The six classes of trees with the largest algebraic connectivity
In this paper, we study the algebraic connectivity α ( T ) of a tree T. We introduce six Classes ( C 1 ) – ( C 6 ) of trees of order n, and prove that if T is a tree of order n ⩾ 15 , then α ( T ) ⩾ 2 - 3 if and only if T ∈ ⋃ i = 1 6 Ci , where the equality holds if and only if T is a tree in the Cl...
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Published in | Discrete Applied Mathematics Vol. 156; no. 5; pp. 757 - 769 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Lausanne
Elsevier B.V
01.03.2008
Amsterdam Elsevier New York, NY |
Subjects | |
Online Access | Get full text |
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Summary: | In this paper, we study the algebraic connectivity
α
(
T
)
of a tree
T. We introduce six Classes
(
C
1
)
–
(
C
6
)
of trees of order
n, and prove that if
T is a tree of order
n
⩾
15
, then
α
(
T
)
⩾
2
-
3
if and only if
T
∈
⋃
i
=
1
6
Ci
, where the equality holds if and only if
T is a tree in the Class
(
C
6
)
. At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that
α
(
T
i
)
>
α
(
T
j
)
if
1
⩽
i
<
j
⩽
6
and
T
i
is any tree in the Class
(
Ci
)
and
T
j
is any tree in the Class
(
Cj
)
. We also give the values of the algebraic connectivity of the trees in these six classes. As a technique used in the proofs of the above mentioned results, we also give a complete characterization of the equality case of a well-known relation between the algebraic connectivity of a tree
T and the Perron value of the bottleneck matrix of a Perron branch of
T. |
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ISSN: | 0166-218X 1872-6771 |
DOI: | 10.1016/j.dam.2007.08.014 |