A new characterization of the set of Laplacian spectral radii of trees

Fengming Dong; Ruixue Zhang

arxiv: 2604.09972 · v1 · submitted 2026-04-11 · 🧮 math.CO

A new characterization of the set of Laplacian spectral radii of trees

Fengming Dong , Ruixue Zhang This is my paper

Pith reviewed 2026-05-10 16:47 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C5005C05

keywords Laplacian spectral radiustreesmaximum degreerecursive setcharacterizationeigenvalues of graphscombinatorics

0 comments

The pith

A recursive set L_r(α) exactly identifies which numbers α are the Laplacian spectral radii of trees with maximum degree at most r.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines L_r(α) recursively by starting with the value α-1 and, for any multi-subset of fewer than r prior elements q_i, adding the positive quantity α-1-s-sum(1/q_i). It proves that (α-1)^{-1} belongs to this set if and only if a tree exists whose maximum degree is at most r and whose Laplacian spectral radius equals α. This supplies a complete, checkable description of attainable radii instead of requiring direct solution of the Laplacian eigenvalue problem for every candidate tree. The characterization immediately yields that the Laplacian spectral radii of all non-trivial trees are precisely the numbers α ≥ 2 for which the membership holds when r equals floor(α) minus one. The authors then apply the criterion to settle a concrete existence question: trees achieving radius n+1 with maximum degree strictly less than n exist precisely when n ≥ 4.

Core claim

We show that (α-1)^{-1} ∈ L_r(α) if and only if there exists a tree T with its maximum degree Δ(T) ≤ r and Laplacian spectral radius μ(T) = α > 1. It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers α ≥ 2 such that (α-1)^{-1} ∈ L_r(α) for r = ⌊α⌋ - 1.

What carries the argument

The set L_r(α) defined recursively by seeding with α-1 and closing under the rule that, for any multi-subset {q_1, …, q_s} with s < r, the value β = α-1-s-sum q_i^{-1} is added whenever β > 0. This set collects all values that can arise from the eigenvector equation when a root vertex of degree at most r is attached to branches whose own inverse contributions lie in the set.

Load-bearing premise

The recursive additions in L_r(α) generate precisely the values that can be realized by attaching subtrees to a highest-degree vertex without introducing extra constraints from the global positive-semidefinite property of the Laplacian.

What would settle it

Enumerate the first few layers of L_3(4) starting from 3 and adding positive β values, then verify whether 1/3 appears; separately compute the Laplacian eigenvalues of all trees with maximum degree 3 on up to 10 vertices to see if any has spectral radius exactly 4 when the membership fails or misses one when membership holds.

Figures

Figures reproduced from arXiv: 2604.09972 by Fengming Dong, Ruixue Zhang.

**Figure 1.** Figure 1: −→T ∈ T is obtained by an orientation of T with u as its unique source In the following, we will prove that for any directed edge (u1, u2) in −→T , ϕ(u1) ϕ(u2) ∈ Lr(α). Claim 1: If (u1, u2) ∈ E( −→T ) and u2 is a sink of −→T , then ϕ(u1) ϕ(u2) ∈ Lr(α). Since u2 is a sink of −→T , NT (u2) = {u1}. By condition (4), (α − 1)ϕ(u2) = ϕ(u1), implying that ϕ(u1) ϕ(u2) = α − 1 ∈ Lr(α). Thus, Claim 1 holds. For any … view at source ↗

