Bounds on the $\alpha$-distance spectrum of graphs

Changjiang Bu; Lizhu Sun; Yang Yang

REVIEW 1 major objections 4 minor 30 references

Bounds on the $\alpha$-distance spectrum of graphs

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The $\alpha$-distance spectrum of a connected graph obeys sharp bounds tied to its Wiener index.

desk verdict The inequality in Theorem 3.10 is true but its equality case is false, and Proposition 2.3 is contradicted by P_3 with α=3/5. read the letter →

arxiv 1908.03893 v1 pith:UPUUUUVT submitted 2019-08-11 math.CO

Yang Yang , Lizhu Sun , Changjiang Bu This is my paper · ORCID

classification math.CO MSC 05C50

keywords alpha-distancematrixdistancespectrumgraphenergyEstradaindexspectralradiusWienertransmissionregularcomplete

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the $\alpha$-distance matrix $D_{\alpha}(G)=\alpha\operatorname{Tr}(G)+(1-\alpha)D(G)$, a one-parameter family that interpolates between the distance matrix, the distance signless Laplacian, and the transmission matrix of a connected graph. It proves bounds on the largest eigenvalue (the spectral radius), on the $\alpha$-distance energy, and on a newly defined $\alpha$-distance Estrada index, all expressed through the Wiener index, vertex transmissions, and the Frobenius norm of $D_{\alpha}(G)$. The sharp results identify complete graphs as the unique extremal cases: equality in the spectral-radius upper bound and in the energy lower bound holds exactly for $K_n$. The paper matters because these bounds estimate spectral properties without computing the full spectrum and unify earlier distance-matrix and distance-signless-Laplacian inequalities as the special cases $\alpha=0$ and $\alpha=1/2$.

What carries the argument

The load-bearing object is the $\alpha$-distance matrix $D_{\alpha}(G)=\alpha\operatorname{Tr}(G)+(1-\alpha)D(G)$, a real symmetric nonnegative matrix whose Perron root is the spectral radius $\sigma_1(G)$. The proofs repeatedly center the eigenvalues by subtracting their mean $2\alpha W(G)/n$ and then feed them through variance-type inequalities, chiefly Lemma 3.9: if $x_1,\dots,x_m$ have zero sum, then $x_1\leq \sqrt{\frac{m-1}{m}\sum_i x_i^2}$. Row-sum estimates for polynomials in $D_{\alpha}(G)$, obtained from the Perron-Frobenius theorem, convert bounds on vertex transmissions into eigenvalue bounds. A quotient-matrix interlacing argument (Lemma 3.11) produces the spectral-spread estimate in Proposition 3.12.

What would settle it

For $\alpha=1/2$, compute the eigenvalues of $D_{\alpha}$ for every connected graph on up to six vertices, or for the cycle $C_5$ and the Petersen graph; if any non-complete graph has exactly two distinct eigenvalues, Proposition 2.3 is false and the equality statement of Theorem 3.10 collapses. Equivalently, evaluate both sides of Theorem 3.10 on a non-complete graph such as $C_5$; a single equality example would refute the claimed characterization.

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Extended reading notes

Core claim

The central discovery is a set of sharp inequalities for the $\alpha$-distance spectrum. Theorem 3.10 states that for every connected $n$-vertex graph $G$, $$\sigma_1(G)\leq \frac{2\$\alpha$ W(G)}{n}+\sqrt{\frac{n-1}{n}\left(\|D_{\$\alpha$}(G)\|$_F^{2}$-\frac{4\$alpha^{2}$W(G)^2}{n}\right)},$$ with equality if and only if $G$ is the complete graph $K_n$. Theorem 2.7 gives the companion lower bound for the $\alpha$-distance energy, $\varsigma_{\alpha}(G)\geq 2(1-\alpha)(n-1)$ for $\alpha\in[1/2,1)$, again with equality exactly for $K_n$. The paper also defines the $\alpha$-distance Estrada index $\operatorname{DEE}_{\alpha}(G)=\sum_i e^{\sigma_i(G)}$ and proves upper and lower bounds for it, with transmission-regular graphs appearing as extremal cases. The proofs shift the eigenvalues by their mean $2\alpha W(G)/n$ and apply variance-type inequalities, Perron-Frobenius row-sum estimates, and quotient-matrix interlacing.

