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Some Inequalities on Weighted Vertex Degrees, Eigenvalues, and Laplacian Eigenvalues of Weighted

Graphs

Behzad Torkian

Dept. of Mathematical Sciences

University of South Carolina Aiken

Advisor : Dr. Rao Li

Francis Marion Undergraduate Mathematics Conference Francis Marion University

April 11, 2008

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1. Introduction

A graph G = (V, E).

A weight function W: E -> R, where R is the set of real numbers and W(e) > 0 for each e in E.

The weighted graph G(W) = G(V, E, W).

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d(vi, G(W)) denotes the weighted degree

of vertex vi in G(W). We define d(vi, G(W))

as the sum of weights of edges which are

incident with vi.

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We assume that

A(G(W)) is the weighted adjacency matrix

of G(W), where if

is an edge of G; otherwise.

)).(,(...))(,())(,( 2211 WGvddWGvddWGvdd nn =≥≥=≥=

)())((, jiji vvWWGa =jivv

0))((, =WGa ji

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The eigenvalues μ1 = μ1(G(W)) ≥ μ2 =

μ2(G(W)) ≥ … ≥ μn = μn(G(W)) of A(G(W))

are called the weighted eigenvalues of G(W).

D(G(W)) denotes the diagonal matrix diag[di].

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L(G(W)) = D(G(W)) - A(G(W)) is called

weighted Laplacian matrix of G(W).

The eigenvalues λ1 = λ1(G(W)) ≥ λ2

= λ2(G(W)) ≥ … ≥ λn = λn(G(W)) of

L(G(W)) are called the weighted

Laplacian eigenvalues of G(W).

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2. Result

,)1(2

n

ncd kk

−≤−λ

Theorem 1. Let G(W) be a weighted graph of order

n such that W(e) > 0 for each edge e of G. Then

.))((2

1:

1 1

2∑∑= =

=n

ij

n

jivvWcwhere

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Theorem 1 generalizes the following

Theorem A proved by Rao Li.

Theorem A [1]. Let G be a graph of n

vertices and e edges. Then

.)1(2

n

ned kk

−≤−λ

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Since if W(e) = 1 for each edge e, then

c = e and in Theorem 1 become respectively the ordinary vertex

degrees and Laplacian eigenvalues in

Theorem A.

,1 ,, nid ii ≤≤λ

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3. A Theorem of Hoffman and Wielandt

If C = A + B, where A, B, and C are symmetric matrices of order n having respectively the eigenvalues arranged in non-increasing order

,,, iii γβα

€

1≤ i ≤ n

.)(1

2

1

2 ∑∑==

≤−n

ii

n

iii βαγ

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4. Proof

Note that the sum of eigenvalues of a matrix M is

equal to the sum of diagonal entries of matrix M.

Hence

.0)(1

=−∑=

i

n

iid λ

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,1 nk ≤≤

.)()(,1∑

≠=

−=−−n

kiiiikk dd λλ

For each k,

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By Cauchy-Schwarz inequality,

.)(1))((,1

2

,1 ,1

22 ∑∑ ∑≠=≠= ≠=

−≤−n

kiiii

n

kii

n

kiiii dd λλ

Note:

€

(x1y1 + x2y2 + ...+ xnyn )2 ≤ (x1

2 + x22 + ...+ xn

2)(y12 + y2

2 + ...+ yn2)

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.)()1())(()(,1

222 ∑≠=

−−≤−−=−n

kiiiikkkk dndd λλλ

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Notice that D(G(W)) = L(G(W)) + A(G(W)).

By the theorem of Hoffman and Wielandt,

we have that

∑

∑∑∑

=

==≠=

==

−−≤−−−=−

n

ii

n

ikkikk

n

iii

n

kiiii

cGAtrace

dddd

1

22

1

222

1

2

,1

2

2)))(((

)()()()(

μ

λμλλλ

€

a b c ⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥×

a

b

c

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

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€

(dk − λ k )2 ≤ (n −1)(2c − (dk − λ k )

2),

n(dk − λ k )2 ≤ 2c(n −1).

.)1(2

n

ncd kk

−≤−λ

Therefore

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5. Reference

[1]. Rao Li, Some Inequalities on Vertex

Degrees, Eigenvalues, and Laplacian

Eigenvalues of Graphs, to appear.

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Thanks.