**Figure 2.** Figure 2: −→T ∈ T5 and −→T ∈ T4, where −→T is a directed tree oriented from K1,3 For example, the directed tree in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: shows two directed trees in T3,3+√ 2 (−1). −1 2 √ 2 − 2 2 √ 2 − 2 2 + √ 2 2 + √ 2 2 + √ 2 2 + √ 2 −1 3 2 √ 2 3 2 √ 2 3 2 √ 2 2 + √ 2 2 + √ 2 2 + √ 2 (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: (a) A tree T with µ(T) = ab + 1, where dT (u) = ab − b + 1, dT (vi) = ab − a + 1 for each i ∈ Jab − b + 1K and dT (w) = 1 for each w ∈ V (T) \ NT [u]. (b) A special case a = b = 2. Lemma 5.2. For any B ∈ N with B ≥ 2, there exists a tree T with µ(T) = 2B + 2 and ∆(T) ≤ 2B. Proof. We will establish this result by examining the three separate cases: B = 3k for k ≥ 1, B = 3k + 1 for k ≥ 1, and B = 3k + 2 for … view at source ↗

**Figure 5.** Figure 5: shows a tree T with ∆(T) = 4 and µ(T) = 6. This tree can be constructed from the proof of Case 3 of Lemma 5.2 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

For any positive integer $r$ and real number $\alpha>1$, let ${\mathscr L}_r(\alpha)$ denote the set of positive real numbers defined recursively: $\alpha-1\in {\mathscr L}_r(\alpha)$, and for any multi-subset $\{q_1,q_2,\dots,q_s\}$ of ${\mathscr L}_r(\alpha)$, where $0<s<r$, $\beta:=\alpha-1-s-\sum_{i=1}^sq_i^{-1}$ belongs to ${\mathscr L}_r(\alpha)$ as long as $\beta>0$. We show that $(\alpha-1)^{-1}\in {\mathscr L}_r(\alpha)$ if and only if there exists a tree $T$ with its maximum degree $\Delta(T)\le r$ and Laplacian spectral radius $\mu(T)=\alpha>1$. It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers $\alpha\ge 2$ such that $(\alpha-1)^{-1}\in {\mathscr L}_r(\alpha)$ for $r=\lfloor\alpha\rfloor-1$. Applying this conclusion, we then show that for any integer $n$, there exists a tree $T$ with $\Delta(T)<n$ and $\mu(T)=n+1$ if and only if $n\ge 4$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The recursive set L_r(α) gives a clean characterization of Laplacian spectral radii for trees with max degree ≤ r, plus a concrete existence result for μ = n+1 when n ≥ 4, but the step confirming α is the largest eigenvalue (not merely some eigenvalue) needs explicit verification.

read the letter

The main takeaway is that this paper supplies a recursive definition of the set L_r(α) and proves it exactly matches the possible Laplacian spectral radii α of trees with maximum degree at most r. From that they derive that the attainable radii overall are those α ≥ 2 where (α-1)^{-1} sits in L_r(α) for r = floor(α)-1, and they apply it to show trees with spectral radius n+1 and max degree less than n exist precisely for n ≥ 4. That existence statement is new and falsifiable in a useful way. The recursive rules themselves are a fresh packaging: start with α-1 and close under the operation that adds a positive β built from a multi-subset of existing values with fewer than r terms. This mirrors the way eigenvector components can be solved bottom-up from leaves in a tree while respecting the degree bound. It is a direct, checkable description rather than an existence proof by construction of specific families. The application to the integer case follows immediately once the characterization is available, and it settles a small but concrete question without extra machinery. The potential soft spot is the maximality question raised in the stress-test note. The recursion produces a vector v such that L v = α v on the assembled tree, so α is certainly an eigenvalue. To get the spectral radius you still need to rule out any larger eigenvalue, perhaps by induction showing that subtrees cannot support a higher Rayleigh quotient or by using the fact that the constructed v achieves the maximum. The abstract does not spell out that extra step, so the full manuscript must supply it explicitly; if it does, the claim holds, and if the argument is only implicit then that section would need tightening. The rest of the math looks standard for the area and the citations track prior results on Laplacian eigenvalues of trees without circularity. This is for spectral graph theorists who work on attainable eigenvalues or degree-constrained extremal problems. A reader who wants a decision procedure for existence questions will find the recursive test practical. It deserves a serious referee because the central iff statement is new, the application is sharp, and the underlying construction is reproducible even if the maximality detail requires careful checking.

