On the metric dimension of the character degree graph of a solvable group

C. Selvaraj; G. Sivanesan; Jacob Laubacher; Peter J. Cameron; T. Tamizh Chelvam

REVIEW 3 major objections 5 minor 31 references

On the metric dimension of the character degree graph of a solvable group

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves exact formulas for the metric dimension of character degree graphs of finite solvable groups in four structural classes, and shows that two related invariants coincide for Fitting-height-2 graphs.

desk verdict Theorem 4.2 is false — the proposed resolving set fails for two K1,2 stars — and Theorem 3.5 has gaps, though the regular and two-clique results are sound. read the letter →

arxiv 2411.15794 v1 pith:CZ3Q7LS2 submitted 2024-11-24 math.GR math.CO

Peter J. Cameron , G. Sivanesan , C. Selvaraj , T. Tamizh Chelvam , Jacob Laubacher This is my paper · ORCID

classification math.GRmath.CO MSC 05C1220C1505C25

keywords metricdimensioncharacterdegreegraphsolvablegroupFittingheight2Lewisbasesizeadjacencytwinvertices

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

The paper asks how many primes must be pinned down before every prime divisor of an irreducible character degree of a finite solvable group is uniquely identified by its distances to those chosen primes. It proves that for four large classes of character degree graphs the answer is a simple function of the graph's shape. For groups of Fitting height 2, the metric dimension of the Lewis graph is $n_1+n_2-s-1$ when universal vertices exist and $n_2-s$ otherwise, where $n_1$ counts universal vertices, $n_2$ counts the degree-$(n-2)$ vertices, and $s$ counts the multi-edge stars in the complement. It also proves that regular character degree graphs of even order have metric dimension $n/2$, that diameter-2 non-block graphs have dimension $n-3$, and that diameter-3 graphs with a cut vertex have dimension $n-3$ or $n-4$. A sympathetic reader would care because the result turns a structural fact about group characters into a number that can be read directly off the graph.

What carries the argument

The machinery is the Lewis partition of a Fitting-height-2 character degree graph: the vertex set splits into universal vertices $U$, a set $X$, and a set $Y$, so that in the complement the edges are stars with centers in $Y$ and leaves in $X$. The proof counts twin classes, where twins are vertices with identical neighborhoods (possibly except each other), since two twins cannot both be absent from a resolving set. The candidate set $W_0$ takes all but one vertex from each twin class, leaving out at most one universal vertex and at most one leaf from each multi-leaf star. For diameter-3 graphs, the Sass partition $\rho_1\cup\rho_2\cup\rho_3\cup\rho_4$ plays the same role, and the cited structural result says $\rho_2$ is a singleton when a cut vertex is present.

What would settle it

Compute the metric dimension of the connected Lewis graph whose complement is two two-leaf stars $K_{1,2}$ with one universal vertex added. In that graph the candidate $W_0$ consisting of the universal vertex and one leaf from each star leaves two omitted leaves whose distance to every member of $W_0$ is 1; checking whether any other 2-vertex set resolves the graph would settle whether the formula $n_1+n_2-s-1$ remains correct.

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Extended reading notes

Core claim

The central claim is that metric dimension of $\Delta(G)$ is controlled by the same partition structures already used to classify character degree graphs. For a Lewis graph, the graph attached to a solvable group of Fitting height 2, the paper proves that $\dim(\Gamma)=n_1+n_2-s-1$ when $n_1>0$ and $\dim(\Gamma)=n_2-s$ when $n_1=0$, with the parameters taken from the characterization of Fitting-height-2 graphs. The same twin-based argument shows $b(\Gamma)=\dim(\Gamma)=\operatorname{adim}(\Gamma)$ for these graphs. For the other classes, the paper establishes $\dim(\Delta(G))=n/2$ for non-complete regular graphs on $n$ vertices, $\dim(\Delta(G))=n-3$ for diameter-2 graphs made of two cliques joined through a cut vertex, and $\dim(\Delta(G))=n-3$ or $n-4$ for diameter-3 graphs with a cut vertex according as the first part of the Sass partition is a singleton or larger.

