The p-Laplacian spectral radius of weighted trees with a degree sequence and a weight set

Electronic Journal of Linear Algebra, Sep 2018

Guang-Jun Zhang, Xiao-Dong Zhang

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The p-Laplacian spectral radius of weighted trees with a degree sequence and a weight set

Volume 1081-3810 The p -Laplacian spectral radius of weighted trees with a degree sequence and a weight set Guang-Jun Zhang Xiao-Dong Zhang Follow this and additional works at; http; //repository; uwyo; edu/ela - THE P -LAPLACIAN SPECTRAL RADIUS OF WEIGHTED TREES WITH A DEGREE SEQUENCE AND A WEIGHT SET∗ GUANG-JUN ZHANG† AND XIAO-DONG ZHANG† AMS subject classifications. 05C50, 05C05, 05C22. 1. Introduction. In the last decade, the p-Laplacian, which is a natural nonlinear generalization of the standard Laplacian, plays an increasing role in geometry and partial differential equations. Recently, the discrete p-Laplacian, which is the analogue of the p-Laplacian on Riemannian manifolds, has been investigated by many researchers. For example, Amghibech in [ 1 ] presented several sharp upper bounds for the largest p-Laplacian eigenvalues of graphs. Takeuchi in [ 7 ] investigated the spectrum of the p-Laplacian and p-harmonic morphism of graphs. Luo et al. in [ 6 ] used the eigenvalues and eigenvectors of the p-Laplacian to obtain a natural global embedding for multi-class clustering problems in machine learning and data mining areas. Based on the increasing interest in both theory and application, the spectrum of the discrete p-Laplacian should be further investigated. The main purpose of this paper is to investigate some properties of the spectral radius and eigenvectors of the p-Laplacian 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 p-Laplacian △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)|x|q. When p = 2, △2(G) is the well-known (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 p-Laplacian 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 p-Laplacian 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 p-Laplacian 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 p-Laplacian 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 p-Laplacian for weighted graphs. The proofs The p-Laplacian 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 well-known v Rayleigh-Ritz theorem. Proposition 2.1. ([ 6 ]) λp(G) = ||fm||apx=1 RGp(f ) = max X ||f||p=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 p-Laplacian 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 p-Laplacian spectral radius. Then we have the following. Proposition 2.2. ([ 2 ]) μp(G) = ||fm||apx=1 ΛpG(f ) = max X ||f||p=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 p-Laplacian 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 p-Laplacian 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 well-ordering ≺ of the vertices is called a weighted breadth-first-search ordering (WBFS-ordering 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 p-Laplacian 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 WBFS-tree if its vertices have a WBFS-ordering. For a given degree sequence and a positive weight set, it is easy to see that the WBFS-tree 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 zero-th 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 t-st 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 p-Laplacian 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 WBFS-ordering. 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 p-Laplacian 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 p-Laplacian 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 WBFS-tree. 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 WBFS-ordering ≺ 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 p-Laplacian spectral radius in Tn,k. Acknowledgment. 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Guang-Jun Zhang, Xiao-Dong Zhang. The p-Laplacian spectral radius of weighted trees with a degree sequence and a weight set, Electronic Journal of Linear Algebra, 2018,