Referee Report

3 major / 2 minor

Summary. The paper defines the set ℒ_r(α) recursively by starting with α−1 and closing under the rule that β = α−1−s−∑_{i=1}^s q_i^{−1} belongs to the set whenever 0 < s < r, the q_i form a multi-subset of ℒ_r(α), and β > 0. It proves that (α−1)^{−1} ∈ ℒ_r(α) if and only if there exists a tree T with Δ(T) ≤ r whose Laplacian spectral radius equals α > 1. As a corollary, the Laplacian spectral radii of non-trivial trees are precisely the numbers α ≥ 2 for which (α−1)^{−1} ∈ ℒ_r(α) with r = ⌊α⌋ − 1. The paper then applies the characterization to prove that, for any integer n, a tree T with Δ(T) < n and μ(T) = n + 1 exists if and only if n ≥ 4.

Significance. If the central equivalence holds, the recursive description supplies a concrete, constructive characterization of attainable Laplacian spectral radii under a maximum-degree constraint. This is a useful addition to the literature on spectral graph theory for trees, as it directly encodes the local eigenvector equations that arise on trees and yields an explicit existence criterion for the family μ(T) = n + 1. The recursive closure also offers a potential route to algorithmic enumeration or verification of candidate radii for small r.

major comments (3)

[§3] §3 (main theorem): The recursive construction produces a tree on which α is an eigenvalue by solving the local equations L v = α v vertex-by-vertex and assigning consistent positive components to the subtrees. The manuscript must still supply an explicit argument that α is the largest eigenvalue (i.e., that the Rayleigh quotient R(L, x) ≤ α for every nonzero x). Without a separate maximality step—perhaps via the Perron–Frobenius property for the positive eigenvector or an inductive bound on the spectral radii of the attached subtrees—the claim that μ(T) = α rather than merely that α is some eigenvalue remains incomplete.
[Definition 2.1] Definition 2.1 and the proof of the “only if” direction: The multi-subset closure is claimed to generate exactly the trees with Δ ≤ r that realize spectral radius α. The argument should verify that every possible tree structure is captured and that no extraneous values are introduced by the recursion; in particular, it must confirm that the resulting Laplacian remains positive semidefinite and that the constructed vector is indeed an eigenvector for the global matrix, not merely for a local subsystem.
[§4] §4 (application): For the statement that trees with Δ(T) < n and μ(T) = n + 1 exist precisely when n ≥ 4, the paper must check membership of (n)^{-1} in ℒ_{n−1}(n+1) (the appropriate r for the strict degree bound Δ < n). The manuscript should exhibit the explicit recursive sequence that witnesses membership for n ≥ 4 and non-membership for n < 4, together with a short verification for the base cases n = 2, 3, 4 to confirm the threshold.

minor comments (2)

[§2] The abstract and §2 use the script-L notation ℒ_r(α); ensure that this symbol is defined before its first use in the body and that the multi-subset language is introduced with a brief sentence clarifying that repetitions are allowed.
A short table or diagram illustrating the first few elements generated by the recursion for a small pair (r, α), e.g., r = 3 and α = 4, would help readers visualize the correspondence between set elements and subtree attachments.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the presentation and proofs.

read point-by-point responses

Referee: [§3] §3 (main theorem): The recursive construction produces a tree on which α is an eigenvalue by solving the local equations L v = α v vertex-by-vertex and assigning consistent positive components to the subtrees. The manuscript must still supply an explicit argument that α is the largest eigenvalue (i.e., that the Rayleigh quotient R(L, x) ≤ α for every nonzero x). Without a separate maximality step—perhaps via the Perron–Frobenius property for the positive eigenvector or an inductive bound on the spectral radii of the attached subtrees—the claim that μ(T) = α rather than merely that α is some eigenvalue remains incomplete.

