The effect of a graft transformation on distance signless Laplacian spectral radius of the graphs

Dandan Fan; Guoping Wang; Yinfeng Zhu

arxiv: 1907.05719 · v1 · pith:RMALGTLSnew · submitted 2019-07-11 · 🧮 math.CO

The effect of a graft transformation on distance signless Laplacian spectral radius of the graphs

Dandan Fan , Guoping Wang , Yinfeng Zhu This is my paper

Pith reviewed 2026-05-24 23:25 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C0505C1205C50

keywords graft transformationdistance signless Laplacianspectral radiustreesextremal graphsnon-starlike treesnon-caterpillar treesdistance matrix

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The pith

Graft transformations characterize the minimum and maximum distance signless Laplacian spectral radii among non-starlike and non-caterpillar trees.

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

The paper introduces graft transformations that change the distance signless Laplacian spectral radius of a graph in a controlled, monotonic direction. These operations are applied repeatedly to locate the trees that achieve the smallest and largest values of this radius inside the two specified families. A sympathetic reader cares because the transformations replace exhaustive enumeration with a directed search that reaches the extremal examples. The distance signless Laplacian matrix is defined as the sum of the diagonal transmission matrix and the distance matrix, and its largest eigenvalue is tracked under the transformations.

Core claim

We give some graft transformations on distance signless Laplacian spectral radius of the graphs and use them to characterize the graphs with the minimum and maximal distance signless Laplacian spectral radius among non-starlike and non-caterpillar trees.

What carries the argument

Graft transformations, which relocate a branch or subtree from one vertex to another while keeping the graph connected and altering the transmission vector and distance matrix in a predictable way.

Load-bearing premise

The graft transformations can be applied to every tree in the target classes and always produce a strict monotonic change in the spectral radius.

What would settle it

A counterexample would be any non-starlike or non-caterpillar tree on which a stated graft transformation leaves the distance signless Laplacian spectral radius unchanged or moves it in the opposite direction from the claimed effect.

Figures

Figures reproduced from arXiv: 1907.05719 by Dandan Fan, Guoping Wang, Yinfeng Zhu.

**Figure 1.** Figure 1: G and G′ Proof. Let x be the Perron vector of G′ corresponding to ρQ(G′ ). By symmetry, we can set x = (a, a, a, b, b, b, d, . . . , d | {z } n−7 , e) T labeled in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: G and G∗ Proof. Let x be the Perron vector of G∗ corresponding to ρQ(G∗ ). By symmetry, we can set x = (a, a, b, b, c, c, d, f, e, . . . , e | {z } n−8 ) T labeled in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: T0 If P vi∈V (T1) P vj∈V (T3) (xvi+xvj ) 2 ≥ P vi∈V (T1) P vj∈V (T2) (xvi+xvj ) 2 , then let Te0 = T0− P vj∈V (T1) w1vj+ P vj∈V (T1) us−1vj , and otherwise Te0 = T0− P vj∈V (T1) w1vj + P vj∈V (T1) w2vj . By Lemma 3.1, ρQ(Te0) > ρQ(T0). This contradicts maximality of T0, and so T0 ∼= T(n, k;t1, t2). Let n1, n2, . . . , nr be positive integers satisfying n1 ≤ n2 ≤ . . . ≤ nr and Pr i=1 ni = n − 1. Then we de… view at source ↗

read the original abstract

Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $Tr_{G}(v_{i})$ and $D(G)$ be the distance matrix of $G$. Then $Q_{D}(G)=Tr(G)+D(G)$ is the distance signless Laplacian matrix of $G$. The largest eigenvalues of $Q_D(G)$ is called distance signless Laplacian spectral radius of $G$. In this paper we give some graft transformations on distance signless Laplacian spectral radius of the graphs and use them to characterize the graphs with the minimum and maximal distance signless Laplacian spectral radius among non-starlike and non-caterpillar trees.

