vol13_pp175-186. 203KB Jun 04 2011 12:05:53 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
www.math.technion.ac.il/iic/ela
ALGEBRAIC CONNECTIVITY OF TREES WITH A PENDANT
∗
EDGE OF INFINITE WEIGHT
A. BERMAN†
AND
K.-H. FÖRSTER‡
Abstra
t.
Let G be a weighted graph. Let v be a vertex of G and let Gv
ω denote the graph
obtained by adding a vertex u and an edge {v, u} with weight ω to G. Then the algebrai
onne
tivity
µ(Gvω ) of Gvω is a nonde
reasing fun
tion of ω and is bounded by the algebrai
onne
tivity µ(G) of
G. The question of when lim µ(Gvω ) is equal to µ(G) is
onsidered and answered in the
ase that
ω→∞
G is a tree.
Key words.
Weighted graphs, Trees, Lapla
ian matrix, Algebrai
onne
tivity, Pendant edge.
AMS subje
t
lassi
ations.
5C50, 5C40, 15A18.
1. Introdu
tion. A weighted graph on n verti
es is an undire
ted simple graph
G on n verti
es su
h that with ea
h edge e of G, there is an asso
iated positive number
ω(e) whi
h is
alled the weight of e.
The Lapla
ian matrix of a weighted graph G on n verti
es is the n × n matrix
L(G) = L = (lij ), where for ea
h i, j = 1, . . . , n,
−ω(e)
lij =
0
l
k=i ik
if i = j and e = {i, j},
if i = j and i is not adja
ent to j,
if i = j.
Clearly L is a singular M -matrix and positive semide nite, so λ1 (L) = 0, where
for a symmetri
matrix A we arrange the eigenvalues in nonde
reasing order
λ1 (A) ≤ λ2 (A) ≤ . . .
Fiedler [3℄ showed that λ2 (L) is positive i G is
onne
ted and
alled it the
algebrai
onne
tivity of G. The algebrai
onne
tivity of G will be denoted by µ(G).
In this paper G always denotes a
onne
ted weighted graph without loops.
Let G be a graph with n verti
es. Let v be a vertex of G and let Gvω be the graph
with n + 1 verti
es obtained by adding to G a vertex u and an edge e = {v, u} with
weight ω .
∗ Re
eived by the editors 21 July 2003. A
epted for publi
ation 24 June 2005. Handling Editor:
Stephen J. Kirkland.
† Department of Mathemati
s, Te
hnion, Haifa 32000, Israel,(bermante
hunix.te
hnion.a
.il).
Resear
h supported by the New York Metropolitan Fund for Resear
h at the Te
hnion. Most of
the work was done during a visit to TU Berlin; part of the work was done in FU Berlin, Charite Benjamin Franklin.
‡ Department of Mathemati
s, Te
hni
al University Berlin, Sekr. MA 6-4, Strasse des 17. Juni
136, 10623 Berlin, Germany (foerstermath.tu-berlin.de).
175
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
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176
A. Berman and K.-H. Förster
Theorem 1.1. The algebrai
onne
tivity
ω
and for every
ω
and
µ(Gvω )
is a nonde
reasing fun
tion of
n>1
µ(Gvω ) ≤ µ(G).
v
Proof. Let Lω be the Lapla
ian matrix of Gω and let 0 < ω1 ≤ ω2 . Then
B = Lω2 − Lω1 is a singular rank one positive semide nite matrix. By [7, Th. 4.3.1℄
λk (Lω1 ) ≤ λk (Lω2 )
and for k = 2,
for k = 1 . . . n,
µ(Gvω1 )
≤ µ(Gvω2 ).
To show that µ(Gvω ) is bounded, write Lω as the sum of two blo
k matri
es
L(G) 0
Lω =
0
0
+
0
0
0
ω
−ω
−ω
ω
,
where L(G) is n×n and the left upper zero blo
k in the se
ond matrix is (n−1)×(n−1).
By [7, Th. 4.3.4 (a), the
ase k = 2℄,
µ(Gvω ) = λ2 (Lω ) ≤ λ3 (L(G) ⊕ (0))
= λ2 (L(G)) = µ(G).
Remark 1.2. The theorem is essentially a
onsequen
e of Cor. 4.2 of [6℄. It is
proved for trees in [8℄.
Example 1.3. For the
omplete graphs Kn , n > 1, with all weights equal to 1
lim µ((Kn )vω ) =
ω→∞
Example 1.4.
n+1
< n = µ(Kn ).
2
For the
y
les Cn , n > 2, with weights equal to 1
lim µ((Cn )vω ) = µ(Cn+1 ) < µ(Cn ).
ω→∞
Example 1.5. Let G be the graph obtained from K4 by deleting an edge and
let all the weights of G be equal to 1. If the degree of v is 3,
lim µ(Gvω ) = 2 = µ(G).
ω→∞
If the degree of v is 2
lim µ(Gvω ) < µ(G).
ω→∞
Sin
e µ(Gvω ) is bounded by µ(G), it is natural to ask when does
lim µ(Gvω ) = µ(G).
ω→∞
We answer this question in Se
tion 3, in the
ase that G is a tree. The needed
ba
kground on the algebrai
onne
tivity of trees is des
ribed in Se
tion 2.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
177
2. Results on trees. Our paper relies heavily on the work of [12℄ so in this
se
tion we des
ribe their main results and basi
ba
kground on trees needed for these
results and for the next se
tion. In some
ases we
hange the notation of [12℄.
Theorem 2.1. [4, Th. 3.11℄ Let T be a weighted tree with Lapla
ian matrix L
and algebrai
onne
tivity µ. Let y be an eigenve
tor of L asso
iated with µ. Then
exa
tly one of the following two
ases o
ur:
(a) Some entry of y is 0.
(b) All entries of y are nonzero.
In the rst
ase there exists a unique vertex c su
h that yc = 0 and c is adja
ent to
a vertex d with yd = 0. In the se
ond
ase there is a unique pair of verti
es i and j
adja
ent in T su
h that yi yj < 0.
Definition 2.2. A weighted tree T is said to be of type I with a
hara
teristi
vertex c if
ase (a) of Theorem 2.1 holds, and of type II with
hara
teristi
verti
es i
and j in
ase (b). We use also the notation Ic in the rst
ase and IIi,j in the se
ond
ase.
The name
hara
teristi
verti
es was
oined in [11℄ by R. Merris who showed that
if µ is not a simple eigenvalue, then all the
orresponding eigenve
tors yield the same
type of tree and the same
hara
teristi
verti
es.
Definition 2.3. Let v be a vertex of a tree T . Let Lv be the matrix obtained
by deleting the row and
olumn of the Lapla
ian matrix of T that
orrespond to v.
The matrix Mv,T := L−1
v is
alled the bottlene
k matrix of T at v .
In [9℄ and [10℄, it is shown that the entry of Mv,T that
orresponds to the verti
es
k and l is
mkl =
1
ω(g)
where the summation is on all edges g that lie on the interse
tion of the path between
k and v and the path between l and v . The matrix Mv,T is permutationally similar
to a blo
k diagonal matrix, where the number of blo
ks is the degree of v and ea
h
blo
k is a positive matrix whi
h
orresponds to a unique bran
h at v.