Load-bearing premise

The load-bearing premise is Proposition 2.3, stated without proof: for $\alpha\in[0,1)$, the matrix $D_{\alpha}(G)$ has exactly two distinct eigenvalues only when $G$ is complete. Theorem 3.10 uses this equivalence to turn its algebraic equality condition into the graph-theoretic equality case $G\cong K_n$; if the proposition is false, the inequality can still hold but the equality characterization fails.

Editorial extensions

If this is right

For complete graphs the new inequalities become equalities, so $K_n$ is the unique extremal connected graph for the $\alpha$-distance spectral-radius bound and the $\alpha$-distance energy lower bound.
Because $\alpha=0$ gives the ordinary distance matrix and $\alpha=1/2$ gives half the distance signless Laplacian, the bounds specialize to known and new inequalities for those matrices.
The $\alpha$-distance Estrada index bounds provide computable estimates of $\operatorname{DEE}_{\alpha}(G)$ from the Wiener index, vertex transmissions, and the sum of squared distances, without computing eigenvalues.
The explicit star formulas (Theorem 2.8 and Proposition 2.5) give benchmarks for testing how close other trees come to extremal behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the equality characterization in Theorem 3.10 holds, complete graphs are the unique maximizers of $\sigma_1(G)$ among connected graphs with fixed order and Wiener index; a similar extremal principle may extend to other distance-based spectral invariants by the same mean-shift argument.
Because the Estrada index is built from the whole spectrum, comparing the $\alpha=0$ specialization of these bounds with known distance-Estrada bounds would quantify how much the parameter $\alpha$ expands the theory.
A direct testable extension is to determine which graphs minimize $\sigma_1(G)$ for fixed $n$ and $W(G)$; stars are natural candidates by analogy with Lemma 2.4, but this paper does not prove that.
A small exhaustive check of non-complete graphs could verify whether Proposition 2.3's 'two distinct eigenvalues iff complete' claim is true; if it fails, the inequalities survive but the equality cases need re-identification.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my take on arXiv:1908.03893 (Yang, Sun, Bu). The paper defines the α-distance Estrada index and proves a set of bounds on the α-distance spectral radius, energy, and Estrada index. The new index is a legitimate extension of the distance Estrada index, and the closed-form star energy formula in Theorem 2.8 is a concrete new result. Most proofs are standard: trace identities, Cauchy-Schwarz, Perron-Frobenius, and quotient-matrix interlacing. The bounds in Theorems 2.7, 2.11, 2.13, 3.8, and 3.14 look correct to me.

But the paper has a load-bearing false statement. Proposition 2.3 claims that for α∈[0,1), Dα(G) has two distinct eigenvalues only if G is complete. That is false. Take P3 with α=3/5. Dα has eigenvalues 14/5 and 1 (multiplicity 2), so two distinct eigenvalues, but P3 is not complete. This is not a curiosity: Theorem 3.10 uses Proposition 2.3 to conclude that equality in its spectral-radius bound occurs only for Kn. For the same P3 and α=3/5, equality in Theorem 3.10 does hold, so the claimed equality characterization is wrong. The inequality itself survives; the "if and only if" does not.

Other soft spots: Theorem 3.15 asserts monotonicity of a function f(x) on [0,4] without proof; that step needs checking. Cross-references are broken in places (Theorem 3.16 cites Lemma ??). The typesetting is rough, making some formulas hard to verify, but that is secondary.

The derivations are otherwise self-contained and rely on external lemmas rather than circular reasoning. The novelty is extension-level, not conceptual.

Is this worth a serious referee? Yes. The mistake in Proposition 2.3 is serious but localized; the principal inequalities appear to hold. A referee can send the authors back to correct equality conditions, prove or remove the false proposition, and fix the missing details. If revised, the paper could become a reasonable spectral graph theory contribution. As it stands, I would not accept it.

Referee Report

1 major / 4 minor

Summary. The paper studies the α-distance matrix Dα(G)=αTr(G)+(1−α)D(G) of a connected graph and defines the α-distance Estrada index. It derives upper and lower bounds for the α-distance energy, for the spectral radius of Dα(G), and for the α-distance Estrada index, using trace identities, Perron-Frobenius theory, quotient-matrix interlacing, and standard inequalities. The α-distance spectrum and energy of stars are computed explicitly, and equality conditions are stated for several of the bounds.