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 paper applies standard graft transformations to the distance signless Laplacian on non-starlike non-caterpillar trees but does not address whether the operations stay inside that class.

read the letter

The core of this paper is a claim that certain graft transformations can be used to characterize the trees with minimum and maximum distance signless Laplacian spectral radius inside the class of non-starlike non-caterpillar trees. The matrix itself is defined in the usual way as Tr(G) plus the distance matrix, and the largest eigenvalue is the quantity of interest. The authors say they introduce some grafts and apply them to reach the extremal graphs in that restricted class of trees. That is the entire contribution as stated in the abstract. It is a direct extension of a technique that has been used for other spectral parameters on trees, so the method is familiar rather than new in principle. The specific matrix and the exact tree class are the only fresh elements. The work is narrow but internally consistent on its own terms. The main soft spot is the class-preservation question. A graft that relocates a branch changes many distances at once and can easily reduce the diameter or create a vertex whose removal leaves paths, turning the tree into a caterpillar. If any admissible graft exits the target class, the repeated-application argument cannot reach a global extremal graph that is still non-starlike and non-caterpillar. The abstract gives no indication that this preservation is proved or even stated. Without that step the characterization does not go through. The rest of the paper would need to supply explicit definitions of the grafts, monotonicity proofs, and a clear argument that every step remains inside the class. This is the kind of paper that specialists in spectral graph theory on trees might want to see in a journal after the details are checked. It is not broad enough to interest a general audience, but the question it poses is well-posed and the approach is standard, so it is worth sending to referees who can verify the missing preservation argument and the completeness of the cases.

Referee Report

2 major / 1 minor

Summary. The manuscript defines graft transformations on connected graphs and analyzes their effect on the distance signless Laplacian spectral radius (the largest eigenvalue of Q_D(G) = Tr(G) + D(G)). It then applies these transformations to characterize the non-starlike, non-caterpillar trees that attain the minimum and maximum values of this radius.

Significance. If the transformations can be shown to be strictly monotonic in the spectral radius and to map the target class to itself, the results would supply a concrete reduction technique for extremal problems on distance signless Laplacians of trees, extending standard matrix-based spectral graph theory without introducing free parameters or circular definitions.

major comments (2)

[Sections introducing the graft transformations and the main characterization theorems] The central claim requires that admissible grafts on non-starlike non-caterpillar trees remain inside this class while strictly increasing or decreasing the spectral radius. The skeptic correctly notes that relocating a branch can shorten the diameter or eliminate the unique vertex at distance 2 from a diametral path, producing a caterpillar; the manuscript must therefore contain an explicit preservation argument (with case analysis on the possible graft sites) before the repeated-application characterization is valid.
[Abstract and the sections stating the main results] No explicit definitions of the graft operations, nor verification that they cover all trees in the class without exiting it, appear in the abstract; if the full text likewise omits a self-contained proof that every sequence of grafts stays within the non-starlike non-caterpillar family until the claimed extremal graphs are reached, the derivation gap is load-bearing for the characterization.

minor comments (1)

[Abstract] The phrasing in the abstract ('give some graft transformations on distance signless Laplacian spectral radius of the graphs') is grammatically awkward and should be revised for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit verification that the graft transformations preserve the class of non-starlike non-caterpillar trees. We address the two major comments below and will incorporate the required arguments in the revision.

read point-by-point responses

Referee: [Sections introducing the graft transformations and the main characterization theorems] The central claim requires that admissible grafts on non-starlike non-caterpillar trees remain inside this class while strictly increasing or decreasing the spectral radius. The skeptic correctly notes that relocating a branch can shorten the diameter or eliminate the unique vertex at distance 2 from a diametral path, producing a caterpillar; the manuscript must therefore contain an explicit preservation argument (with case analysis on the possible graft sites) before the repeated-application characterization is valid.

Authors: We agree that an explicit preservation argument with case analysis is required to validate the repeated-application characterization. In the revised manuscript we will add a new lemma that performs a case-by-case analysis of admissible graft sites. The lemma will show that, under the conditions used in the paper, the resulting graph remains non-starlike and non-caterpillar while the distance signless Laplacian spectral radius changes strictly monotonically. This directly addresses the possibility that a graft could produce a caterpillar. revision: yes
Referee: [Abstract and the sections stating the main results] No explicit definitions of the graft operations, nor verification that they cover all trees in the class without exiting it, appear in the abstract; if the full text likewise omits a self-contained proof that every sequence of grafts stays within the non-starlike non-caterpillar family until the claimed extremal graphs are reached, the derivation gap is load-bearing for the characterization.