For verti
es u, v of a tree T let v → u denote the bran
h of T at v, that
ontains u.
We denote by Mv→u,T the blo
k of Mv,T that
orresponds to v → u, and by Mv→u,T
the matrix obtained from Mv,T by deleting the rows and the
olumns
orresponding
to Mv→u,T .
Definition 2.4. A diagonal blo
k of Mv,T whose spe
tral radius is equal to
ρ(Mv,T ), where ρ(A) denotes the spe
tral radius of the matrix A, is
alled a Perron
blo
k and the
orresponding bran
h of T at v is
alled a Perron bran
h.
Theorem 2.5. [9, Cor. 2.1℄ Let T be a weighted tree. Then T is of type I with a
hara
teristi
vertex c, if and only if at c, T has more than one Perron bran
h.
In this
ase, µ(T ), the algebrai
onne
tivity of T is equal to ρ(M1c,T ) .
Let e be an edge of a graph G. Repla
e the weight at e by ω and denote the
resulting graph by Geω . Observe that sin
e e = {v, u} is a pendant edge of Gvω , then
(Gvω )eω = Gvω . Let Ge∞ denote the family of weighted graphs {Geω , ω > 0}, and let Gv∞
denote the family of weighted graphs {Gvω , ω > 0}.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
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178
A. Berman and K.-H. Förster
Theorem 2.6. [12, Corollary 1.1℄ Let T be a weighted tree and let e be an edge
of T . Then there exists a positive number ω0 su
h that all the trees Tωe , ω0 < ω < ∞,
are of the same type and have the same
hara
teristi
verti
es.
The following de nitions are used in [12℄.
e
is a type I tree at in nity with
hara
The family of trees T∞
teristi
vertex c if there exists an ω0 > 0 su
h that for all ω ∈ [ω0 , ∞), Tωe is of type
e
Ic . Similarly, T∞
is a type II tree at in nity with
hara
teristi
verti
es i and j if
there exists an ω0 > 0 su
h that for all ω ∈ [ω0 , ∞), Tωe is of type IIi,j .
We now
an state the main result of [12℄.
Theorem 2.8. [12, Th.1.8℄ Let e = {v, u} be an edge that is not a pendant edge
Definition 2.7.
of a tree T . Let T1 and T2 be the resulting
omponents arising from the deletion of
e. Suppose v ∈ T1 , u ∈ T2 and µ(T1 ) ≤ µ(T2 ). Then lim µ(Tωe ) = µ(T1 ) i T1 is a
ω→∞
tree of type I with a
hara
teristi
vertex, say, c, and one of the following
onditions
holds:
e
is of type I with a
hara
teristi
vertex c.
(a) T∞
(b) c is in
ident to e and ρ(Mu,T2 ) ≤ µ(T1 1 ) .
We
on
lude the ba
kground results with the analogue of Theorem 2.5 for type
II trees and two propositions that will be used in proving the main result.
Theorem 2.9. [9, Th.1℄ A weighted tree T is of type II i at every vertex T has
a unique Perron bran
h. If the
hara
teristi
verti
es, i and j , of T are joined by an
edge of weight θ, then there exists a number 0 < γ < 1, su
h that
ρ(Mi→j,T −
µ(T ) =
1−γ
γ
J) = ρ(Mj→i,T −
J), and
θ
θ
1
ρ(Mi→j,T −
γ
θ J)
=
1
ρ(Mj→i,T −
,
1−γ
θ J)
where J denotes an all ones matrix.
Proposition 2.10.
[8, Cor. 1.1℄ The
hara
teristi
verti
es of Tωv lie on the path
between the
hara
teristi
verti
es of T and u.
Proposition 2.11. [12, Claim 3.2℄ Let T be a tree. Let {ik , jk } be edges in T
with weights αk , for k = 1, 2 , su
h that the path from i1 to j2
ontains j1 and i2 , and
let 0 < γ1 , γ2 < 1. Then
ρ(Mj1 →i1 ,T −
γ1
γ2
J) < ρ(Mj1 →i1 ,T ) < ρ(Mj2 →i2 ,T −
J).
α1
α2
3. Assigning an arbitrarily large weight to a pendant edge of a tree. In
this se
tion we
onsider the
ase where T is a tree and u is a pendant vertex of T ∪ e
where e = {v, u} and v ∈ T .
In some sense the question in this
ase may be
onsidered as a spe
ial
ase of
the dis
ussion in [12℄. To do this, {u} is to be
onsidered as a "tree with algebrai
onne
tivity ∞" and the spe
tral radius of an empty matrix has to be de ned (for
example as 0).
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
179
Our dis
ussion is based on the analysis of the limits of the bottlene
k matri
es of
Tωv
when
ω
in
reases to
∞;
namely
M
v,Tωv
= Mv,T
1
⊕
,
ω
1
J
ω
Mu,Tωv = (Mv,T ⊕ (0)) +
and if
s = v
is a vertex of
T,
Ms,Tωv =
where
of
M (v)
Ms,T ,
not
ontain
v,
lim Ms,Tωv
by
ω→∞
Ms,T
is the
olumn of
orresponding to
v.
M (v)
mvv + ω1
Ms,T
M (v)t
orresponding to
and
mvv
is the diagonal entry
In parti
ular, for all the bran
hes of
Ms,Tωv
s∈
/e
the diagonal blo
ks of
v
Ms,T∞
v
,
we see that for
v =
Ms,T∞
and of
M (v)
Ms,T
M
Ms,T
(v)t
mvv
T
at
are the same.
s
that do
Denoting
(3.1)
and
v = Mu,T v = Mv,T ⊕ (0).
Mv,T∞
∞
The reader should not be
onfused by the fa
t that
while
v
Ms,T∞
v
T∞
(3.2)
denotes a family of trees
denotes a single matrix (up to permutation similarity).
Example 3.1.
◦@@
@@1/2
@@
@
~◦c
~
~
~
~
b ~~ 1/2
◦
a
v = Mu,T v
Mv,T∞
∞
v
Mc,T∞
1
◦v
ω
◦u
3
1
=
1
0
2
0
=
0
0
0
2
0
0
1
3
1
0
0
0
1
1
1
1
1
0
0
0
,
0
0
0
0
,
1
1
Electronic Journal of Linear Algebra ISSN 1081-3810
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A. Berman and K.-H. Förster
180
4
2
=
2
2
v = Mb,T v
Ma,T∞
∞
Remark 3.2. The matri
es
information on
limω→∞ µ(Tωv ).
v
Ms,T∞
2
2
2
2
2
2
3
3
2
2
.
3
3
are of
ourse singular, but they do
ontain
As in the
ase of nonsingular bottlene
k matri
es we
all the diagonal blo
ks of
v whose spe
tral radius is maximal, Perron blo
ks. The
orresponding bran
hes
Ms,T∞
v
v
of Tω do not depend on ω ; they will be
alled the Perron bran
hes of the family T∞ .
v
Ms,T∞
Lemma 3.3. If
has more than one Perron blo
k, then
lim µ(Tωv ) =
ω→∞
1
.
v )
ρ(Ms,T∞
Proof. Consider the prin
ipal submatrix
trix of
Tωv
Ls,Tωv
obtained from the Lapla
ian ma-
by deleting the row and
olumn
orresponding to
s.