Significance. If the results were fully correct, the paper would provide a useful parameterized family of bounds interpolating between distance and signless-Laplacian spectral invariants, together with a new Estrada-type invariant and explicit formulas for stars. The main inequalities are derived transparently from standard tools, and many of the trace computations (Lemma 2.12, Eq. (4)), Cauchy-Schwarz steps, and quotient-matrix arguments are sound in their inequality form. However, the equality characterization in Theorem 3.10 depends on a false proposition, so the paper needs a substantial correction before its central equality claims can be accepted.

major comments (1)

[§2, Proposition 2.3; §3, Theorem 3.10] Proposition 2.3 is false as stated. For G=P3 and α=3/5, Dα(G)=[[9/5,2/5,4/5],[2/5,6/5,2/5],[4/5,2/5,9/5]] has characteristic polynomial (x−14/5)(x−1)^2, so it has exactly two distinct eigenvalues although P3 is not complete. This proposition is exactly what converts the eigenvalue equality condition in Lemma 3.9 into the graph-theoretic equality case in Theorem 3.10, so the equality characterization G≅Kn is incorrect. Indeed, for this example μ=2αW/n=8/5, σ1−μ=6/5, and σ2−μ=σ3−μ=−3/5, which satisfies the equality condition of Lemma 3.9, so equality in Theorem 3.10 holds for a non-complete graph. The inequality in Theorem 3.10 appears correct, but the equality case must be restated in terms of the eigenvalue multiplicity condition or proved by a different argument.

minor comments (4)

[§3, Theorem 3.15] The claim that f(x) is monotonically decreasing on [0,4] is stated without proof and is in fact false; for α=1/2, n=3, W(G)=3, f(0)≈7.303 and f(2)≈7.506. The derived lower bound is already obtained by taking δ=0, so the monotonicity assertion should be removed or corrected.
[§3, Theorem 3.16 and Corollary 3.17] The reference 'Lemma ??' should be to Lemma 2.1, and Corollary 3.17 cites 'Theorem 2.1', but no Theorem 2.1 exists in the paper.
[§3, Proposition 3.12] In the proof of Proposition 3.12, 'λ1(BM), λ1(BM)' should read 'λ1(BM), λ2(BM)', and the displayed discriminant appears to contain typographical errors: 'αnTr2(vi)' should presumably be 'αnTr(vi)' and 'Tr2(vi)' should be 'Tr(vi)^2'; as printed, the formula does not match the characteristic polynomial of the displayed quotient matrix (for example, for K3 with α=1/2).
[Throughout] There are numerous typographical issues, including 'transimission' for 'transmission', 'greaterorequalslant' for '≥', and inconsistent notation 'T r' versus 'Tr'; these should be cleaned up before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bounds follow from standard inequalities and externally cited lemmas; Proposition 2.3 is unsupported/false but that is a correctness issue, not circularity.

full rationale

All derived inequalities (Theorems 2.7, 2.11, 2.13, 3.8, 3.10, 3.14, 3.15, 3.16, 3.18) are obtained by combining trace identities (Eq. (3), Lemma 2.12), Cauchy-Schwarz, AM-GM, Perron-Frobenius row-sum arguments, and externally cited lemmas ([3], [8], [9], [22], [26], etc.). No parameter is fitted to data and no quantity being bounded is inserted as an assumption. For example, Theorem 3.10's bound follows from the zero-sum identity sum_i(sigma_i - 2alpha W/n) = 0 and Lemma 3.9, with the right-hand side computed from the invariant ||D_alpha(G)||_F^2 and 2alpha W/n; the equality case invokes Proposition 2.3. That proposition is stated without proof ('From Lemma 2.2, we have the following result') and appears false: for P_3 with alpha = 3/5, D_alpha has eigenvalues 14/5, 1, 1, so two distinct eigenvalues occur while P_3 is not complete. This is a correctness defect in the equality characterization, not a circular dependence: it does not reduce Theorem 3.10 to its own inputs. The only 'from Lemma 2.2' inference in Proposition 2.3 is an omitted-converse gap, again a correctness concern rather than circularity. No load-bearing self-citations by the present authors appear, and the cited lemmas come from other research groups. Accordingly, no circular step can be exhibited; the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The paper introduces no fitted constants. All bounds are expressed in terms of graph invariants (n, W, Tr, S) and the parameter α. The main external dependencies are cited lemmas from [3], [4], [8], [14], and [20]; accepting these is the price of entry. Proposition 2.3 is new but unproved. The new index DEE_α is a definition, not a postulated entity requiring independent evidence.