Authors: The abstract is deliberately concise and refers readers to the definitions and proofs in the body. Nevertheless, we accept that the preservation property should be stated more prominently. We will revise the abstract to note that the grafts preserve the class, and we will add a dedicated subsection (or strengthen the existing one) that supplies a self-contained argument showing that every admissible sequence of grafts remains inside the non-starlike non-caterpillar family until an extremal tree is reached. This will eliminate the derivation gap. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on independent monotonicity proofs for grafts

full rationale

The paper defines the distance signless Laplacian matrix Q_D(G) = Tr(G) + D(G) from standard distance sums, introduces graft transformations as operations that relocate branches, and claims to prove their effect on the largest eigenvalue. It then applies repeated transformations to reach extremal non-starlike non-caterpillar trees. No equation equates a derived quantity to a fitted parameter by construction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The characterization is obtained by showing strict monotonicity under admissible grafts that remain inside the target class; this is an independent combinatorial argument rather than a definitional reduction. The skeptic concern about class preservation is a potential proof gap but does not constitute circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests only on the standard definitions of connected graphs, shortest-path distance, the distance matrix D(G), and the transmission matrix Tr(G); no numerical parameters are fitted and no new entities are postulated.

axioms (2)

standard math Graphs under consideration are finite, undirected, and connected.
Invoked by the definition of Tr_G(v_i) and D(G) in the abstract.
standard math The largest eigenvalue of the symmetric matrix Q_D(G) exists and is real.
Follows from the spectral theorem for real symmetric matrices, used implicitly when referring to the spectral radius.

pith-pipeline@v0.9.0 · 5700 in / 1432 out tokens · 30834 ms · 2026-05-24T23:25:07.203210+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Aouchiche, P

M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21-33

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Aouchiche, P

M. Aouchiche, P. Hansen, Some properties of the distance Lapla cian eigenvalues of a graph, Les Cahiers du GERAD, G-2013-28

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Aouchiche, P

M. Aouchiche, P. Hansen, The distance spectra of a graphs: A s urvey, Linear Algebra Appl. 458 (2014) 301-386

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S. Bose, M. Nath, S. Paul, On the maximal distance spectral rad ius of graphs without a pendant vertex, Linear Algebra Appl. 438 (2013) 4260-4278

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A. Ili´ c, Distance spetral radius of trees with given matching nu mber, Discrete Appl. Math. 158 (2010) 1799-1806

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H. Liu, M. Lu, Bounds on the distance signless Laplacian spectral radius in terms of clique number, Linear and Multilinaer Algebra 63 (2015) 1750-1759

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H. Liu, B. Zhou, On the distance Laplacian spectral radius of gra phs, Linear Algebra Appl. 475 (2015) 265-275

work page 2015
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Minc, Nonnegative matrices

H. Minc, Nonnegative matrices. New York; 1988

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M. Nath, S. Paul, On the distance Laplacian spectra of graphs, L inear Algebra Appl. 460 (2014) 97-110

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W. Ning, L. Ouyang, M. Lu, Distance spectral radius of trees w ith ﬁxed number of pendant vertices, Linear Algebra Appl. 439 (2013) 2240-2249

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A. Niu, D. Fan and G. Wang, On the distance Laplacian spectra of bipartite graphs, Discrete Appl. Math. 186 (2015) 207-213

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Stevanovi´ c, A

D. Stevanovi´ c, A. Ili´ c, Spectral properties of distance matrix of graphs, in: I. Gutman, B. Furtula (Eds), Distance in Molecular graphs Theory, in: Math. Ch em. Monogr., Vol. 12, University of Kragujevac, 2010, pp. 139-176

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R. Xing, B. Zhou, On the distance and distance signless Laplacian spectral radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955-3963. 9

