The pLaplacian spectral radius of weighted trees with a degree sequence and a weight set
Volume
10813810
The p Laplacian spectral radius of weighted trees with a degree sequence and a weight set
GuangJun Zhang
XiaoDong Zhang
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THE P LAPLACIAN SPECTRAL RADIUS OF WEIGHTED TREES
WITH A DEGREE SEQUENCE AND A WEIGHT SET∗
GUANGJUN ZHANG† AND XIAODONG ZHANG†
AMS subject classifications. 05C50, 05C05, 05C22.
1. Introduction. In the last decade, the pLaplacian, which is a natural
nonlinear generalization of the standard Laplacian, plays an increasing role in geometry
and partial differential equations. Recently, the discrete pLaplacian, which is the
analogue of the pLaplacian on Riemannian manifolds, has been investigated by many
researchers. For example, Amghibech in [
1
] presented several sharp upper bounds
for the largest pLaplacian eigenvalues of graphs. Takeuchi in [
7
] investigated the
spectrum of the pLaplacian and pharmonic morphism of graphs. Luo et al. in [
6
]
used the eigenvalues and eigenvectors of the pLaplacian to obtain a natural global
embedding for multiclass clustering problems in machine learning and data mining
areas. Based on the increasing interest in both theory and application, the spectrum
of the discrete pLaplacian should be further investigated. The main purpose of this
paper is to investigate some properties of the spectral radius and eigenvectors of the
pLaplacian of weighted trees.
In this paper, we only consider simple weighted graphs with a positive weight
set. Let G = (V (G), E(G), W (G)) be a weighted graph with vertex set V (G) =
{v0, v1, . . . , vn−1}, edge set E(G) and weight set W (G) = {wk > 0, k = 1, 2, . . . ,
E(G)}. Let wG(uv) denote the weight of an edge uv. If uv ∈/ E(G), define wG(uv) =
0. Then uv ∈ E(G) if and only if wG(uv) > 0. The weight of a vertex u, denoted by
∗Received by the editors on September 1, 2010. Accepted for publication on February 27, 2011.
Handling Editor: Michael Neumann.
†Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan road, Shanghai,
200240, P.R. China (). This work is supported by the National Natural
Science Foundation of China (No 10971137), the National Basic Research Program (973) of China
(No 2006CB805900), and a grant of Science and Technology Commission of Shanghai Municipality
(STCSM, No 09XD1402500)
by
wG(u), is the sum of weights of all edges incident to u in G.
G.J. Zhang and X.D. Zhang
Let p > 1. Then the discrete pLaplacian △p(G) of a function f on V (G) is given
△p(G)f (u) =
X
(f (u) − f (v))[p−1]wG(uv),
where x[q] = sign(x)xq. When p = 2, △2(G) is the wellknown (combinatorial) graph
Laplacian (see [
4
]), i.e., Δ2(G) = L(G) = D(G)−A(G), where A(G) = (wG(vivj ))n×n
denotes the weighted adjacency matrix of G and D(G) = diag(wG(v0), wG(v1), . . . ,
wG(vn−1)) denotes the weighted diagonal matrix of G (see [
8
]).
A real number λ is called an eigenvalue of △p(G) if there exists a function f 6= 0
on V (G) such that for u ∈ V (G),
Δp(G)f (u) = λf (u)[p−1].
The function f is called the eigenfunction corresponding to λ. The largest eigenvalue
of Δp(G), denoted by λp(G), is called the pLaplacian spectral radius. Let d(v)
denote the degree of a vertex v, i.e., the number of edges incident to v. A nonincreasing
sequence of nonnegative integers π = (d0, d1, · · · , dn−1) is called graphic degree
sequence if there exists a simple connected graph having π as its vertex degree sequence.
Zhang [
9
] in 2008 determined all extremal trees with the largest spectral radius of the
Laplacian matrix among all trees with a given degree sequence. Further, Bıyıko˘glu,
Hellmuth, and Leydold [
2
] in 2009 characterized all extremal trees with the largest
pLaplacian spectral radius among all trees with a given degree sequence. Let Tπ,W be
the set of trees with a given graphic degree sequence π and a positive weight set W .