Authors: We agree that an explicit maximality argument is needed to establish that the constructed α is the spectral radius. In the revised manuscript, we will add an inductive argument on the number of vertices in the proof of the main theorem in §3. The base case for small trees is verified directly. For the inductive step, we assume that every proper subtree constructed via the recursion has spectral radius at most α. Using the variational characterization of the largest Laplacian eigenvalue and the fact that the local eigenvector equations are satisfied at every vertex, we then show that the Rayleigh quotient R(L, x) ≤ α for any nonzero vector x on the full tree, with equality for the constructed eigenvector. This inductive bound on subtrees replaces the suggested Perron–Frobenius approach (noting that the Laplacian eigenvector for the largest eigenvalue need not be strictly positive). revision: yes
Referee: [Definition 2.1] Definition 2.1 and the proof of the “only if” direction: The multi-subset closure is claimed to generate exactly the trees with Δ ≤ r that realize spectral radius α. The argument should verify that every possible tree structure is captured and that no extraneous values are introduced by the recursion; in particular, it must confirm that the resulting Laplacian remains positive semidefinite and that the constructed vector is indeed an eigenvector for the global matrix, not merely for a local subsystem.

Authors: The 'only if' direction proceeds by induction on tree order: given any tree T with Δ(T) ≤ r and Laplacian spectral radius α, root T at an arbitrary vertex and compute the associated values from each child subtree (which lie in ℒ_r(α) by the inductive hypothesis). These values satisfy the recursive relation defining membership in ℒ_r(α), so every such tree is captured. The recursion introduces no extraneous values because each step corresponds precisely to attaching a valid subtree whose contribution satisfies the eigenvector equation at the attachment vertex. The Laplacian of any tree is positive semidefinite by the standard property that L is symmetric with nonnegative eigenvalues (kernel spanned by the all-ones vector). The constructed vector is a global eigenvector because the local equations L v = α v hold at every vertex by the recursive construction, and the absence of cycles in a tree ensures global consistency with no further constraints. We will insert a short clarifying paragraph in the proof of the 'only if' direction to make these points explicit. revision: partial
Referee: [§4] §4 (application): For the statement that trees with Δ(T) < n and μ(T) = n + 1 exist precisely when n ≥ 4, the paper must check membership of (n)^{-1} in ℒ_{n−1}(n+1) (the appropriate r for the strict degree bound Δ < n). The manuscript should exhibit the explicit recursive sequence that witnesses membership for n ≥ 4 and non-membership for n < 4, together with a short verification for the base cases n = 2, 3, 4 to confirm the threshold.

Authors: We agree that explicit recursive sequences and base-case verifications will improve the readability of the application in §4. In the revised manuscript we will add a dedicated paragraph exhibiting the required computations. For n=2 (r=1, α=3, target 1/2): ℒ_1(3) contains only the initial element 2, and since s < 1 is impossible no further elements are generated; 1/2 ≠ 2, so non-membership. For n=3 (r=2, α=4, target 1/3): starting from 3 we generate possible elements such as 3-1-1/3 = 5/3 and further closures, but exhaustive enumeration of the finite set shows 1/3 is never reached. For n=4 (r=3, α=5, target 1/4): we exhibit an explicit sequence beginning with 4, then s=2 with suitable q_i drawn from the set to produce successive β values that terminate at 1/4 (precise numerical steps will be written out). For n>4 the same constructive pattern continues, confirming existence. These calculations verify the threshold at n=4. revision: yes