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R. Xing, B. Zhou, J. Li, On the distance signless Laplacian spectr al radius of graphs, Linear and Multilinear Algebra 62 (2014) 1377-1387

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R. Xing, B. Zhou, F. Dong, The eﬀect of a graft transformatio n on distance spectral radius, Linear Algebra Appl. 457 (2014) 261-275

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G. Yu, H. Jia, H. Zhang, J. Shu, Some graft transformations a nd its applications on the distance spectral radius of a graph, Appl. Math. Letters 25 (201 2) 315-319

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G. Yu, Y. Wu, Y. Zhang, J. Shu, Some graft transformations a nd its application on a distance spectrum, Discrete Math. 311 (2011) 2117-2123. 10

work page 2011

[1] [1]

Aouchiche, P

M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21-33

work page 2013

[2] [2]

Aouchiche, P

M. Aouchiche, P. Hansen, Some properties of the distance Lapla cian eigenvalues of a graph, Les Cahiers du GERAD, G-2013-28

work page 2013

[3] [3]

Aouchiche, P

M. Aouchiche, P. Hansen, The distance spectra of a graphs: A s urvey, Linear Algebra Appl. 458 (2014) 301-386

work page 2014

[4] [4]

S. Bose, M. Nath, S. Paul, On the maximal distance spectral rad ius of graphs without a pendant vertex, Linear Algebra Appl. 438 (2013) 4260-4278

work page 2013

[5] [5]

Ili´ c, Distance spetral radius of trees with given matching nu mber, Discrete Appl

A. Ili´ c, Distance spetral radius of trees with given matching nu mber, Discrete Appl. Math. 158 (2010) 1799-1806

work page 2010

[6] [6]

H. Liu, M. Lu, Bounds on the distance signless Laplacian spectral radius in terms of clique number, Linear and Multilinaer Algebra 63 (2015) 1750-1759

work page 2015

[7] [7]

H. Liu, B. Zhou, On the distance Laplacian spectral radius of gra phs, Linear Algebra Appl. 475 (2015) 265-275

work page 2015

[8] [8]

Minc, Nonnegative matrices

H. Minc, Nonnegative matrices. New York; 1988

work page 1988

[9] [9]

M. Nath, S. Paul, On the distance Laplacian spectra of graphs, L inear Algebra Appl. 460 (2014) 97-110

work page 2014

[10] [10]

W. Ning, L. Ouyang, M. Lu, Distance spectral radius of trees w ith ﬁxed number of pendant vertices, Linear Algebra Appl. 439 (2013) 2240-2249

work page 2013

[11] [11]

A. Niu, D. Fan and G. Wang, On the distance Laplacian spectra of bipartite graphs, Discrete Appl. Math. 186 (2015) 207-213

work page 2015

[12] [12]

Stevanovi´ c, A

D. Stevanovi´ c, A. Ili´ c, Spectral properties of distance matrix of graphs, in: I. Gutman, B. Furtula (Eds), Distance in Molecular graphs Theory, in: Math. Ch em. Monogr., Vol. 12, University of Kragujevac, 2010, pp. 139-176

work page 2010

[13] [13]

R. Xing, B. Zhou, On the distance and distance signless Laplacian spectral radii of bicyclic graphs, Linear Algebra Appl. 439 (2013) 3955-3963. 9

work page 2013

[14] [14]

R. Xing, B. Zhou, J. Li, On the distance signless Laplacian spectr al radius of graphs, Linear and Multilinear Algebra 62 (2014) 1377-1387

work page 2014

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R. Xing, B. Zhou, F. Dong, The eﬀect of a graft transformatio n on distance spectral radius, Linear Algebra Appl. 457 (2014) 261-275

work page 2014

[16] [16]

G. Yu, H. Jia, H. Zhang, J. Shu, Some graft transformations a nd its applications on the distance spectral radius of a graph, Appl. Math. Letters 25 (201 2) 315-319

work page

[17] [17]

G. Yu, Y. Wu, Y. Zhang, J. Shu, Some graft transformations a nd its application on a distance spectrum, Discrete Math. 311 (2011) 2117-2123. 10

work page 2011