Recently, Tan [
8
] determined the extremal trees with the largest spectral radius of the
weight Laplacian matrix in Tπ,W . Moreover, the adjacency, Laplacian and signless
Laplacian eigenvalues of graphs with a given degree sequence have been studied (for
example, see [
3
] and [
10
]). Motivated by the above results, we investigate the largest
pLaplacian spectral radius of trees in Tπ,W . The main result of this paper can be
stated as follows:
Theorem 1.1. For a given degree sequence π of some tree and a positive weight
set W , Tπ∗,W (see in Section 3) is the unique tree with the largest pLaplacian spectral
radius in Tπ,W , which is independent of p.
The rest of this paper is organized as follows. In Section 2, some notations and
results are presented. In Section 3, we give a proof of Theorem 1.1 and a majorization
theorem for two tree degree sequences.
2. Preliminaries. The following are several propositions and lemmas about the
Rayleigh quotient and eigenvalues of the pLaplacian for weighted graphs. The proofs
The pLaplacian Spectral Radius of Weighted Trees With a Degree Sequence
269
are similar to unweighted graphs (see [
2
]). So we only present the result and omit the
proofs.
Let f be a function on V (G) and
P
RGp(f ) = uv∈E(G)
 f (u) − f (v) p wG(uv)
k f kpp
where kf kp = qpP  f (v) p. The following Proposition 2.1 generalizes the wellknown
v
RayleighRitz theorem.
Proposition 2.1. ([
6
])
λp(G) = fmapx=1 RGp(f ) =
max X
fp=1 uv∈E(G)
 f (u) − f (v) p wG(uv).
Moreover, if RGp(f ) = λp(G), then f is an eigenfunction corresponding to the
pLaplacian spectral radius λp(G).
Define the signless pLaplacian Qp(G) of a function f on V (G) by
and its Rayleigh quotient by
Qp(G)f (u) =
X
(f (u) + f (v))[p−1]wG(uv)
The largest eigenvalue of Qp(G), denoted by μp(G), is called the signless pLaplacian
spectral radius. Then we have the following.
Proposition 2.2. ([
2
])
μp(G) = fmapx=1 ΛpG(f ) =
max X
fp=1 uv∈E(G)
 f (u) + f (v) p wG(uv).
Moreover, if ΛpG(f ) = μp(G), then f is an eigenfunction corresponding to μp(G).
Corollary 2.3. Let G be a connected weighted graph. Then the signless
pLaplacian spectral radius μp(G) of Qp(G) is positive. Moreover, if f is an
eigenfunction of μp(G), then either f (v) > 0 for all v ∈ V (G) or f (v) < 0 for all v ∈ V (G).
,
.
270
G.J. Zhang and X.D. Zhang
Let f be an eigenfunction of μp(G). We call f a Perron vector of G if f (v) > 0
for all v ∈ V (G).
Lemma 2.4. Let G = (V1, V2, E, W ) be a bipartite weighted graph with bipartition
V1 and V2. Then λp(G) = μp(G).
Clearly, trees are bipartite graphs. So, Lemma 2.4 also holds for trees.
3. Main result. Let G − uv denote the graph obtained from G by deleting an
edge uv and G + uv denote the graph obtained from G by adding an edge uv. The
following lemmas will be used in the proof of the main result, Theorem 1.1.
Lemma 3.1. Let T ∈ Tπ,W with u, v ∈ V (T ) and f be a Perron vector of T .
Assume uui ∈ E(T ) and vui ∈/ E(T ) such that ui is not in the path from u to v for
k k
i = 1, 2, . . . , k. Let T ′ = T − S uui + S vui, wT ′ (vui) = wT (uui) for i = 1, 2, . . . ,
i=1 i=1
k, and wT ′ (e) = wT (e) for e ∈ E(T ) \ {uu1, uu2, . . . , uuk}. In other words, T ′ is the
weighted tree obtained from T by deleting the edges uu1, . . . , uuk and adding the edges
vu1, . . . , vuk with their weights wT (uu1), . . . , wT (uuk), respectively. If f (u) ≤ f (v),
then μp(T ) < μp(T ′).
Proof. Without loss of generality, assume k f kp= 1. Then
μp(T ′) − μp(T ) ≥ ΛpT ′ (f ) − ΛpT (f )
k
= X[(f (v) + f (ui))p − (f (u) + f (ui))p]wT (uui)
i=1
≥ 0.