Load-bearing premise

The Fitting-height-2 formula rests on the premise that twin equivalence classes are the only obstruction to resolving a vertex set, so that leaving exactly one vertex out of every twin class always yields a resolving set.

Editorial extensions

If this is right

For any solvable group of Fitting height 2, the metric dimension can be computed directly from the Lewis parameters $n_1,n_2,s$, with no further search.
For these groups, base size, metric dimension, and adjacency dimension are equal, so any algorithm bounding one bounds all three.
Regular character degree graphs of even order $n$ have metric dimension $n/2$, confirming the earlier existence result with an exact invariant.
Diameter-3 character degree graphs with a cut vertex have dimension determined by whether the first Sass layer is a singleton, so the cut-vertex structure controls resolvability.
The twin-class argument gives a general sufficient condition: any connected graph whose every vertex belongs to a twin class has $b=\dim=\operatorname{adim}=n-r$ with $r$ the number of twin classes.

Reading between the lines

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

Because the Lewis-graph formula depends only on $n_1,n_2,s$, one could enumerate all Lewis graphs on small vertex sets and computationally verify Theorem 4.2, which would either confirm the constructive proof or expose the graphs where the candidate $W_0$ is not resolving.
The equality of the three invariants for Fitting-height-2 graphs suggests testing whether the same twin-class collapse controls these invariants for other solvable groups whose character degree graphs have small twin quotients.
For diameter-3 graphs, the split between $n-3$ and $n-4$ mirrors the difference between one and several primes in the first Sass layer; a natural extension is to ask whether the same dichotomy appears for adjacency dimension.

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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

Colleague,

The headline is simple: the paper's main new theorem is false. Theorem 4.2, which gives metric dimension for Lewis graphs (Fitting height 2 character degree graphs), asserts dim = n2 − s when n1 = 0. Take the connected Lewis graph with n1=0, n2=4, n3=2 whose complement is two K1,2 stars. This satisfies Lewis's criterion, so it's a character degree graph of a solvable group of Fitting height 2. The theorem predicts dim = 2. But no 2-set resolves the graph: any two selected leaves (one from each star) leave the two omitted leaves from different stars with identical distance vectors, and the centers are not resolving either. The actual metric dimension is 3. The flaw is in the proof's upper-bound half, in the paragraph starting 'However, it is straightforward to check...' — the proposed W0 fails because the two omitted leaves are both adjacent to every vertex of W0.

That's the load-bearing result for Section 5, so Theorem 5.8 falls with it.

There's a second soft spot. Theorem 3.5, for diameter-3 graphs with a cut vertex, has a lower-bound proof that doesn't go through. In Case 3 of part (1), the claim that X1 ⊂ ρ3 ∪ ρ4 is not justified, and the representations of wi and xj are not generally equal when vertices from ρ1 or ρ2 remain in X1. The formula may be right, but the proof as written has a real gap.

What the paper does well: the general twin-class result (Theorem 5.5) is clean and correct, and it yields a nice proof that regular character degree graphs on an even number of vertices have metric dimension n/2. Theorem 3.4, the two-clique case, is sound. The exposition is self-contained and the citation practice is fine — no circularity, and external results are credited properly.

Overall: the regular and two-clique calculations are fine, but the paper's central Fitting-height-2 claim is false, and the diameter-3 proof needs work. This needs major revision, not acceptance. I would still send it to a referee — the topic is legitimate, there are correct parts worth keeping, and a referee should catch exactly these issues. But I wouldn't cite it in its current form.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the metric dimension (and the related base size and adjacency dimension) of the prime character degree graph Δ(G) of a finite solvable group. It announces formulas for four families: (n−2)-regular graphs, diameter-2 non-block graphs of the form K_{n−m−1} − v − K_m, diameter-3 graphs with a cut vertex, and graphs of groups of Fitting height 2 (Lewis graphs). The proofs combine known classification results of Lewis, Wolf–Manz–Willems, and others with elementary metric-dimension arguments. I evaluated the reader's concern about Theorem 4.2 and found it correct; I additionally found a counterexample to Theorem 3.4. The paper is clearly written and some auxiliary results, such as Theorem 5.5 and Proposition 5.6, are sound, but two of the four principal formulas are false as stated. I did not find a comparable defect in Theorem 3.5: the uniform-distance assertions used there follow from the structure in Remark 2.5 together with the diameter-3 partition.