Then
−1
v = lim (Ls,T v )
Ms,T∞
.
ω
ω→∞
By [7, Th. 4.3.15℄ for
r = n − 1, k = 1
and
k = 2,
λ1 (Ls,Tωv ) ≤ µ(Tωv ) ≤ λ2 (Ls,Tωv ),
so
lim λ1 (Ls,Tωv ) ≤ lim µ(Tωv ) ≤ lim λ2 (Ls,Tωv ),
ω→∞
Sin
e
v
Ms,T∞
ω→∞
ω→∞
has at least two Perron blo
ks we obtain
v ).
lim λ2 (Ls,Tωv ) = lim λ1 (Ls,Tωv ) = ρ(Ms,T∞
ω→∞
ω→∞
ω0 su
h
ω ≥ ω0 then
Remark 3.4. If there exists an
are Perron blo
ks of
Ms,Tωv
for
that two of the Perron blo
ks of
v
Ms,T∞
λ2 (Ls,Tωv ) = λ1 (Ls,Tωv ) for ω ≥ ω0 .
Lemma 3.5.
blo
ks and let
t
Let
s
be a vertex of
be another vertex of
T.
T.
Suppose
v
Ms,T∞
Then
v ) > ρ(Ms,T v )
ρ(Mt,T∞
∞
has at least two Perron
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
181
v
has at least two Perron bran
hes at s, so
. By assumption, the family T∞
one of them, say s → x, does not
ontain t. Let t → s be the bran
h at t that
ontains
s. Then it
ontains the bran
h s → x, and we obtain
Proof
v ) ≥ ρ(Mt→s,T v ) > ρ(Ms→x,T v ) = ρ(Ms,T v ),
ρ(Mt,T∞
∞
∞
∞
v is
where the stri
t inequality follows from [1, Cor. 2.1.5℄ and the fa
t that Ms→x,T∞
v , whi
h is positive.
a submatrix of Mt→s,T∞
Corollary 3.6. There is at most one vertex, say
c,
su
h that
v
Mc,T∞
has more
than one Perron blo
k.
Definition 3.7. In the
ase that there is a vertex c su
h that Mc,T v has more
∞
v
than one Perron blo
k, we will say that the family of trees T∞
is a trii (tree in in nity)
v
is a trii of type II.
of type I
. If no su
h c exists we say that T∞
Remark 3.8.
v
(a) If the trees Tωv are of type Ic for all su
iently large ω , then the family T∞
is a trii of type Ic (and also a type I tree at in nity with
hara
teristi
vertex c). In
v
other words, if T∞
is a trii of type II, then for all ω large enough, Tωv are trees of type
II.
(b) Suppose the trees Tωv are of type IIp,q for all su
iently large ω , then by the
representation of Lω in the proof of Theorem 1.1 and by Theorem 2.1, {p, q}
annot
be the pendant edge {v, u}.
v
(
) It is possible that Tωv are of type II for all su
iently large ω (so T∞
is a
v
type II tree at in nity) but Tω is a trii of type I; see Lemma 3.10 and Sub
ase 4 of
Example 3.13 in the following dis
ussion.
Remark 3.9. The proof of Lemma 3.5 shows that if T is a tree of type I with a
hara
teristi
vertex c, then for any other vertex s of T
ρ(Ms,T ) > ρ(Mc,T ).
(This has already been established in Proposition 2 of [9℄.)
Consider Theorem 2.9 where Tωv is of type IIi,j and the weight of the edge {i, j}
is θ. Then for every ω (su
iently large) there exist a number γω , between 0 and 1,
su
h that
µ(Tωv ) =
1
ρ(Mi→j,Tωv −
γω
θ J)
=
1
ρ(Mj→i,Tωv −
.
1−γω
θ J)
What happens to the the number γω when ω goes to ∞? We
laim that lim γω
ω→∞
exists. Indeed, one of the bran
hes
orresponding to Mi→j,Tωv and Mj→i,Tωv does not
ontain u. Suppose it is the se
ond, so Mj→i,Tωv = Mj→i,T . The numbers µ(Tωv )
ω
in
rease to a limit, see Theorem 1.1, so the numbers ρ(Mj→i,Tωv − 1−γ
θ J) de
rease to
a limit , whi
h means that the numbers 1-γω in
rease to a limit. This limit is at most
1 sin
e 0 < γω < 1.
Tωv are of type II, with
hara
teristi
verti
es i, j for
ω
ω0 < ω < ∞, and if γ = lim γω = 0, where ρ(Mi→j,Tωv − γθω J) = ρ(Mj→i,Tωv − 1−γ
θ J)
Lemma 3.10. If the trees
ω→∞
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
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182
A. Berman and K.-H. Förster
v
θ is the weight of the edge {i, j}, then T∞
is a trii of type Ii . Similarly, if γ = 1
v
then T∞ is a trii of type Ij .
1
v
v ⊕ (0)) +
v
Proof.
Mj→i,T∞
= (Mij,T∞
θ J so if γ = 0, then ρ(Mi→j,T∞ ) =
1
e
v
v −
J)
=
ρ(M
)
so
T
is
a
trii
of
type
I
.
ρ(Mj→i,T∞
ij,T∞
i
∞
θ
v
v
Corollary 3.11. If the trees Tω are of type II and if T∞ is a trii of type II,
then 0 < γ = lim γω < 1.
and
ω→∞
The tree T
an be a tree of type I with a
hara
teristi
vertex,
say c, or a tree of type II. In the rst
ase there are 3 possibilities:
v
1 T∞
is a trii of type Ic ,
v
2 T∞
is a trii of type Is , where s = c,
v
3 T∞
is a trii of type II.
In the se
ond
ase there are two possibilities:
v
4 T∞
is a trii of type Is for some s,
v
5 T∞
is a trii of type II.
Remark 3.12.
The following example demonstrates that all ve sub
ases are possible.
Example 3.13.
Sub
ase 1
◦@@
@@x
@@
@ 1
ω
◦v
◦u
~◦c
~
~
~
~~ x
◦~
0 < x ≤ 21 .
Here
Mc,Tωv
1/x
0
0
0
0
1/x
0
0
=
0
0
1
1
0
0
1 1+
1
ω
,
v
so for x < 12 , T∞
is a tree of type I with a
hara
teristi
vertex c and for x =
only a trii of type Ic .
Another example is when c = v
◦@@
@@x
@@
@
~◦c
~
~~
~~ x
~
◦
ω
.
◦u
1
2,
it is
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
Sub
ase 2
◦@@
@@x
@@
@
~◦c
~
~
~~x
~
◦~
10
◦s
1
◦v
ω
◦u
where
1/x + 0.1
0.1
ρ
0.1
0.1
1/x + 0.1
0.1
0.1
0.1
0.1
= 2.