If μp(T ′) = μp(T ), then f must be an eigenfunction of μp(T ′). Clearly, by computing
the values of the function f on V (T ) and V (T ′) at the vertex u, we have
Qp(T )f (u) =
X
(f (x) + f (u))[p−1]wT (ux)
x,xu∈E(T )
=
X
x,xu∈E(T ′)
k
(f (x) + f (u))[p−1]wT (ux) + X(f (u) + f (ui))[p−1]wT (uui)
i=1
and
X
x,xu∈E(T ′)
Qp(T ′)f (u) =
(f (x) + f (u))[p−1]wT (ux).
Moreover, Qp(T )f (u) = μp(T )f (u)[p−1] = μp(T ′)f (u)[p−1] = Qp(T ′)f (u). Hence
k
P (f (u) + f (ui))[p−1]wT (uui) = 0, which implies f (u) + f (ui) = 0 for i = 1, 2, . . . , k.
i=1
This is impossible. So the assertion holds.
The pLaplacian Spectral Radius of Weighted Trees With a Degree Sequence
271
From Lemma 3.1 we can easily get the following corollary.
Corollary 3.2. Let T be a weighted tree with the largest pLaplacian spectral
radius in Tπ,W and u, v ∈ V (T ). Suppose that f is a Perron vector of T . Then we
have the following:
(1) if f (u) ≤ f (v), then d(u) ≤ d(v);
(2) if f (u) = f (v), then d(u) = d(v).
Lemma 3.3. ([
2
]) Let 0 ≤ ε ≤ δ ≤ z and p > 1. Then (z + ǫ)p + (z − ǫ)p ≤
(z + δ)p + (z − δ)p. Equality holds if and only if ǫ = δ.
Lemma 3.4. Let T ∈ Tπ,W and uv, xy ∈ E(T ) such that v and y are not in the
path from u to x. Let f be a Perron vector of T and T ′ = T − uv − xy + uy + xv with
wT ′ (uy) = max{wT (uv), wT (xy)}, wT ′ (xv) = min{wT (uv), wT (xy)}, and wT ′ (e) =
wT (e) for e ∈ E(T ) \ {uv, xy}. If f (u) ≥ f (x) and f (y) ≥ f (v), then T ′ ∈ Tπ,W and
μp(T ) ≤ μp(T ′). Moreover, μp(T ) < μp(T ′) if one of the two inequalities is strict.
Proof. Without loss of generality, assume k f kp= 1.
Claim : (f (u) + f (y))p + (f (x) + f (v))p ≥ (f (u) + f (v))p + (f (x) + f (y))p.
Assume f (u) + f (y) = z + δ, f (x) + f (v) = z − δ, max{f (u) + f (v), f (x) + f (y)} =
z + ǫ, min{f (u) + f (v), f (x) + f (y)} = z − ǫ. Without loss of generality, assume
f (u) + f (v) ≥ f (x) + f (y). Then δ − ǫ = f (y) − f (v) ≥ 0. By Lemma 3.3, the Claim
holds. Without loss of generality, assume wT (uv) ≥ wT (xy). Then, by the Claim and
wT ′ (uy) = wT (uv) and wT ′ (xv) = wT (xy), we have
μp(T ′) − μp(T ) ≥ ΛpT ′ (f ) − ΛpT (f )
= (f (u) + f (y))pwT ′ (uy) + (f (x) + f (v))pwT ′ (xv)
−(f (u) + f (v))pwT (uv) − (f (x) + f (y))pwT (xy)
= [(f (u) + f (y))p − (f (u) + f (v))p]wT (uv)
+[(f (x) + f (v))p − (f (x) + f (y))p]wT (xy)
≥ [(f (u) + f (y))p + (f (x) + f (v))p − (f (u) + f (v))p
−(f (x) + f (y))p]wT (uv)
≥ 0.
If μp(T ′) = μp(T ), then ǫ = δ by Lemma 3.3, and f must be an eigenfunction of
z,zy∈E(T )\{xy}
X
X
z,zy∈E(T )\{xy}
ut∈E(T )\{uv}
= Qp(T )f (u)
= X
ut∈E(T )\{uv}
μp(T ′). So f (y) = f (v). Moreover, since wT ′ (uy) = wT (uv) ≥ wT (xy) and
Qp(T )f (y) =
(f (z) + f (y))[p−1]wT (zy) + (f (x) + f (y))[p−1]wT (xy)
= μp(T )f (y)[p−1] = μp(T ′)f (y)[p−1] = Qp(T ′)f (y)
=
(f (z) + f (y))[p−1]wT (zy) + (f (u) + f (y))[p−1]wT ′ (uy),
we have f (x) ≥ f (u). Hence f (x) = f (u), and the assertion holds.