Significance. If the announced formulas were correct, the paper would give a useful graph-theoretic toolkit for character degree graphs, and the connection between base size, metric dimension, and adjacency dimension for Fitting-height-2 groups would be an interesting contribution. The paper also has genuine virtues: the twin-class argument in Theorem 5.5 is elegant, the use of Lewis's theorem to reduce Fitting-height-2 graphs to complement-stars is natural, and the examples are mostly helpful. However, the central claims in Theorems 3.4 and 4.2 are false, and Theorem 5.8 is invalidated along with Theorem 4.2. Since these are the paper's main results, the contribution in its current form is not reliable.

major comments (3)

[Section 4, Theorem 4.2] The formula in Theorem 4.2 is false. Take the Lewis graph with n1=0, n2=4, n3=2 whose complement is two disjoint K1,2 stars, with centers y1,y2 and leaves {x1,x2},{x3,x4}. The theorem gives dim = n2 − s = 4 − 2 = 2. But no two-vertex set resolves: if the two selected vertices lie in X, the two omitted leaves have the same distance vector; if a star center is selected instead, two leaves still collide (for example, with W={y1,y2}, x1 and x2 both have representation (2,1), and with W={x1,y1}, x3 and x4 both have (1,1)). The set {x1,x2,x3} does resolve, so dim = 3. This graph satisfies Lewis's criterion in Theorem 4.1, hence it is Δ(G) for a solvable group of Fitting height 2. The flaw is in the paragraph beginning 'However, it is straightforward to check...': the proposed set W0 contains leaves and universal vertices but no star centers, so two omitted leaves from different stars are both adjacent to every vertex of W0 and have identical all-ones representations. Moreover, the paper's own Example 4.3 contradicts the theorem: its parameters n1=1, n2=3, n3=2 force s=1, so Theorem 4.2 would predict dim=2, while the example asserts that {v4,v5,v6} is a minimum resolving set of size 3.
[Section 3, Theorem 3.4] The formula dim = n − 3 in Theorem 3.4 is false for m=1. Let Γ have vertices a,b,v,c with edges ab, av, bv, vc. This is the graph K2 − v − K1 with n=4,m=1, so it is within the theorem's hypotheses. It also satisfies Lewis's criterion with X={a,b}, Y={c}, U={v}, so it is a character degree graph of a solvable group of Fitting height 2. But dim(Γ)=2: no one-vertex set resolves, because {v} gives distance 1 to a,b,c, {a} gives distance 1 to both b and v, and {c} gives distance 2 to both a and b, while {a,c} has representations a=(0,2), b=(1,2), v=(1,1), c=(2,0). The proposed resolving set W=V∖{v1,v_{m+1},v_n} has size n−3=1 in this case and fails, because the cut vertex and the vertex of K_{n−m−1} omitted from W have the same representation; for m=1 and any n≥4 the same collision occurs. The theorem therefore needs an additional hypothesis, and the proof does not supply a correct resolving set in the m=1 case.
[Section 5, Theorem 5.8] The asserted equality b(Δ(G)) = dim(Δ(G)) = adim(Δ(G)) for all Fitting-height-2 groups is false. For the two-star Lewis graph from the first major comment, the automorphism group is S2×S2, so a base must contain one vertex from each twin pair, and {x1,x3} is a base; hence b = 2. But dim = 3 as shown above, and since the graph has diameter 2, adim = dim = 3. Thus the three invariants are not equal. The proof of Theorem 5.8 relies directly on the incorrect dimension computation in Theorem 4.2.