Sub
ase 3
◦@@
@@x
@@
@
~◦c
~
~
~~x
~
◦~
1
2
1
◦v
ω
◦u
< x ≤ 1,
or
◦
1
◦c
1
◦v
ω
◦.
Sub
ase 4
◦
1
◦
x
◦s
1
◦v
ω
◦, ρ
1 + 1/x 1/x
1/x
1/x
Sub
ase 5
Here we suggest 3 examples:
◦@@
@@x
@@
@ 1
◦
~◦
~
~
~
~~ x
◦~
x > 1,
◦
x
1
ω
◦
ω
◦
◦
◦
x∈
/ {1/2, 1},
= 2.
183
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184
A. Berman and K.-H. Förster
◦@@
@@1
@@
@ ω
◦u
~◦v
~
~
~
~~ x
◦~
x = 1.
We are now ready to state and prove the main result.
Theorem 3.14. Let
T
be a tree. Then
(3.3)
lim µ(Tωv ) = µ(T )
ω→∞
if and only if
(a) T
is a tree of type I with
hara
teristi
vertex say
c,
and
1
v ) ≤ ρ(Mc,T ) =
(b) ρ(Mc→u,T∞
µ(T ) .
Proof. We prove the theorem by
onsidering the ve sub
ases of Remark 3.12,
and showing that (3), (a) and (b) hold in Sub
ase 1 and only in this
ase, i.e. if and
v
are of type Ic for some vertex c.
only if T and T∞
Sub
ase 1: Obviously (a) holds. From (1) and (2) follows that ρ(Mc,Tωv ) ≥
ρ(Mc,T ). But if T and Tωv are of type Ic , then equality holds. Thus
v ) ≤ ρ(Mc,T ),
ρ(Mc→u,T∞
proving (b), and
µ(T ) =
1
1
=
= lim µ(Tωv ),
ρ(Mc,T )
ρ(Mc,Tωv ) ω→∞
proving (3). This
ompletes the proof in Sub
ase 1.
If T is a tree of type Ic and (b) holds, then it follows easily that Tωv is a trii of
type Ic. Therefore (b) does not hold in Sub
ases 2 and 3, while (a) obviously does
not hold in Sub
ases 4 and 5. Now we will prove that (3) does not hold in the last
four sub
ases.
Sub
ase 2: We have to show that (3) does not hold. Indeed
1
1
>
ρ(Mc,T )
ρ(Ms,T )
1
, by Lemma 3.5
=
ρ(Ms,Tωv )
µ(T ) =
= lim µ(Tωv ) , by Lemma 3.3.
ω→∞
Sub
ase 3: By Remark 3.8(a) the trees Tωv are for su
iently large ω of type II ,
say of type IIp,q , see Theorem 2.6, and the edge {p, q} of T has weight θ, by Remark
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
3.8(b) it does not depend on
q
ω.
Without loss of generality,
p lies on the
185
path between
and c.
p and q lie
c su
h that c, p and q lie on the
of T . Then we obtain
By Proposition 2.10 the verti
es
be a neighbor of
Perron bran
h
1
lim µ(Tωv ) = lim
c
on the path between
path between
i
and
u
, by Theorem 2.9,
ρ(Mq→p,Tωv − γθω J)
1
= lim
, sin
e q → p is in T ,
γ
ω→∞ ρ(Mq→p,T − ω J)
θ
1
=
, where 0 < γ < 1, by Corollary
ρ(Mq→p,T − γθ J)
1
<
, by Proposition 2.11,
ρ(Mc→i,T )
1
=
, sin
e c → i is a Perron bran
h of T,
ρ(Mc,T )
= µ(T ) , sin
e T is of type Ic ,
ω→∞
u. Let i
c → i is a
and
and
ω→∞
3.11,
so (3) does not hold.
T is of
IIij , where j lies on the path from i to u. Let θ and γ be as in Theorem 2.9.
Sin
e Tωv is a trii of type Is , the by Proposition 2.10, s lies on the path from i to u.
Sub
ase 4: Here again we have to show that (3) does not hold. Suppose
type
Therefore
1
1
= lim
ρ(Ms→i,Tωv ) ω→∞ ρ(Ms→i,T )
1
1
≤
<
= µ(T ).
ρ(Mj→i,T )
ρ(Mj→i,T − γθ J)
lim µ(Tωv ) = lim
ω→∞
ω→∞
Sub
ase 5: Here T is of, say, type IIij and for
ω
large enough,
Tωv
are of, say, type
p and q to
{i, j} in T and
Observe that θˆ does
IIpq , where by Proposition 2.10, we may take, without loss of generality,
i
lie between
let
θˆ and γω
and
Let
θ
and
γ
be as in Theorem 2.9 for the edge
{p, q} in Tωv .
γˆ = limω→∞ γω . By Corollary
be the
orresponding pair for the edge
not depend on
0 < γ < 1.
q.
ω
by Remark 3.8(b). Let
3.13 we have
Now
lim µ(Tωv ) = lim
ω→∞
ω→∞
=
1
ρ(Mq→p,Tωv −
1
ρ(Mq→p,T −
γ
ˆ
J)
θˆ
γω
J)
θˆ
<
= lim
ω→∞
1
ρ(Mq→p,T −
γω
J)
θˆ
1
= µ(T ),
ρ(Mj→i,T − γθ J)
where the inequality follows from Proposition 2.11.
A
knowledgments.
We are indebted to Mr. Felix Goldberg [5℄ for suggesting
Example 1.5 whi
h shows that
G
does not have to be a tree for
lim µ(Gvω ) = µ(G)
ω→∞
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186
A. Berman and K.-H. Förster
lim µ(Gvω ) = µ(G), when G is a general graph,
to hold. The question of when does ω→∞
seems to be mu
h more di
ult than the one in the
ase that G is a tree. We
are grateful to the referee for his or her important remarks and for suggesting that
Propositions 1.3 and 1.4, as well as Lemma 2.2 of [2℄, may be useful in dealing with
the general
ase.
REFERENCES
[1℄ A. Berman and R.J. Plemmons. Nonnegative Matri
es in the Mathemati
al S
ien
es, SIAM,
Philadelphia, 1994.
[2℄ S. Fallat and S. Kirkland. Extremizing algebrai
onne
tivity subje
t to graph theoreti
onstraints. Ele
troni
Journal of Linear Algebra, 3:4874, 1998.
[3℄ M. Fiedler. Algebrai
onne
tivity of graphs. Cze
hoslovak Mathemati
al Journal, 23: 298305,
1973.
[4℄ M. Fiedler. A property of eigenve
tors of nonnegative symmetri
matri
es and its appli
ations
to graph theory. Cze
hoslovak Mathemati
al Journal, 25: 619633, 1975.
[5℄ F. Goldberg. Private
ommuni
ation.
[6℄ R. Grone, R. Merris, and V. Sunder. The Lapla
ian spe
trum of a graph. SIAM Journal on
Matrix Analysis and Appli
ations, 11: 218-238, 1990.
[7℄ R.J. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985.
[8℄ S. Kirkland and M. Neumann. Algebrai
onne
tivity of weighted trees under perturbation.