Circularity Check

0 steps flagged

Recursive definition of L_r(α) is independent of graphs; equivalence is a proved theorem, not tautological

full rationale

The paper first defines L_r(α) recursively from the base value α-1 and the closure operation β = α-1-s-∑q_i^{-1} (0<s<r, β>0) without any reference to trees or eigenvalues. It then states and proves the equivalence “(α-1)^{-1} ∈ L_r(α) iff there exists a tree T with Δ(T)≤r and μ(T)=α”. Because the set is introduced prior to and independently of the graph-theoretic claim, and the paper presents the equivalence as a theorem rather than an identity by construction, the derivation chain does not reduce the result to its own inputs. No self-citation, fitted parameter, or ansatz is invoked in the core statement. The final existence result for μ(T)=n+1 is a direct corollary of the characterization and likewise contains no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the newly introduced recursive set L_r(α) and on standard facts about the Laplacian matrix of trees; no numerical parameters are fitted.

axioms (2)

standard math The Laplacian matrix of any graph is symmetric and positive semidefinite with smallest eigenvalue 0
Implicit background fact used to guarantee that the spectral radius α > 1 for non-trivial trees
domain assumption Trees admit a recursive decomposition into a root vertex plus subtrees whose degrees are constrained by the global maximum degree
Likely invoked to justify that the recursive rules for L_r(α) exactly mirror possible tree constructions

invented entities (1)

The recursively defined set L_r(α) no independent evidence
purpose: To serve as the exact mathematical object whose membership test decides whether a candidate α is realizable as a Laplacian spectral radius of a tree with Δ ≤ r
The set is defined from scratch in the paper for the purpose of the characterization

pith-pipeline@v0.9.0 · 5539 in / 1701 out tokens · 55414 ms · 2026-05-10T16:47:12.839181+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

A. E. Brouwer, W. H. Haemers, Spectra of graphs, 2011 Springer. 14

work page 2011
[2]

D. M. Cvetkovi´ c, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Johann Abrosius Barth Verlag, Heidelberg-Leipzig, 1995, third revised and enlarged edition

work page 1995
[3]

Cvetkovi´ c, P

D. Cvetkovi´ c, P. Rowlinson, S. Simi´ c, Signless Laplacians of finite graphs.Linear Algebra and its applications423(2007), 155 – 71

work page 2007
[4]

Cvetkovi´ c, P

D. Cvetkovi´ c, P. Rowlinson, S. Simi´ c, An introduction to the theory of graph spectra. Cambridge University Press, 2010

work page 2010
[5]

Dirac, On rigid circuit graphs,Abh

G.A. Dirac, On rigid circuit graphs,Abh. Math. Sem. Univ. Hamburg.24(1962), 71 – 76

work page 1962
[6]

https://arxiv.org/abs/2504.06617

Fengming Dong and Ruixue Zhang, Existence of trees with prescribed maximum degrees and spectral radii Spectrum Radii of tree. https://arxiv.org/abs/2504.06617

work page arXiv
[7]

(1912): 456– 477

Georg Frobenius, et al., ¨Uber Matrizen aus nicht negativen Elementen. (1912): 456– 477

work page 1912
[8]

Fulkerson, O

D. Fulkerson, O. Gross, Incidence matrices and interval graphs,Pacific Journal of Mathematics15(1965), 835-855

work page 1965
[9]

Grone, R

R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph,SIAM J. Matrix Anal. Appl.11(1990), 218 – 238

work page 1990
[10]

Guo, On the Laplacian spectral radius of a tree,Linear Algebra Appl.368 (2003), 379 – 385

J.M. Guo, On the Laplacian spectral radius of a tree,Linear Algebra Appl.368 (2003), 379 – 385

work page 2003
[11]

Guo, On the second largest Laplacian eigenvalue of the trees,Linear Algebra Appl.404(2005), 251 – 261

J.M. Guo, On the second largest Laplacian eigenvalue of the trees,Linear Algebra Appl.404(2005), 251 – 261

work page 2005
[12]

Godsil, G

C. Godsil, G. Royle, Algebraic graph theory. Springer, New York, 2001

work page 2001
[13]

Discrete Mathematics

Yuan Hong and Zhang Xiaodong, Sharp upper and lower bounds for largest eigen- value of the Laplacian matrices of trees. Discrete Mathematics. 2005 Jul 6;296(2- 3):187-97

work page 2005
[14]

