), which only contains two parameter Ni and Mi. Our result extends recent known results.

Topics: Bound graph (64%), Vertex (geometry) (55%), Spectral radius (54%), Upper and lower bounds (51%)

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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
ji
t
j
and M
i
=
P
ji
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
ji
t
j
and M
i
=
P
ji
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
ji
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
ji
N
j
= ρ(G)N
i
. Since
P
ji
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
ji
bN
j
= ρ(G)aN
i
, i = 1, 2, . . . , s
and
P
ji
aN
j
= ρ(G)bN
i
, i = s + 1, s + 2, . . . , n.
Since
P
ji
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
ji
N
j
2
=
n
P
i=1
t
i
P
ji
P
kj
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
ji
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.

Citations
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01 Jan 2013
Abstract: Sharp upper bounds for the energy and Randic energy of a (bipartite) graph are established. From these, some previously known results could be deduced.

27 citations


Journal ArticleDOI
16 Jan 2012
TL;DR: The Sandwich Theorem is applied to show monotonicity in this and a related family of lower bounds of Nikiforov, which leads to generalized upper bounds for the energy of graphs.
Abstract: We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed walks of length k starting at a given vertex v. The first is a direct implication of a matrix inequality by Marcus and Newman and generalizes two inequalities by Lagarias et al. and Dress & Gutman. We then use an inequality of Blakley and Dixon to show the inequality wk2e+p ≤ w2e+pk · wk−12e which also generalizes the inequality by Dress and Gutman and also an inequality by Erdos and Simonovits. Both results can be translated directly into the corresponding forms using the higher order densities, which extends former results. In the second part, we provide a new family of lower bounds for the largest eigenvalue λ1 of the adjacency matrix based on closed walks and apply the before mentioned inequalities to show monotonicity in this and a related family of lower bounds of Nikiforov. This leads to generalized upper bounds for the energy of graphs. In the third part, we demonstrate that a further natural generalization of the inequality w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) is not valid for general graphs. We show that wa+b · wa+b+c ≤ wa · wa+2b+c does not hold even in very restricted cases like w1 · w2 ≤ w0 · w3 (i.e., d · w2 ≤ w3) in the context of bipartite or cycle free graphs. In contrast, we show that surprisingly this inequality is always satisfied for trees and show how to construct worst-case instances (regarding the difference of both sides of the inequality) for a given degree sequence. We also provide a proof for the inequality w1 · w4 ≤ w0 · w5 (i.e., d · w4 ≤ w5) for trees and conclude with a corresponding conjecture for longer walks.

13 citations


Additional excerpts

  • ...Other papers with lower bounds, namely ∑ v∈V w2(v)(2)/ ∑ v∈V d2 v ≤ λ(2)1 [53], ∑ v∈V w3(v)(2)/ ∑ v∈V w2(v)(2) ≤ λ(2)1 [30], ∑ v∈V w4(v)(2)/ ∑ v∈V w3(v)(2) ≤ λ(2)1 [32], and ∑ v∈V wk+1(v)(2)/ ∑ v∈V wk(v)(2) ≤ λ(2)1 [31] consider the sum of squares of walk numbers, but do not mention the corresponding number of walks of the double length (w4/w2 ≤ λ(2)1, w6/w4 ≤ λ(2)1, w8/w6 ≤ λ(2)1, and w2k+2/w2k ≤ λ(2)1)....

    [...]


01 Jan 2014
Abstract: We consider the number of walks in undirected and directed graphs and, more generally, the weighted sum of entries of matrix powers. In this respect, we generalize an earlier result for Hermitian matrices. By using these inequalities for the entry sum of matrix powers, we deduce similar inequalities for iterated kernels. For further conceivable inequalities, we provide counterexamples in the form of graphs that contradict the corresponding statement for the number of walks. For the largest eigenvalue of adjacency matrices, we generalize a bound of Nikiforov that uses the number of walks. Furthermore, we relate the number of walks in graphs to the number of nodes and the number of edges in iterated directed line graphs.

1 citations


Additional excerpts

  • ...In several other papers, the sum of squares of walk numbers was considered to obtain the lower bounds ∑ v∈V w2(v) 2/ ∑ v∈V d 2 v ≤ λ21 [YLT04], ∑ v∈V w3(v) 2/ ∑ v∈V w2(v) 2 ≤ λ21 [HZ05],∑ v∈V w4(v) 2/ ∑ v∈V w3(v) 2 ≤ λ21 [Hu09], and ∑ v∈V wk+1(v) 2/ ∑ v∈V wk(v) 2 ≤ λ21 [HTW07]....

    [...]


References
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01 Jan 2009
TL;DR: The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.
Abstract: Graphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.

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6,318 citations


"A sharp lower bound of the spectral..." refers background in this paper

  • ...By the Perron-Frobenius theorem [1, 2], the spectral radius ρ(G) is simple and there is a unique positive unit eigenvector....

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Book
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Abstract: Introduction. Basic Concepts of the Spectrum of a Graph. Operations on Graphs and the Resulting Spectra. Relations Between Spectral and Structural Properties of Graphs. The Divisor of a Graph. The Spectrum and the Group of Automorphisms. Characterization of Graphs by Means of Spectra. Spectra Techniques in Graph Theory and Combinatories. Applications in Chemistry an Physics. Some Additional Results. Appendix. Tables of Graph Spectra Biblgraphy. Index of Symbols. Index of Names. Subject Index.

2,094 citations


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"A sharp lower bound of the spectral..." refers background in this paper

  • ...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....

    [...]


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