Linear and Multilinear Algebra, 42: 187203, 1997.
[9℄ S. Kirkland, M. Neumann, and B. Shader. Chara
teristi
verti
es of weighted trees via Perron
values. Linear and Multilinear Algebra, 40: 311325, 1996.
[10℄ S. Kirkland, M. Neumann, and B. Shader. Distan
es in weighted trees and group inverse of
Lapla
ian Matri
es. SIAM Journal on Matrix Analysis and Appli
ations, 18: 827841,
1997.
[11℄ R. Merris. Chara
teristi
verti
es of trees. Linear and Multilinear Algebra, 22: 115131, 1987.
[12℄ J.J. Molitierno and M. Neumann. The algebrai
onne
tivity of two trees
onne
ted by an edge
of in nite weight. Ele
troni
Journal of Linear Algebra, 8: 113, 2001.
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
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ALGEBRAIC CONNECTIVITY OF TREES WITH A PENDANT
∗
EDGE OF INFINITE WEIGHT
A. BERMAN†
AND
K.-H. FÖRSTER‡
Abstra
t.
Let G be a weighted graph. Let v be a vertex of G and let Gv
ω denote the graph
obtained by adding a vertex u and an edge {v, u} with weight ω to G. Then the algebrai
onne
tivity
µ(Gvω ) of Gvω is a nonde
reasing fun
tion of ω and is bounded by the algebrai
onne
tivity µ(G) of
G. The question of when lim µ(Gvω ) is equal to µ(G) is
onsidered and answered in the
ase that
ω→∞
G is a tree.
Key words.
Weighted graphs, Trees, Lapla
ian matrix, Algebrai
onne
tivity, Pendant edge.
AMS subje
t
lassi
ations.
5C50, 5C40, 15A18.
1. Introdu
tion. A weighted graph on n verti
es is an undire
ted simple graph
G on n verti
es su
h that with ea
h edge e of G, there is an asso
iated positive number
ω(e) whi
h is
alled the weight of e.
The Lapla
ian matrix of a weighted graph G on n verti
es is the n × n matrix
L(G) = L = (lij ), where for ea
h i, j = 1, . . . , n,
−ω(e)
lij =
0
l
k=i ik
if i = j and e = {i, j},
if i = j and i is not adja
ent to j,
if i = j.
Clearly L is a singular M -matrix and positive semide nite, so λ1 (L) = 0, where
for a symmetri
matrix A we arrange the eigenvalues in nonde
reasing order
λ1 (A) ≤ λ2 (A) ≤ . . .
Fiedler [3℄ showed that λ2 (L) is positive i G is
onne
ted and
alled it the
algebrai
onne
tivity of G. The algebrai
onne
tivity of G will be denoted by µ(G).
In this paper G always denotes a
onne
ted weighted graph without loops.
Let G be a graph with n verti
es. Let v be a vertex of G and let Gvω be the graph
with n + 1 verti
es obtained by adding to G a vertex u and an edge e = {v, u} with
weight ω .
∗ Re
eived by the editors 21 July 2003. A
epted for publi
ation 24 June 2005. Handling Editor:
Stephen J. Kirkland.
† Department of Mathemati
s, Te
hnion, Haifa 32000, Israel,(bermante
hunix.te
hnion.a
.il).
Resear
h supported by the New York Metropolitan Fund for Resear
h at the Te
hnion. Most of
the work was done during a visit to TU Berlin; part of the work was done in FU Berlin, Charite Benjamin Franklin.
‡ Department of Mathemati
s, Te
hni
al University Berlin, Sekr. MA 6-4, Strasse des 17. Juni
136, 10623 Berlin, Germany (foerstermath.tu-berlin.de).
175
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176
A. Berman and K.-H. Förster
Theorem 1.1. The algebrai
onne
tivity
ω
and for every
ω
and
µ(Gvω )
is a nonde
reasing fun
tion of
n>1
µ(Gvω ) ≤ µ(G).
v
Proof. Let Lω be the Lapla
ian matrix of Gω and let 0 < ω1 ≤ ω2 . Then
B = Lω2 − Lω1 is a singular rank one positive semide nite matrix. By [7, Th. 4.3.1℄
λk (Lω1 ) ≤ λk (Lω2 )
and for k = 2,
for k = 1 . . . n,
µ(Gvω1 )
≤ µ(Gvω2 ).
To show that µ(Gvω ) is bounded, write Lω as the sum of two blo
k matri
es
L(G) 0
Lω =
0
0
+
0
0
0
ω
−ω
−ω
ω
,
where L(G) is n×n and the left upper zero blo
k in the se
ond matrix is (n−1)×(n−1).
By [7, Th. 4.3.4 (a), the
ase k = 2℄,
µ(Gvω ) = λ2 (Lω ) ≤ λ3 (L(G) ⊕ (0))
= λ2 (L(G)) = µ(G).
Remark 1.2. The theorem is essentially a
onsequen
e of Cor. 4.2 of [6℄. It is
proved for trees in [8℄.
Example 1.3. For the
omplete graphs Kn , n > 1, with all weights equal to 1
lim µ((Kn )vω ) =
ω→∞
Example 1.4.
n+1
< n = µ(Kn ).
2
For the
y
les Cn , n > 2, with weights equal to 1
lim µ((Cn )vω ) = µ(Cn+1 ) < µ(Cn ).
ω→∞
Example 1.5. Let G be the graph obtained from K4 by deleting an edge and
let all the weights of G be equal to 1. If the degree of v is 3,
lim µ(Gvω ) = 2 = µ(G).
ω→∞
If the degree of v is 2
lim µ(Gvω ) < µ(G).
ω→∞
Sin
e µ(Gvω ) is bounded by µ(G), it is natural to ask when does
lim µ(Gvω ) = µ(G).
ω→∞
We answer this question in Se
tion 3, in the
ase that G is a tree. The needed
ba
kground on the algebrai
onne
tivity of trees is des
ribed in Se
tion 2.
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
177
2. Results on trees. Our paper relies heavily on the work of [12℄ so in this
se
tion we des
ribe their main results and basi
ba
kground on trees needed for these
results and for the next se
tion. In some
ases we
hange the notation of [12℄.
Theorem 2.1. [4, Th. 3.11℄ Let T be a weighted tree with Lapla
ian matrix L
and algebrai
onne
tivity µ. Let y be an eigenve
tor of L asso
iated with µ. Then
exa
tly one of the following two
ases o
ur:
(a) Some entry of y is 0.
(b) All entries of y are nonzero.
In the rst
ase there exists a unique vertex c su
h that yc = 0 and c is adja
ent to
a vertex d with yd = 0. In the se
ond
ase there is a unique pair of verti
es i and j
adja
ent in T su
h that yi yj < 0.
Definition 2.2. A weighted tree T is said to be of type I with a
hara
teristi
vertex c if
ase (a) of Theorem 2.1 holds, and of type II with
hara
teristi
verti
es i
and j in
ase (b). We use also the notation Ic in the rst
ase and IIi,j in the se
ond
ase.