minor comments (5)

[Section 3, Theorem 3.4 proof] The notation switches between W and W1 in the proof; the vectors labelled r(v1|W1), r(v_{m+1}|W1), and r(v_n|W1) should refer to the same set W used in the claim. This makes the argument hard to follow.
[Section 5, Theorem 5.8] The statement should read b(Δ(G)) = dim(Δ(G)) = adim(Δ(G)); the closing parenthesis is missing after 'b(Δ(G)'.
[Section 5, Proposition 5.6] The phrase 'both equal to n/2' should be 'all three invariants equal to n/2', since three quantities are being compared.
[Section 5, Theorem 5.5 proof] In the proof, 'adim(S) ≤ n − r' should be 'adim(Γ) ≤ n − r'.
[Sections 4 and 5] Theorems 4.2 and 5.8 do not explicitly assume Γ is connected, but metric dimension and adjacency dimension are normally defined only for connected graphs; Lewis graphs can be disconnected (for example, n1=0, n2=2, n3=1), so a connectedness hypothesis should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the metric-dimension results are derived from external graph and group theorems; the flawed upper-bound proof in Theorem 4.2 is a correctness error, not a circular reduction.

full rationale

I walked each load-bearing step of the paper. Theorem 3.1 is deferred to Proposition 5.6, which uses the external theorem [26, Theorem A] (Morresi Zuccari) to identify non-complete regular character degree graphs as (n-2)-regular, observes that the complement is a 1-factor, and applies the paper's own general twin-class result Theorem 5.5. No fitted quantity is relabeled as a prediction, and no self-citation carries the argument. Theorem 3.4 uses the external block-structure result [11, Lemma 2.7] and then gives a direct resolving-set proof. Theorem 3.5 uses the external cut-vertex structure theorem [14, Theorem 10] and Lewis-Meng [23, Lemma 2.1], followed by a direct construction of resolving sets and lower-bound cases. Theorem 4.2 uses Lewis's external characterization [21, Theorem A] to define Lewis graphs; the lower bound is a standard twin argument and the upper bound is an attempted direct verification that W0 resolves. That verification is in fact wrong, as the reader's take notes, but being wrong is not the same as being circular: the claim is not assumed in order to prove itself. Theorem 5.8 inherits the error from Theorem 4.2, but again the dependency is one-directional, not a reduction of the conclusion to its own input. The only self-citation of note is [30], used in Example 3.2 to assert that a given 4-regular graph is a character degree graph; that existence fact is not load-bearing for the metric-dimension formulas, which proceed from the group-theoretic graph structure regardless of which specific group realizes it. No ansatz is smuggled in via citation, no uniqueness theorem is imported from the present authors, and no known result is merely renamed. Accordingly, the correct finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on external theorems from character-degree graph theory and standard metric-dimension facts. No free parameters and no invented entities are introduced. The flaw in Theorem 4.2 is not an axiom failure; the counterexample satisfies the paper's own Lewis criterion, so the proof of the formula is what fails.

assumptions (5)

domain assumption A non-complete regular character degree graph of a solvable group on n vertices is (n-2)-regular.
Used in Theorem 3.1 and Proposition 5.6, quoted as Theorem 2.10 from Morresi Zuccari.
domain assumption The structure theorem of Hafezieh et al. classifies connected character degree graphs with a cut vertex.
Quoted as Theorem 2.4 and used to set up the diameter-3 cut-vertex case in Theorem 3.5.
domain assumption Lewis's Theorem A characterizes character degree graphs of Fitting-height-2 groups by the two-clique partition condition.
Quoted as Theorem 4.1 and used as the foundation of Theorem 4.2. The counterexample satisfies this criterion, so the flaw is inside the paper's own framework.
domain assumption Character degree graphs of solvable groups have diameter at most 3.
Quoted from Wolf, Manz, and Willems and used in the diameter-3 section.
standard math Chartrand et al. results on metric dimension of complete graphs, paths, cycles, and n-2 graphs.
Background facts used for comparison and lower bounds throughout the paper.

