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arxiv: 2604.14195 · v1 · submitted 2026-04-02 · 🧮 math.CO · math.SP

RD_α-Spectra of Joined Union Graphs with Applications to Power Graphs of Finite Groups

Aditya Singh , Yogendra Singh , Anand Kumar Tiwari This is my paper

Pith reviewed 2026-05-13 20:55 UTC · model grok-4.3

classification 🧮 math.CO math.SP MSC 05C50
keywords reciprocal distance matrixRD alpha spectrumjoined union graphspower graphsfinite groupscharacteristic polynomialgraph spectradihedral groups
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The pith

Joined unions of regular graphs have closed-form characteristic polynomials for their generalized reciprocal distance matrices RD_α.

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

The paper defines the matrix RD_α as a convex combination of the reciprocal transmission matrix and the reciprocal distance matrix of a graph. It derives explicit formulas for the characteristic polynomial of this matrix on any joined union constructed from regular graphs. These formulas produce the complete spectra for several families of graphs without requiring direct computation. The same structural results are used to express the power graphs of cyclic groups, dihedral groups, generalized quaternion groups, elementary abelian p-groups, and certain groups of order pq as joined unions, thereby giving their RD_α spectra in closed form.

Core claim

For a joined union of regular graphs, the characteristic polynomial of RD_α factors into an explicit product involving the spectra of the component graphs, the sizes of the components, and the joining edges; this identity supplies closed-form RD_α spectra for the power graphs of D_{2n}, Q_{4n}, Z_n, and selected non-abelian groups by first showing that each such power graph admits a joined-union representation.

What carries the argument

The joined-union construction on regular graphs, which produces a block-structured RD_α matrix whose eigenvalues are obtained by combining the component spectra with a single auxiliary quadratic factor that accounts for the complete bipartite connections between blocks.

If this is right

  • The RD_α spectra of all power graphs of dihedral groups D_{2n} are given by explicit algebraic expressions in n and α.
  • Analogous closed-form spectra hold for the power graphs of generalized quaternion groups Q_{4n} and elementary abelian p-groups.
  • The characteristic polynomials factor completely into linear and quadratic terms whose roots are written in radicals.
  • The same technique yields the spectra for power graphs of all cyclic groups and certain non-abelian groups of order pq.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit spectra could be used to compare energy or Wiener-index-type invariants across different families of group power graphs.
  • The block-diagonal technique may extend to other linear combinations of distance-based matrices on the same joined-union graphs.
  • Numerical verification on small-order groups would immediately confirm or refute the algebraic expressions for their power-graph spectra.

Load-bearing premise

The input graphs to the joined union must be regular so that their individual RD_α matrices are simultaneously diagonalizable with the all-ones vector and combine without extra interaction terms.

What would settle it

For the joined union of two copies of the cycle C_4, compute the 8-by-8 RD_α matrix explicitly for any fixed α and compare its characteristic polynomial with the formula obtained by substituting the known C_4 spectrum into the joined-union expression.

read the original abstract

The \emph{generalized reciprocal distance matrix} of a graph $\mathscr{G}$, denoted by $RD_\alpha(\mathscr{G})$, is defined as $RD_\alpha(\mathscr{G})=\alpha\,RT_r(\mathscr{G})+(1-\alpha)\,RD(\mathscr{G}), \, \alpha\in[0,1],$ where $RT_r(\mathscr{G})$ represents the diagonal matrix of reciprocal vertex transmissions, and $RD(\mathscr{G})$ is the Harary (reciprocal distance) matrix of $\mathscr{G}$. In this paper, we investigate the $RD_\alpha$-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of $RD_\alpha(\mathscr{G})$ when $\mathscr{G}$ is formed as a joined union of regular graphs. These results provide closed-form expressions for the corresponding spectra of several important graph classes. Moreover, we show that the power graphs of the dihedral group $D_{2n}$ and the generalized quaternion group $Q_{4n}$ admit representations as joined union graphs. Using this structural characterization, we determine the $RD_\alpha$-spectra of power graphs arising from various classes of finite groups, including cyclic groups $\mathbb{Z}_n$, dihedral groups $D_{2n}$, generalized quaternion groups $Q_{4n}$, elementary abelian $p$-groups, and certain non-abelian groups of order $pq$.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines the generalized reciprocal distance matrix RD_α(G) = α RT_r(G) + (1-α) RD(G) for α in [0,1] and derives explicit formulas for its characteristic polynomial when G is the joined union of regular graphs. It then identifies power graphs of groups such as D_{2n}, Q_{4n}, Z_n, elementary abelian p-groups, and certain groups of order pq as joined unions, and computes their RD_α-spectra in closed form.