The name
hara
teristi
verti
es was
oined in [11℄ by R. Merris who showed that
if µ is not a simple eigenvalue, then all the
orresponding eigenve
tors yield the same
type of tree and the same
hara
teristi
verti
es.
Definition 2.3. Let v be a vertex of a tree T . Let Lv be the matrix obtained
by deleting the row and
olumn of the Lapla
ian matrix of T that
orrespond to v.
The matrix Mv,T := L−1
v is
alled the bottlene
k matrix of T at v .
In [9℄ and [10℄, it is shown that the entry of Mv,T that
orresponds to the verti
es
k and l is
mkl =
1
ω(g)
where the summation is on all edges g that lie on the interse
tion of the path between
k and v and the path between l and v . The matrix Mv,T is permutationally similar
to a blo
k diagonal matrix, where the number of blo
ks is the degree of v and ea
h
blo
k is a positive matrix whi
h
orresponds to a unique bran
h at v.
For verti
es u, v of a tree T let v → u denote the bran
h of T at v, that
ontains u.
We denote by Mv→u,T the blo
k of Mv,T that
orresponds to v → u, and by Mv→u,T
the matrix obtained from Mv,T by deleting the rows and the
olumns
orresponding
to Mv→u,T .
Definition 2.4. A diagonal blo
k of Mv,T whose spe
tral radius is equal to
ρ(Mv,T ), where ρ(A) denotes the spe
tral radius of the matrix A, is
alled a Perron
blo
k and the
orresponding bran
h of T at v is
alled a Perron bran
h.
Theorem 2.5. [9, Cor. 2.1℄ Let T be a weighted tree. Then T is of type I with a
hara
teristi
vertex c, if and only if at c, T has more than one Perron bran
h.
In this
ase, µ(T ), the algebrai
onne
tivity of T is equal to ρ(M1c,T ) .
Let e be an edge of a graph G. Repla
e the weight at e by ω and denote the
resulting graph by Geω . Observe that sin
e e = {v, u} is a pendant edge of Gvω , then
(Gvω )eω = Gvω . Let Ge∞ denote the family of weighted graphs {Geω , ω > 0}, and let Gv∞
denote the family of weighted graphs {Gvω , ω > 0}.
Electronic Journal of Linear Algebra ISSN 1081-3810
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178
A. Berman and K.-H. Förster
Theorem 2.6. [12, Corollary 1.1℄ Let T be a weighted tree and let e be an edge
of T . Then there exists a positive number ω0 su
h that all the trees Tωe , ω0 < ω < ∞,
are of the same type and have the same
hara
teristi
verti
es.
The following de nitions are used in [12℄.
e
is a type I tree at in nity with
hara
The family of trees T∞
teristi
vertex c if there exists an ω0 > 0 su
h that for all ω ∈ [ω0 , ∞), Tωe is of type
e
Ic . Similarly, T∞
is a type II tree at in nity with
hara
teristi
verti
es i and j if
there exists an ω0 > 0 su
h that for all ω ∈ [ω0 , ∞), Tωe is of type IIi,j .
We now
an state the main result of [12℄.
Theorem 2.8. [12, Th.1.8℄ Let e = {v, u} be an edge that is not a pendant edge
Definition 2.7.
of a tree T . Let T1 and T2 be the resulting
omponents arising from the deletion of
e. Suppose v ∈ T1 , u ∈ T2 and µ(T1 ) ≤ µ(T2 ). Then lim µ(Tωe ) = µ(T1 ) i T1 is a
ω→∞
tree of type I with a
hara
teristi
vertex, say, c, and one of the following
onditions
holds:
e
is of type I with a
hara
teristi
vertex c.
(a) T∞
(b) c is in
ident to e and ρ(Mu,T2 ) ≤ µ(T1 1 ) .
We
on
lude the ba
kground results with the analogue of Theorem 2.5 for type
II trees and two propositions that will be used in proving the main result.
Theorem 2.9. [9, Th.1℄ A weighted tree T is of type II i at every vertex T has
a unique Perron bran
h. If the
hara
teristi
verti
es, i and j , of T are joined by an
edge of weight θ, then there exists a number 0 < γ < 1, su
h that
ρ(Mi→j,T −
µ(T ) =
1−γ
γ
J) = ρ(Mj→i,T −
J), and
θ
θ
1
ρ(Mi→j,T −
γ
θ J)
=
1
ρ(Mj→i,T −
,
1−γ
θ J)
where J denotes an all ones matrix.
Proposition 2.10.
[8, Cor. 1.1℄ The
hara
teristi
verti
es of Tωv lie on the path
between the
hara
teristi
verti
es of T and u.
Proposition 2.11. [12, Claim 3.2℄ Let T be a tree. Let {ik , jk } be edges in T
with weights αk , for k = 1, 2 , su
h that the path from i1 to j2
ontains j1 and i2 , and
let 0 < γ1 , γ2 < 1. Then
ρ(Mj1 →i1 ,T −
γ1
γ2
J) < ρ(Mj1 →i1 ,T ) < ρ(Mj2 →i2 ,T −
J).
α1
α2
3. Assigning an arbitrarily large weight to a pendant edge of a tree. In
this se
tion we
onsider the
ase where T is a tree and u is a pendant vertex of T ∪ e
where e = {v, u} and v ∈ T .
In some sense the question in this
ase may be
onsidered as a spe
ial
ase of
the dis
ussion in [12℄. To do this, {u} is to be
onsidered as a "tree with algebrai
onne
tivity ∞" and the spe
tral radius of an empty matrix has to be de ned (for
example as 0).
Electronic Journal of Linear Algebra ISSN 1081-3810
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
179
Our dis
ussion is based on the analysis of the limits of the bottlene
k matri
es of
Tωv
when
ω
in
reases to
∞;
namely
M
v,Tωv
= Mv,T
1
⊕
,
ω
1
J
ω
Mu,Tωv = (Mv,T ⊕ (0)) +
and if
s = v
is a vertex of
T,
Ms,Tωv =
where
of
M (v)
Ms,T ,
not
ontain
v,
lim Ms,Tωv
by
ω→∞
Ms,T
is the
olumn of
orresponding to
v.
M (v)
mvv + ω1
Ms,T
M (v)t
orresponding to
and
mvv
is the diagonal entry
In parti
ular, for all the bran
hes of
Ms,Tωv
s∈
/e
the diagonal blo
ks of
v
Ms,T∞
v
,
we see that for
v =
Ms,T∞
and of
M (v)
Ms,T
M
Ms,T
(v)t
mvv
T
at
are the same.
s
that do
Denoting
(3.1)
and
v = Mu,T v = Mv,T ⊕ (0).
Mv,T∞
∞
The reader should not be
onfused by the fa
t that
while
v
Ms,T∞
v
T∞
(3.2)
denotes a family of trees
denotes a single matrix (up to permutation similarity).
Example 3.1.