Lemma 3.5. Let T ∈ Tπ,W with uv, xy ∈ E(T ) and f be a Perron vector of T . If
f (u) + f (v) ≥ f (x) + f (y) and wT (uv) < wT (xy), then there exists a tree T ′ ∈ Tπ,W
such that μp(T ′) > μp(T ).
Proof. Without loss of generality, assume k f kp= 1. Let T ′ be the tree obtained
from T with vertex set V (T ), edge set E(T ), wT ′ (uv) = wT (xy), wT ′ (xy) = wT (uv)
and wT ′ (e) = wT (e) for e ∈ E(T ) \ {uv, xy}. Then we have
= [(f (u) + f (v))p − (f (x) + f (y))p](wT (xy) − wT (uv))
≥ 0.
If μp(T ′) = μp(T ), then f must be an eigenfunction of μp(T ′). Without loss of
generality, assume u 6= x and u 6= y. Since
(f (u) + f (t))[p−1]wT (ut) + (f (u) + f (v))[p−1]wT (xy)
(f (u) + f (t))[p−1]wT (ut) + (f (u) + f (v))[p−1]wT (uv),
we have wT (uv) = wT (xy), which is a contradiction. So μp(T ′) > μp(T ).
Let v0 be the root of a tree T and h(vi) be the distance between vi and v0.
Definition 3.6. Let T = (V (T ), E(T ), W (T )) be a weighted tree with a positive
weight set W (T ) and root v0. Then a wellordering ≺ of the vertices is called a
weighted breadthfirstsearch ordering (WBFSordering for short) if the following holds
for all vertices u, v, x, y ∈ V (T ):
(1) v ≺ u implies h(v) ≤ h(u);
(2) v ≺ u implies d(v) ≥ d(u);
The pLaplacian Spectral Radius of Weighted Trees With a Degree Sequence
273
(3) Let uv, uy ∈ E(T ) with h(v) = h(y) = h(u) + 1. If v ≺ y, then wT (uv) ≥
wT (uy);
(4) Let uv, xy ∈ E(T ) with h(u) = h(v) − 1 and h(x) = h(y) − 1. If u ≺ x, then
v ≺ y and wT (uv) ≥ wT (xy).
A weighted tree is called a WBFStree if its vertices have a WBFSordering. For
a given degree sequence and a positive weight set, it is easy to see that the WBFStree
is uniquely determined up to isomorphism by Definition 3.6 (for example, see [
9
]).
Let π = (d0, d1, . . . , dn−1) be a degree sequence of tree such that d0 ≥ d1 ≥
· · · ≥ dn−1 and W = {w1, w2, . . . , wn−1} be a positive weight set with w1 ≥ w2 ≥
· · · ≥ wn−1 > 0. We now construct a weighted tree Tπ∗,W with the degree
sequence π and the positive weight set W as follows. Select a vertex v0,1 as the
root and begin with v0,1 of the zeroth layer. Let s1 = d0 and select s1
vertices v1,1, v1,2, . . . , v1,s1 of the first layer such that they are adjacent to v0,1 and
wTπ∗,W (v0,1v1,k) = wk for k = 1, 2, . . . , s1. Assume that all vertices of the tst
layer have been constructed and are denoted by vt,1, vt,2, . . . , vt,st . We construct
all the vertices of the (t + 1)st layer by the induction hypothesis. Let st+1 =
ds1+···+st−1+1 + · · · + ds1+···+st − st and select st+1 vertices vt+1,1, vt+1,2, . . . , vt+1,st+1
of the(t + 1)st layer such that vt,1 is adjacent to vt+1,1, . . . , vt+1,ds1+···+st−1+1−1, . . . ,
vt,st is adjacent to vt+1,st+1−ds1+···+st +2, . . . , vt+1,st+1 and if there exists vt,l with
vt,lvt+1,i ∈ E(Tπ∗,W ),
wTπ∗,W (vt,lvt+1,i) = wd0+d1+···+ds1+s2+···+st−1 −(s1+s2+···+st−1)+i
for 1 ≤ i ≤ st+1. In this way, we obtain only one tree Tπ∗,W with the degree sequence
π and the positive weight set W (see Fig. 3.1 for an example). In the following we
are ready to present a proof of Theorem 1.1.