Significance. If the derivations are correct, the work supplies closed-form spectra for an infinite family of graphs arising from group power graphs, which is a concrete contribution to spectral graph theory with potential applications in algebraic combinatorics. The explicit formulas and structural characterizations are the primary strengths.

major comments (2)
  1. [§3] §3 (main theorem on joined unions): the block-matrix derivation of the characteristic polynomial assumes that all vertices within each regular component share the same internal transmission (i.e., constant row sums of the RD block). Regularity alone guarantees constant degree but not constant transmission; the paper does not prove or assume transmission-regularity (or vertex-transitivity/distance-regularity) of the components, so the claimed closed-form expression does not follow from the stated hypotheses.
  2. [§4–5] Application sections (power graphs of D_{2n} and Q_{4n}): the identification of these power graphs as joined unions of regular graphs is used to invoke the §3 formula, but the same uniformity-of-transmission gap reappears; explicit verification that each component is transmission-regular is missing.
minor comments (2)
  1. [§2] Notation: the symbol RT_r is introduced without an explicit definition of the underlying transmission function; a one-line reminder would help.
  2. [§3] Several displayed equations in §3 contain minor index typos (e.g., summation limits on the all-ones blocks) that do not affect the logic but should be corrected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the precise identification of the gap between degree-regularity and transmission-regularity. We address both major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem on joined unions): the block-matrix derivation of the characteristic polynomial assumes that all vertices within each regular component share the same internal transmission (i.e., constant row sums of the RD block). Regularity alone guarantees constant degree but not constant transmission; the paper does not prove or assume transmission-regularity (or vertex-transitivity/distance-regularity) of the components, so the claimed closed-form expression does not follow from the stated hypotheses.

    Authors: We agree that the block-matrix argument requires constant row sums within each diagonal block of the reciprocal-distance matrix, which holds if and only if each component is transmission-regular. The original statement of the theorem mentioned only regularity; this was an oversight. In the revised version we will restate the main theorem for joined unions of transmission-regular graphs, add a short paragraph recalling the definition of transmission-regularity, and note that many graphs arising in the applications (vertex-transitive graphs, distance-regular graphs, and the specific components we encounter) satisfy the stronger condition. The algebraic derivation itself remains unchanged once the hypothesis is corrected. revision: yes

  2. Referee: [§4–5] Application sections (power graphs of D_{2n} and Q_{4n}): the identification of these power graphs as joined unions of regular graphs is used to invoke the §3 formula, but the same uniformity-of-transmission gap reappears; explicit verification that each component is transmission-regular is missing.

    Authors: We will add an explicit verification step. For each family (power graphs of D_{2n}, Q_{4n}, Z_n, elementary abelian p-groups, and groups of order pq) we will insert a short lemma showing that the natural partition into joined-union components consists of transmission-regular graphs. The argument proceeds by direct computation of the reciprocal transmissions using the known distance formulas in these power graphs (which depend only on element orders and conjugacy) together with the symmetry of the underlying groups; all vertices inside a given component have identical transmission values. These lemmas will be placed immediately before the spectral formulas so that the invocation of the corrected main theorem is justified. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives explicit characteristic polynomials for RD_α on joined unions of regular graphs directly from the matrix definition RD_α = α RT_r + (1-α) RD and standard block-matrix eigenvalue techniques. No fitted parameters are renamed as predictions, no self-citations bear the central claim, and no uniqueness theorems or ansatzes are smuggled in. The derivations remain self-contained using only the given definitions and linear-algebra identities; the result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard facts from linear algebra (characteristic polynomials of block matrices) and graph theory (definitions of distance matrix, transmission, and joined union). No free parameters are introduced. No new entities are postulated.

axioms (1)
  • standard math The characteristic polynomial of a block matrix formed by the joined-union construction can be expressed in closed form using the spectra of the component graphs and the joining parameters.
    Invoked when deriving the explicit formulas for joined unions of regular graphs.

pith-pipeline@v0.9.0 · 5563 in / 1264 out tokens · 26134 ms · 2026-05-13T20:55:04.617437+00:00 · methodology

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Reference graph

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