On the spectral moments of trees with a given bipartition

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Publication date: 20 November 2012

Abstract: For two given positive integers

p

and

q

with

p l e q s l a n t q

, we denote

is a tree of order

n

with a

(p, q)

-bipartition}. For a graph

G

with

n

vertices, let

A (G)

be its adjacency matrix with eigenvalues

l a m b d a_{1} (G), l a m b d a_{2} (G), . . ., l a m b d a_{n} (G)

in non-increasing order. The number

S_{k} (G) : = s u m_{i = 1}^{n} l a m b d a_{i}^{k} (G), (k = 0, 1, . . ., n - 1)

is called the

k

th spectral moment of

G

. Let

S (G) = (S_{0} (G), S_{1} (G), . . ., S_{n - 1} (G))

be the sequence of spectral moments of

G

. For two graphs

G_{1}

and

G_{2}

, one has

G_{1} p r e c_{s} G_{2}

if for some

k i n 1, 2, . . ., n - 1

,

S_{i} (G_{1}) = S_{i} (G_{2}) (i = 0, 1, . . ., k - 1)

and

S_{k} (G_{1}) < S_{k} (G_{2})

holds. In this paper, the last four trees, in the

S

-order, among

m a t h s c r T_{n}^{p, q} (4 l e q s l a n t p l e q s l a n t q)

are characterized.

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