On (distance) Laplacian characteristic polynomials of power graphs

Abstract: The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order $ pqr $, where $ p,q $ and $ r $ are { primes,} are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

• The paper provides explicit forms for Laplacian and distance Laplacian characteristic polynomials of power graphs, proving the spectra are fully integral for groups of order pqr.
• It establishes rigorous eigenvalue inequalities and bounds that link algebraic connectivity with structural properties such as vertex connectivity and transmission.
• The study also examines proper power graphs for cyclic and dicyclic groups, presenting complete spectral characterizations with applications in spectral graph theory.

Laplacian and Distance Laplacian Characteristic Polynomials of Power Graphs

Introduction and Motivation

The paper "On (distance) Laplacian characteristic polynomials of power graphs" (2604.17607) systematically investigates the spectral properties of Laplacian and distance Laplacian matrices associated with power graphs for finite groups, especially those of order $pqr$ (with $p$, $q$, $r$ primes), as well as proper power graphs for cyclic and dicyclic groups. The analysis focuses on explicit computation of characteristic polynomials, spectral integrality, eigenvalue inequalities, and structural implications tied to group-theoretic provenance. Existing literature has addressed spectral analysis for select classes, but the present work provides comprehensive results for several families previously lacking full spectral characterization.

Definitions and Preliminaries

Let $G$ be a simple, finite, connected graph. The Laplacian matrix $L(G) = \text{Deg}(G) - A(G)$ encapsulates combinatorial connectivity, where $\text{Deg}(G)$ is the degree matrix and $A(G)$ is the adjacency matrix. The distance Laplacian matrix $D^L(G) = \text{Tr}(G) - D(G)$ utilizes the transmission matrix $\text{Tr}(G)$, defined by vertex distances, and the distance matrix $p$0 where $p$1 equals the shortest-path length between $p$2 and $p$3. The associated characteristic polynomials encode spectral invariants and are central to the analysis.

Power graphs, introduced for semigroups and subsequently for groups, are constructed such that two vertices (group elements) are adjacent if one is a power of the other. Proper power graphs exclude the group identity from the vertex set. For groups of composite order—especially those with $p$4 structure—the interplay between algebraic and combinatorial properties becomes highly nontrivial.

Explicit Characteristic Polynomials for Groups of Order $p$5

Main Results

The paper derives explicit forms for the distance Laplacian and Laplacian characteristic polynomials for several classes:

• For $p$6, the distance Laplacian characteristic polynomial is given explicitly by:

$p$7

All zeros are integers, establishing Laplacian and distance Laplacian integrality.

• For $p$8, the characteristic polynomial simplifies further:

$p$9

• Analogous complete forms are given for power graphs of semi-direct products, such as $q$0, and for other group constructs details relating to the joined union structures.
• Structural classification for groups of order $q$1 is leveraged to derive the respective spectral results, including factorization properties following module partitioning and the means formula for equitable quotient matrices.

Proper Power Graphs

The proper power graphs of cyclic groups $q$2 and dicyclic groups $q$3 are analyzed:

• For cyclic groups, the proper power graph is constructed as a join graph of $q$4 (where $q$5 is Euler's totient function) with additional complete graphs corresponding to proper divisors.
• For dicyclic groups, characteristic polynomials for their proper power graphs are given with full multiplicities and explicit integer zeros.

Strong Numerical Claims

The characteristic polynomials for these cases exhibit integral spectra, i.e., all roots are integer-valued. This is formally proven for groups of order $q$6, their semi-direct products, and certain extensions. Immediate consequences regarding spectral integrality are derived for both Laplacian and distance Laplacian spectra, exploiting Lemma results bounding diameters.

Eigenvalue Inequalities and Structural Observations

The work presents important inequalities and extremal values for Laplacian and distance Laplacian eigenvalues:

• For any power graph of a finite group $q$7, the largest Laplacian and distance Laplacian eigenvalues equal the group order ($q$8).
• Spectral decrease inequalities:

$q$9

with equality if and only if $r$0 is cyclic of prime power order.

• Several bounds connecting transmission, vertex connectivity, and algebraic connectivity are established, with values computed for classes such as cyclic, dihedral and dicyclic groups.

Conjectures and Further Analytical Implications

Two conjectures are advanced on spectral integrality for cyclic order:

1. The Laplacian spectrum is integral if and only if $r$1 is a prime power or a product of two distinct primes.
2. Analogous criterion holds for distance Laplacian spectrum.

The paper establishes strong theoretical connections between algebraic group structure and spectral graph invariants, underpinning the deep interplay between combinatorial and algebraic properties.

Additionally, explicit bounds are computed for algebraic connectivity and Estrada index derivatives. The analysis exemplifies how structural features such as vertex connectivity $r$2, clique partitioning, and independent sets translate to precise spectral properties.

Practical and Theoretical Implications

Spectral integrality and explicit understanding of Laplacian and distance Laplacian polynomials have direct applications in graph energy, spectral radius computation, and combinatorial optimization in algebraic and chemical graph theory. The results provide new tools for extremal graph characterization in group-theoretic families, enable deeper investigation of spectral invariants, and facilitate more robust analysis of connectivity and transmission properties.

The theoretical implications extend to spectral graph theory, group theory, and their intersection via the study of power graphs. These insights can prompt further exploration of spectral extremality, classification of graphs with integral spectra, and generalization to broader families of algebraic graphs (e.g., rings, semigroups).

Speculation on Future Developments

Future research may focus on characterizing extremal graphs in $r$3 with respect to spectral invariants such as energy, spectral radius, algebraic connectivity, or Estrada index. Further generalizations to other algebraic structures and deeper links with connectivity and structural topology are possible, accompanying automated symbolic computation of spectra for larger classes.

Conclusion

The paper provides comprehensive explicit results for Laplacian and distance Laplacian characteristic polynomials of power graphs for finite groups of order $r$4 and proper power graphs for cyclic and dicyclic groups. Strong claims of spectral integrality and precise eigenvalue inequalities are established, connecting group-theoretic properties with spectral graph invariants. The work enriches the analytic toolkit for power graph spectral theory and offers a foundation for further analytic, combinatorial, and computational investigations into algebraic graph spectra.