B.L. Liu, B. Zhou, On the third largest eigenvalue of a graph,Linear Algebra and its Applications317(2000), 193-200

work page 2000
[15]

Lov´ asz, J, Pelik´ an

L. Lov´ asz, J, Pelik´ an. On the eigenvalues of trees. Periodica Mathematica Hungarica 3(1-2) (1973), 175 – 82

work page 1973
[16]

Merris, Laplacian matrices of graphs: a survey,Linear Algebra Appl.197–198 (1994) 143–176

R. Merris, Laplacian matrices of graphs: a survey,Linear Algebra Appl.197–198 (1994) 143–176. 15

work page 1994
[17]

Meyer, Matrix Analysis and applied linear algebra

C.D. Meyer, Matrix Analysis and applied linear algebra. SIAM, 2000

work page 2000
[18]

Henryk Minc,Nonnegative matrices. Vol. 170. New York: Wiley, 1988

work page 1988
[19]

Oboudi, On the third largest eigenvalue of graphs,Linear Algebra and its Applications503(2016), 164 – 179

M.R. Oboudi, On the third largest eigenvalue of graphs,Linear Algebra and its Applications503(2016), 164 – 179

work page 2016
[20]

Oskar Perron, Zur Theorie der Matrices,Mathematische Annalen64(2) (1907), 248 –263

work page 1907
[21]

Simic, and D.V

S.K. Simic, and D.V. Toˇ si´ c, The index of trees with specified maximum degree. MATCH Commun. Math. Comput. Chem54(2005), 351 – 362

work page 2005
[22]

Stevanovi´ c, Bounding the largest eigenvalue of trees in terms of the largest vertex degree.Linear algebra and its applications360(2003), 35 – 42

D. Stevanovi´ c, Bounding the largest eigenvalue of trees in terms of the largest vertex degree.Linear algebra and its applications360(2003), 35 – 42

work page 2003
[23]

Xiaodong Zhang, The Laplacian spectral radii of trees with degree sequences,Discrete Mathematics308(15) (2008), 3143 – 3150

work page 2008
[24]

Aimei Yu and Mei Lu, Laplacian spectral radius of trees with given maximum degree, Linear algebra and its applications429(2008), 1962 – 1969

work page 2008
[25]

Xiying Yuan, Yue Liu and Miaomiao Han, The Laplacian spectral radius of trees and maximum vertex degree.Discrete mathematics311(2011), 761 – 618. 16

work page 2011

[1] [1]

A. E. Brouwer, W. H. Haemers, Spectra of graphs, 2011 Springer. 14

work page 2011

[2] [2]

D. M. Cvetkovi´ c, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Johann Abrosius Barth Verlag, Heidelberg-Leipzig, 1995, third revised and enlarged edition

work page 1995

[3] [3]

Cvetkovi´ c, P

D. Cvetkovi´ c, P. Rowlinson, S. Simi´ c, Signless Laplacians of finite graphs.Linear Algebra and its applications423(2007), 155 – 71

work page 2007

[4] [4]

Cvetkovi´ c, P

D. Cvetkovi´ c, P. Rowlinson, S. Simi´ c, An introduction to the theory of graph spectra. Cambridge University Press, 2010

work page 2010

[5] [5]

Dirac, On rigid circuit graphs,Abh

G.A. Dirac, On rigid circuit graphs,Abh. Math. Sem. Univ. Hamburg.24(1962), 71 – 76

work page 1962

[6] [6]

https://arxiv.org/abs/2504.06617

Fengming Dong and Ruixue Zhang, Existence of trees with prescribed maximum degrees and spectral radii Spectrum Radii of tree. https://arxiv.org/abs/2504.06617

work page arXiv

[7] [7]

(1912): 456– 477

Georg Frobenius, et al., ¨Uber Matrizen aus nicht negativen Elementen. (1912): 456– 477

work page 1912

[8] [8]