uv2,1
w12
u
v3,1
v1,1 Bu
B
w5 Bw6
B
Buv2,2
w13
u
v3,2
w1
w2
w7
v1,2 Bu
B
uv2,3
w14
u
v3,3
Bw8
B
Bu
v2,4
vuH`0w,`H13`H`HvH`1,H3`Bu` `w4````v`1,`4`u
w9
u
v2,5
B
Bw10
B
Bu
v2,6
w11
u
v2,7
P roof of Theorem 1.1. Let T be a weighted tree with the largest pLaplacian
spec274
G.J. Zhang and X.D. Zhang
tral radius in Tπ,W , where π = (d0, d1, . . . , dn−1) with d0 ≥ d1 ≥ · · · ≥ dn−1. Let f be
a Perron vector of T . Without loss of generality, assume V (T ) = {v0, v1, . . . , vn−1}
such that f (vi) ≥ f (vj) for i < j. By Corollary 3.2 we have d(v0) ≥ d(v1) ≥ · · · ≥
d(vn−1). So d(v0) = d0. Let v0 be the root of T . Suppose max h(v) = h(T ). Let
v∈V (T )
Vi = {v ∈ V (T )h(v) = i} and  Vi = si for i = 0, 1, . . . , h(T ). In the following we
will relabel the vertices of T .
Let V0 = {v0,1}, where v0,1 = v0. Obviously, s1 = d0. The vertices of V1 are
relabeled v1,1, v1,2, . . . , v1,s1 such that f (v1,1) ≥ f (v1,2) ≥ · · · ≥ f (v1,s1 ). Assume
that the vertices of Vt have been already relabeled vt,1, vt,2, . . . , vt,st . The vertices of
Vt+1 can be relabeled vt+1,1, vt+1,2, . . . , vt+1,st+1 such that they satisfy the following
conditions: If vt,kvt+1,i, vt,kvt+1,j ∈ E(T ) and i < j, then f (vt+1,i) ≥ f (vt+1,j); if
vt,kvt+1,i, vt,lvt+1,j ∈ E(T ) and k < l, then i < j. In this way we can obtain a well
ordering ≺ of vertices of T as follows:
vi,j ≺ vk,l, if i < k or i = k and j < l.
Clearly, f (v1,1) ≥· · · ≥ f (v1,s1 ), and f (vt+1,i)≥f (vt+1,j) when i < j and vt+1,i, vt+1,j
have the same neighbor.
In the following we will prove that T is isomorphic to Tπ∗,W by proving that the
ordering ≺ is a WBFSordering.
Claim: f (vh,1) ≥ f (vh,2) ≥ · · · ≥ f (vh,sh ) ≥ f (vh+1,1) for 0 ≤ h ≤ h(T ).
We will prove that the Claim holds by induction on h. Obviously, the Claim
holds for h = 0. Assume that the Claim holds for h = r − 1. We now prove that
the assertion holds for h = r. If there exist two vertices vr,i ≺ vr,j with f (vr,i) <
f (vr,j), then there exist two vertices vr−1,k, vr−1,l ∈ Vr−1 with k < l such that
vr−1,kvr,i, vr−1,lvr,j ∈ E(T ). By the induction hypothesis, f (vr−1,k) ≥ f (vr−1,l). Let
T1 = T − vr−1,kvr,i − vr−1,lvr,j + vr−1,kvr,j + vr−1,lvr,i
with
wT1 (vr−1,kvr,j) = max{wT (vr−1,kvr,i), wT (vr−1,lvr,j)},
wT1 (vr−1,lvr,i) = min{wT (vr−1,kvr,i), wT (vr−1,lvr,j)},
and wT1 (e) = wT (e) for e ∈ E(T ) \ {vr−1,kvr,i, vr−1,lvr,j}. Then T1 ∈ Tπ,W . By
Lemma 3.4, μp(T ) < μp(T1), which is a contradiction to our assumption that T has
the largest pLaplacian spectral radius in Tπ,W . So f (vr,i) ≥ f (vr,j). Now assume
f (vr,sr ) < f (vr+1,1). Note that d(v0) ≥ 2. It is easy to see that vr,sr vr−1,sr−1 ,
vr,1vr+1,1 ∈ E(T ). By the induction hypothesis, f (vr−1,sr−1 ) ≥ f (vr,1). Then, by
The pLaplacian Spectral Radius of Weighted Trees With a Degree Sequence
275
similar proof, we can also get a new tree T2 such that T2 ∈ Tπ,W and μp(T2) > μp(T ),
which is also a contradiction. So the Claim holds.