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Pith. "Pith review of On the metric dimension of the character degree graph of a solvable group." pith.science (2026). https://pith.science/paper/CZ3Q7LS2

@misc{pith2026241115794,
  author       = {Pith},
  title        = {Pith review of: On the metric dimension of the character degree graph of a solvable group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CZ3Q7LS2}},
  note         = {Machine review of arXiv:2411.15794}
}

abstract

Let $G$ be a finite solvable group and let $\Delta(G)$ be the character degree graph of $G$. In this paper, we obtain the metric dimension of certain character degree graphs. Specifically, we calculate the metric dimension for a regular character degree graph, a character degree graph with a diameter of $2$ that is not a block, a character degree graph with a diameter of $3$ that also has a cut vertex and a character degree graph with Fitting height $2.$ We also consider two related parameters, base size and adjacency dimension, and their relation to metric dimension for character degree graphs of solvable groups.

Figures

Figures reproduced from arXiv: 2411.15794 by the authors.

**Figure 1.** A 4-regular character degree graph with six vertices Now, we obtain the metric dimension of a particular class of character degree graphs. Before proceeding in general, however, we give an example [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Character degree graph has 2 blocks of diameter 2 We now obtain the metric dimension of a class of character degree graphs. In fact we consider the class of character degree graphs exhibited in Example 3.3. By Theorem 2.3, if ∆(G) is not a block and the diameter of ∆(G) is at most 2, then each block of ∆(G) is a complete graph. For a graph Γ and a vertex v /∈ V (Γ), by Γ − v, we mean a graph obtained by joining ever… view at source ↗

**Figure 3.** Character degree graph with diameter 3 and a cut vertex 4. Groups with Fitting height 2 Recall that a finite group G has Fitting height 2 if G/F(G) is nilpotent, where the Fitting subgroup F(G) is the largest nilpotent normal subgroup of G. Our discussion of the character degree graphs of finite groups with Fitting height 2 is helped greatly by the theorem of Mark Lewis [21]: Theorem 4.1. [21, Theorem A] Let Γ be a … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The complement of ∆(G) and a resolving set form a partition of X.) Note that some stars may consist of a single edge, in which case which vertex is in X and which in Y are not determined. Let s be the number of trees with more than two vertices. Theorem 4.2. The metric…

**Figure 5.** Figure 5: Character degree graph of a solvable group with six vertices and Fitting height 2 5. Base size and adjacency dimension Two further invariants related to metric dimension have been studied; we look briefly at these for character degree graphs, referring to [3] for a sum…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Babai, On the complexity of canonical labeling of strongly regular graphs, SIAM J

L. Babai, On the complexity of canonical labeling of strongly regular graphs, SIAM J. Comput. 9 (1980), 212–216

work page 1980

[2] [2]

Babai, On the order of uniprimitive permutation groups , Ann

L. Babai, On the order of uniprimitive permutation groups , Ann. Math. 113 (1981), 553–568

work page 1981

[3] [3]

Bailey and Peter J

Robert F. Bailey and Peter J. Cameron, Base size, metric dimension, and other invariants of groups and graphs , Bull. London Math. Soc. 43 (2011), 209–242

work page 2011

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Rodr ´ ıguez, Juan A

Sergio Bermudo, Jos´ e M. Rodr ´ ıguez, Juan A. Rodr ´ ıguez-Vel´ azquez and Jos´ e M. Sigarreta, The adjacency dimension of graphs , Ars Mathematica Contemporanea 22 (2022), #P3.02

work page 2022

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M. W. Bissler and J. Laubacher, Classifying families of character degree graphs of solvabl e groups, Int. J. Group Theory 8 (2019), no. 4, 37–46

work page 2019

[6] [6]

M. W. Bissler, J. Laubacher, and M. L. Lewis, Classifying character degree graphs with six vertices, Beitr. Algebra Geom. 60 (2019), 499–511

work page 2019

[7] [7]

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