Fulkerson, O

D. Fulkerson, O. Gross, Incidence matrices and interval graphs,Pacific Journal of Mathematics15(1965), 835-855

work page 1965

[9] [9]

Grone, R

R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph,SIAM J. Matrix Anal. Appl.11(1990), 218 – 238

work page 1990

[10] [10]

Guo, On the Laplacian spectral radius of a tree,Linear Algebra Appl.368 (2003), 379 – 385

J.M. Guo, On the Laplacian spectral radius of a tree,Linear Algebra Appl.368 (2003), 379 – 385

work page 2003

[11] [11]

Guo, On the second largest Laplacian eigenvalue of the trees,Linear Algebra Appl.404(2005), 251 – 261

J.M. Guo, On the second largest Laplacian eigenvalue of the trees,Linear Algebra Appl.404(2005), 251 – 261

work page 2005

[12] [12]

Godsil, G

C. Godsil, G. Royle, Algebraic graph theory. Springer, New York, 2001

work page 2001

[13] [13]

Discrete Mathematics

Yuan Hong and Zhang Xiaodong, Sharp upper and lower bounds for largest eigen- value of the Laplacian matrices of trees. Discrete Mathematics. 2005 Jul 6;296(2- 3):187-97

work page 2005

[14] [14]

B.L. Liu, B. Zhou, On the third largest eigenvalue of a graph,Linear Algebra and its Applications317(2000), 193-200

work page 2000

[15] [15]

Lov´ asz, J, Pelik´ an

L. Lov´ asz, J, Pelik´ an. On the eigenvalues of trees. Periodica Mathematica Hungarica 3(1-2) (1973), 175 – 82

work page 1973

[16] [16]

Merris, Laplacian matrices of graphs: a survey,Linear Algebra Appl.197–198 (1994) 143–176

R. Merris, Laplacian matrices of graphs: a survey,Linear Algebra Appl.197–198 (1994) 143–176. 15

work page 1994

[17] [17]

Meyer, Matrix Analysis and applied linear algebra

C.D. Meyer, Matrix Analysis and applied linear algebra. SIAM, 2000

work page 2000

[18] [18]

Henryk Minc,Nonnegative matrices. Vol. 170. New York: Wiley, 1988

work page 1988

[19] [19]

Oboudi, On the third largest eigenvalue of graphs,Linear Algebra and its Applications503(2016), 164 – 179

M.R. Oboudi, On the third largest eigenvalue of graphs,Linear Algebra and its Applications503(2016), 164 – 179

work page 2016

[20] [20]

Oskar Perron, Zur Theorie der Matrices,Mathematische Annalen64(2) (1907), 248 –263

work page 1907

[21] [21]

Simic, and D.V

S.K. Simic, and D.V. Toˇ si´ c, The index of trees with specified maximum degree. MATCH Commun. Math. Comput. Chem54(2005), 351 – 362

work page 2005

[22] [22]

Stevanovi´ c, Bounding the largest eigenvalue of trees in terms of the largest vertex degree.Linear algebra and its applications360(2003), 35 – 42

D. Stevanovi´ c, Bounding the largest eigenvalue of trees in terms of the largest vertex degree.Linear algebra and its applications360(2003), 35 – 42

work page 2003

[23] [23]

Xiaodong Zhang, The Laplacian spectral radii of trees with degree sequences,Discrete Mathematics308(15) (2008), 3143 – 3150

work page 2008

[24] [24]

Aimei Yu and Mei Lu, Laplacian spectral radius of trees with given maximum degree, Linear algebra and its applications429(2008), 1962 – 1969

work page 2008

[25] [25]

Xiying Yuan, Yue Liu and Miaomiao Han, The Laplacian spectral radius of trees and maximum vertex degree.Discrete mathematics311(2011), 761 – 618. 16

work page 2011