◦@@
@@1/2
@@
@
~◦c
~
~
~
~
b ~~ 1/2
◦
a
v = Mu,T v
Mv,T∞
∞
v
Mc,T∞
1
◦v
ω
◦u
3
1
=
1
0
2
0
=
0
0
0
2
0
0
1
3
1
0
0
0
1
1
1
1
1
0
0
0
,
0
0
0
0
,
1
1
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A. Berman and K.-H. Förster
180
4
2
=
2
2
v = Mb,T v
Ma,T∞
∞
Remark 3.2. The matri
es
information on
limω→∞ µ(Tωv ).
v
Ms,T∞
2
2
2
2
2
2
3
3
2
2
.
3
3
are of
ourse singular, but they do
ontain
As in the
ase of nonsingular bottlene
k matri
es we
all the diagonal blo
ks of
v whose spe
tral radius is maximal, Perron blo
ks. The
orresponding bran
hes
Ms,T∞
v
v
of Tω do not depend on ω ; they will be
alled the Perron bran
hes of the family T∞ .
v
Ms,T∞
Lemma 3.3. If
has more than one Perron blo
k, then
lim µ(Tωv ) =
ω→∞
1
.
v )
ρ(Ms,T∞
Proof. Consider the prin
ipal submatrix
trix of
Tωv
Ls,Tωv
obtained from the Lapla
ian ma-
by deleting the row and
olumn
orresponding to
s.
Then
−1
v = lim (Ls,T v )
Ms,T∞
.
ω
ω→∞
By [7, Th. 4.3.15℄ for
r = n − 1, k = 1
and
k = 2,
λ1 (Ls,Tωv ) ≤ µ(Tωv ) ≤ λ2 (Ls,Tωv ),
so
lim λ1 (Ls,Tωv ) ≤ lim µ(Tωv ) ≤ lim λ2 (Ls,Tωv ),
ω→∞
Sin
e
v
Ms,T∞
ω→∞
ω→∞
has at least two Perron blo
ks we obtain
v ).
lim λ2 (Ls,Tωv ) = lim λ1 (Ls,Tωv ) = ρ(Ms,T∞
ω→∞
ω→∞
ω0 su
h
ω ≥ ω0 then
Remark 3.4. If there exists an
are Perron blo
ks of
Ms,Tωv
for
that two of the Perron blo
ks of
v
Ms,T∞
λ2 (Ls,Tωv ) = λ1 (Ls,Tωv ) for ω ≥ ω0 .
Lemma 3.5.
blo
ks and let
t
Let
s
be a vertex of
be another vertex of
T.
T.
Suppose
v
Ms,T∞
Then
v ) > ρ(Ms,T v )
ρ(Mt,T∞
∞
has at least two Perron
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tivity of Trees with a Pendant Edge of In nite Weight
181
v
has at least two Perron bran
hes at s, so
. By assumption, the family T∞
one of them, say s → x, does not
ontain t. Let t → s be the bran
h at t that
ontains
s. Then it
ontains the bran
h s → x, and we obtain
Proof
v ) ≥ ρ(Mt→s,T v ) > ρ(Ms→x,T v ) = ρ(Ms,T v ),
ρ(Mt,T∞
∞
∞
∞
v is
where the stri
t inequality follows from [1, Cor. 2.1.5℄ and the fa
t that Ms→x,T∞
v , whi
h is positive.
a submatrix of Mt→s,T∞
Corollary 3.6. There is at most one vertex, say
c,
su
h that
v
Mc,T∞
has more
than one Perron blo
k.
Definition 3.7. In the
ase that there is a vertex c su
h that Mc,T v has more
∞
v
than one Perron blo
k, we will say that the family of trees T∞
is a trii (tree in in nity)
v
is a trii of type II.
of type I
. If no su
h c exists we say that T∞
Remark 3.8.
v
(a) If the trees Tωv are of type Ic for all su
iently large ω , then the family T∞
is a trii of type Ic (and also a type I tree at in nity with
hara
teristi
vertex c). In
v
other words, if T∞
is a trii of type II, then for all ω large enough, Tωv are trees of type
II.
(b) Suppose the trees Tωv are of type IIp,q for all su
iently large ω , then by the
representation of Lω in the proof of Theorem 1.1 and by Theorem 2.1, {p, q}
annot
be the pendant edge {v, u}.
v
(
) It is possible that Tωv are of type II for all su
iently large ω (so T∞
is a
v
type II tree at in nity) but Tω is a trii of type I; see Lemma 3.10 and Sub
ase 4 of
Example 3.13 in the following dis
ussion.
Remark 3.9. The proof of Lemma 3.5 shows that if T is a tree of type I with a
hara
teristi
vertex c, then for any other vertex s of T
ρ(Ms,T ) > ρ(Mc,T ).
(This has already been established in Proposition 2 of [9℄.)
Consider Theorem 2.9 where Tωv is of type IIi,j and the weight of the edge {i, j}
is θ. Then for every ω (su
iently large) there exist a number γω , between 0 and 1,
su
h that
µ(Tωv ) =
1
ρ(Mi→j,Tωv −
γω
θ J)
=
1
ρ(Mj→i,Tωv −
.
1−γω
θ J)
What happens to the the number γω when ω goes to ∞? We
laim that lim γω
ω→∞
exists. Indeed, one of the bran
hes
orresponding to Mi→j,Tωv and Mj→i,Tωv does not
ontain u. Suppose it is the se
ond, so Mj→i,Tωv = Mj→i,T . The numbers µ(Tωv )
ω
in
rease to a limit, see Theorem 1.1, so the numbers ρ(Mj→i,Tωv − 1−γ
θ J) de
rease to
a limit , whi
h means that the numbers 1-γω in
rease to a limit. This limit is at most
1 sin
e 0 < γω < 1.
Tωv are of type II, with
hara
teristi
verti
es i, j for
ω
ω0 < ω < ∞, and if γ = lim γω = 0, where ρ(Mi→j,Tωv − γθω J) = ρ(Mj→i,Tωv − 1−γ
θ J)
Lemma 3.10. If the trees
ω→∞
Electronic Journal of Linear Algebra ISSN 1081-3810
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182
A. Berman and K.-H. Förster
v
θ is the weight of the edge {i, j}, then T∞
is a trii of type Ii . Similarly, if γ = 1
v
then T∞ is a trii of type Ij .
1
v
v ⊕ (0)) +
v
Proof.
Mj→i,T∞
= (Mij,T∞
θ J so if γ = 0, then ρ(Mi→j,T∞ ) =
1
e
v
v −
J)
=
ρ(M
)
so
T
is
a
trii
of
type
I
.
ρ(Mj→i,T∞
ij,T∞
i
∞
θ
v
v
Corollary 3.11. If the trees Tω are of type II and if T∞ is a trii of type II,
then 0 < γ = lim γω < 1.
and
ω→∞
The tree T
an be a tree of type I with a
hara
teristi
vertex,
say c, or a tree of type II. In the rst
ase there are 3 possibilities:
v
1 T∞
is a trii of type Ic ,
v
2 T∞
is a trii of type Is , where s = c,
v
3 T∞
is a trii of type II.
In the se
ond
ase there are two possibilities:
v
4 T∞
is a trii of type Is for some s,
v
5 T∞
is a trii of type II.
Remark 3.12.
The following example demonstrates that all ve sub
ases are possible.
Example 3.13.