assumptions (10)

standard math Perron-Frobenius theorem for nonnegative weakly irreducible matrices: Dα has a positive Perron vector and σ1 = ρα.
Invoked in Section 1.2 and in Proposition 3.6 to write Dαx = σ1x with x > 0.
standard math Cauchy-Schwarz inequality and power-mean inequalities for sums of real numbers.
Used in Theorems 2.11, 2.13, and 3.14 to bound sums of absolute eigenvalue deviations.
standard math AM-GM inequality applied to products of exponentials.
Used in Theorem 3.15 to lower-bound pairwise cross terms in the square of the Estrada index.
standard math Quotient-matrix interlacing theorem of Haemers.
Cited as Lemma 3.11 and used in Proposition 3.12 to bound eigenvalue spread.
domain assumption Lemma 2.1 from [3]: ρα(G) ≥ 2W(G)/n for the α-distance spectral radius.
External result accepted without proof; underpins Theorem 2.7 and Theorem 3.16.
domain assumption Lemma 2.2 from [3]: eigenvalue multiplicity for vertices with common neighborhoods.
External result used to derive Proposition 2.3 and the star spectrum in Proposition 2.5.
domain assumption Lemma 2.4 from [4,14,20]: the star minimizes ρα among n-vertex trees, with an explicit closed-form expression.
External result used in Proposition 2.5 and Theorem 2.8 for the exact star energy formula.
domain assumption Lemma 2.9 from [8]: ‖D(G)‖_F² < (1/n)(Σ Tr(vi))².
External inequality used in Proposition 2.10 to bound the Frobenius norm of Dα.
domain assumption Lemma 3.1 from [8] and Lemma 3.3 from [7,19]: bounds on sums of absolute deviations and row sums of nonnegative matrices.
External results used in Proposition 3.2 and Proposition 3.4.
domain assumption Dα(G) is positive semidefinite for α ∈ [1/2,1].
Stated in Section 1.2 without proof; used to ensure nonnegative eigenvalues in Proposition 3.2 and in Estrada-index lower bounds.

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Pith. "Pith review of Bounds on the $\alpha$-distance spectrum of graphs." pith.science (2026). https://pith.science/paper/UPUUUUVT

@misc{pith2026190803893,
  author       = {Pith},
  title        = {Pith review of: Bounds on the $\alpha$-distance spectrum of graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPUUUUVT}},
  note         = {Machine review of arXiv:1908.03893}
}

abstract

For a simple, undirected and connected graph $G$, $D_{\alpha}(G) = \alpha Tr(G) + (1-\alpha) D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $\alpha$-distance energy of $G$ was defined based on the spectra of $D_{\alpha}(G)$. In this paper, we define the $\alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{\alpha}(G)$. And we give some bounds on the spectral radius of $D_{\alpha}(G)$, $\alpha$-distance energy and $\alpha$-distance Estrada index of $G$.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Aouchiche, P

M. Aouchiche, P. Hansen, Two Laplacians for the distance matrix of a graph. Linear Algebra Appl. 439(1)(2013) 21-33

work page 2013

[2] [2]

Brualdi, Energy of a graph, Notes for AIM Workshop On Spec tra of families of matrices described by graphs, digraphs and sign pattern s

R.A. Brualdi, Energy of a graph, Notes for AIM Workshop On Spec tra of families of matrices described by graphs, digraphs and sign pattern s. 2006

work page 2006

[3] [3]

Cui, J.X

S.Y. Cui, J.X. He, G.X. Tian, The generalized distance matrix. Linear Al- gebra Appl. 563(2019) 1-23

work page 2019

[4] [4]

On the generalized distance spectral radius of graphs

S.Y. Cui, G.X. Tian, L. Zheng, On the generalized distance spectra l radius of graphs. arXiv:1901.07695, 2019

work page Pith review arXiv 1901

[5] [5]

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

[6] [6]

K.C. Das, M. Aouchiche, P. Hansen, On (distance) Laplacian ener gy and (distance) signless Laplacian energy of graphs. Discrete Appl. Math. 243(2018) 172-185

work page 2018

[7] [7]

New bounds on the distance Laplacian and distance signless Laplacian spectral radii

R.C. Daz, A.I. Julio, O. Rojo, New bounds on the distance Laplacian and distance signless Laplacian spectral radii. arXiv:1804.06335, 2018

work page Pith review arXiv 2018

[8] [8]

R.C. Daz, O. Rojo, Sharp upper bounds on the distance energies of a graph. Linear Algebra Appl. 54(2018) 55-75

work page 2018