By the Claim and Corollary 3.2, the condition (2) in Definition 3.6 holds.
Assume that uv, uy ∈ E(T ) with h(v) = h(y) = h(u) + 1. If v ≺ y, then f (v) ≥
f (y) and wT (uv) ≥ wT (uy) by Lemma 3.5. So the condition (3) in Definition 3.6
holds.
Let uv, xy ∈ E(T ) with u ≺ x, h(v) = h(u) + 1 and h(y) = h(x) + 1. Then v ≺ y.
By the Claim, f (u) ≥ f (x) and f (v) ≥ f (y), which implies f (u) + f (v) ≥ f (x) + f (y).
Further, by Lemma 3.5, we have wT (uv) ≥ wT (xy). Therefore, “ ≺ ” is a
WBFSordering, i.e., T is a WBFStree. So Tπ∗,W is the unique tree with the largest
pLaplacian spectral radius in Tπ,W . Hence, the proof is completed.
Let π = (d0, d1, . . . , dn−1) and π′ = (d′0, d′1, . . . , d′n−1) be two nonincreasing
posit t n−1 n−1
tive sequences. If P di ≤ P d′i for t = 0, 1, . . . , n − 2 and P di = iP=0 d′i, then π′ is
i=0 i=0 i=0
said to majorize π, and is denoted by π E π′.
Lemma 3.7. ([
5
]) Let π = (d0, d1, . . . , dn−1) and π′ = (d′0, d′1, . . . , d′n−1) be two
nonincreasing graphic degree sequences. If π E π′, then there exist graphic degree
sequences π1, π2, . . . , πk such that π E π1 E π2 E · · · E πk E π′, and only two
components of πi and πi+1 are different by 1.
Theorem 3.8. Let π and π′ be two degree sequences of trees. Let Tπ,W and
Tπ′,W denote the set of trees with the same weight set W and degree sequences π and
π′, respectively. If π E π′, then μp(Tπ∗,W ) ≤ μp(Tπ∗′,W ). The equality holds if and only
if π = π′.
Proof. By Lemma 3.7, without loss of generality, assume π = (d0, d1, . . . , dn−1)
and π′ = (d′0, d′1, . . . , d′n−1) such that di = d′i − 1, dj = d′j + 1 with 0 ≤ i < j ≤ n − 1,
and dk = d′k for k 6= i, j. Then Tπ∗,W has a WBFSordering ≺ consistent with its
Perron vector f such that f (u) ≥ f (v) implies u ≺ v by the proof of Theorem 1.1.
Let v0, v1, . . . , vn−1 ∈ V (Tπ∗,W ) with v0 ≺ v1 ≺ · · · ≺ vn−1. Then f (v0) ≥ f (v1) ≥
· · · ≥ f (vn−1) and d(vt) = dt for 0 ≤ t ≤ n − 1. Since dj = d′j + 1 ≥ 2, there
exists a vertex vs with s > j, vjvs ∈ E(Tπ∗,W ), vivs ∈/ E(Tπ∗,W ) and vs is not in
the path from vi to vj. Let T1 = Tπ∗,W − vjvs + vivs with wT1 (vivs) = wTπ∗,W (vjvs)
and wT1 (e) = wTπ∗,W (e) for e ∈ E(T1) \ {vivs}. Then T1 ∈ Tπ′,W . Since i < j, we
have f (vi) ≥ f (vj). By Lemma 3.1, μp(Tπ∗,W ) < μp(T1) ≤ μp(Tπ∗′,W ). The proof is
completed.
Corollary 3.9. Let Tn,k be the set of trees of order n with k pendent vertices
and the same weight set W . Let π1 = {k, 2, . . . , 2, 1, . . . , 1}, where the number of 1 is
G.J. Zhang and X.D. Zhang
k. Then Tπ∗1,W is the unique tree with the largest pLaplacian spectral radius in Tn,k.
Acknowledgment. The authors would like to thank an anonymous referee very
much for valuable suggestions, corrections and comments which results in a great
improvement of the original manuscript.
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