Sub
ase 1
◦@@
@@x
@@
@ 1
ω
◦v
◦u
~◦c
~
~
~
~~ x
◦~
0 < x ≤ 21 .
Here
Mc,Tωv
1/x
0
0
0
0
1/x
0
0
=
0
0
1
1
0
0
1 1+
1
ω
,
v
so for x < 12 , T∞
is a tree of type I with a
hara
teristi
vertex c and for x =
only a trii of type Ic .
Another example is when c = v
◦@@
@@x
@@
@
~◦c
~
~~
~~ x
~
◦
ω
.
◦u
1
2,
it is
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
Sub
ase 2
◦@@
@@x
@@
@
~◦c
~
~
~~x
~
◦~
10
◦s
1
◦v
ω
◦u
where
1/x + 0.1
0.1
ρ
0.1
0.1
1/x + 0.1
0.1
0.1
0.1
0.1
= 2.
Sub
ase 3
◦@@
@@x
@@
@
~◦c
~
~
~~x
~
◦~
1
2
1
◦v
ω
◦u
< x ≤ 1,
or
◦
1
◦c
1
◦v
ω
◦.
Sub
ase 4
◦
1
◦
x
◦s
1
◦v
ω
◦, ρ
1 + 1/x 1/x
1/x
1/x
Sub
ase 5
Here we suggest 3 examples:
◦@@
@@x
@@
@ 1
◦
~◦
~
~
~
~~ x
◦~
x > 1,
◦
x
1
ω
◦
ω
◦
◦
◦
x∈
/ {1/2, 1},
= 2.
183
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
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184
A. Berman and K.-H. Förster
◦@@
@@1
@@
@ ω
◦u
~◦v
~
~
~
~~ x
◦~
x = 1.
We are now ready to state and prove the main result.
Theorem 3.14. Let
T
be a tree. Then
(3.3)
lim µ(Tωv ) = µ(T )
ω→∞
if and only if
(a) T
is a tree of type I with
hara
teristi
vertex say
c,
and
1
v ) ≤ ρ(Mc,T ) =
(b) ρ(Mc→u,T∞
µ(T ) .
Proof. We prove the theorem by
onsidering the ve sub
ases of Remark 3.12,
and showing that (3), (a) and (b) hold in Sub
ase 1 and only in this
ase, i.e. if and
v
are of type Ic for some vertex c.
only if T and T∞
Sub
ase 1: Obviously (a) holds. From (1) and (2) follows that ρ(Mc,Tωv ) ≥
ρ(Mc,T ). But if T and Tωv are of type Ic , then equality holds. Thus
v ) ≤ ρ(Mc,T ),
ρ(Mc→u,T∞
proving (b), and
µ(T ) =
1
1
=
= lim µ(Tωv ),
ρ(Mc,T )
ρ(Mc,Tωv ) ω→∞
proving (3). This
ompletes the proof in Sub
ase 1.
If T is a tree of type Ic and (b) holds, then it follows easily that Tωv is a trii of
type Ic. Therefore (b) does not hold in Sub
ases 2 and 3, while (a) obviously does
not hold in Sub
ases 4 and 5. Now we will prove that (3) does not hold in the last
four sub
ases.
Sub
ase 2: We have to show that (3) does not hold. Indeed
1
1
>
ρ(Mc,T )
ρ(Ms,T )
1
, by Lemma 3.5
=
ρ(Ms,Tωv )
µ(T ) =
= lim µ(Tωv ) , by Lemma 3.3.
ω→∞
Sub
ase 3: By Remark 3.8(a) the trees Tωv are for su
iently large ω of type II ,
say of type IIp,q , see Theorem 2.6, and the edge {p, q} of T has weight θ, by Remark
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 175-186, July 2005
ELA
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Algebrai
Conne
tivity of Trees with a Pendant Edge of In nite Weight
3.8(b) it does not depend on
q
ω.
Without loss of generality,
p lies on the
185
path between
and c.
p and q lie
c su
h that c, p and q lie on the
of T . Then we obtain
By Proposition 2.10 the verti
es
be a neighbor of
Perron bran
h
1
lim µ(Tωv ) = lim
c
on the path between
path between
i
and
u
, by Theorem 2.9,
ρ(Mq→p,Tωv − γθω J)
1
= lim
, sin
e q → p is in T ,
γ
ω→∞ ρ(Mq→p,T − ω J)
θ
1
=
, where 0 < γ < 1, by Corollary
ρ(Mq→p,T − γθ J)
1
<
, by Proposition 2.11,
ρ(Mc→i,T )
1
=
, sin
e c → i is a Perron bran
h of T,
ρ(Mc,T )
= µ(T ) , sin
e T is of type Ic ,
ω→∞
u. Let i
c → i is a
and
and
ω→∞
3.11,
so (3) does not hold.
T is of
IIij , where j lies on the path from i to u. Let θ and γ be as in Theorem 2.9.
Sin
e Tωv is a trii of type Is , the by Proposition 2.10, s lies on the path from i to u.
Sub
ase 4: Here again we have to show that (3) does not hold. Suppose
type
Therefore
1
1
= lim
ρ(Ms→i,Tωv ) ω→∞ ρ(Ms→i,T )
1
1
≤
<
= µ(T ).
ρ(Mj→i,T )
ρ(Mj→i,T − γθ J)
lim µ(Tωv ) = lim
ω→∞
ω→∞
Sub
ase 5: Here T is of, say, type IIij and for
ω
large enough,
Tωv
are of, say, type
p and q to
{i, j} in T and
Observe that θˆ does
IIpq , where by Proposition 2.10, we may take, without loss of generality,
i
lie between
let
θˆ and γω
and
Let
θ
and
γ
be as in Theorem 2.9 for the edge
{p, q} in Tωv .
γˆ = limω→∞ γω . By Corollary
be the
orresponding pair for the edge
not depend on
0 < γ < 1.
q.
ω
by Remark 3.8(b). Let
3.13 we have
Now
lim µ(Tωv ) = lim
ω→∞
ω→∞
=
1
ρ(Mq→p,Tωv −
1
ρ(Mq→p,T −
γ
ˆ
J)
θˆ
γω
J)
θˆ
<
= lim
ω→∞
1
ρ(Mq→p,T −
γω
J)
θˆ
1
= µ(T ),
ρ(Mj→i,T − γθ J)
where the inequality follows from Proposition 2.11.
A
knowledgments.
We are indebted to Mr. Felix Goldberg [5℄ for suggesting
Example 1.5 whi
h shows that
G
does not have to be a tree for
lim µ(Gvω ) = µ(G)
ω→∞
Electronic Journal of Linear Algebra ISSN 1081-3810
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Volume 13, pp. 175-186, July 2005
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186
A. Berman and K.-H. Förster
lim µ(Gvω ) = µ(G), when G is a general graph,
to hold. The question of when does ω→∞
seems to be mu
h more di
ult than the one in the
ase that G is a tree. We
are grateful to the referee for his or her important remarks and for suggesting that
Propositions 1.3 and 1.4, as well as Lemma 2.2 of [2℄, may be useful in dealing with
the general
ase.
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