[Paper Review] Signless laplacian characteristic polynomials of regular graph transformations

Jianping Li, Bo Zhou|arXiv (Cornell University)|Mar 22, 2013

Graph theory and applications3 references3 citations

TL;DR

This paper derives explicit formulas for the signless Laplacian characteristic polynomials of all 16 $xyz$-transformations of an $r$-regular graph $G$ with $n$ vertices and $m$ edges, expressing them in terms of $n$, $m$, $r$, and the signless Laplacian spectrum of $G$. The key contribution is a systematic spectral characterization of these transformations using matrix block decomposition and eigenvalue analysis, enabling computation of the full signless Laplacian spectrum for all such derived graphs.

ABSTRACT

Let $G$ be a simple $r$-regular graph with $n$ vertices and $m$ vertices. We give the signless Laplacian characteristic polynomials of $xyz$-transformations $G^{xyz}$ of $G$ in terms of $n$, $m$, $r$ and the signless Laplacian spectrum of $G$.

Motivation & Objective

• To systematically characterize the signless Laplacian spectra of all $xyz$-transformations of $r$-regular graphs.
• To extend prior work on adjacency and Laplacian spectra to the signless Laplacian matrix.
• To provide closed-form expressions for the signless Laplacian characteristic polynomials of $G^{xyz}$ in terms of $n$, $m$, $r$, and the eigenvalues of $G$.
• To unify and generalize existing results on graph transformations under the signless Laplacian framework.

Proposed method

• Utilizes the vertex-edge incidence matrix $R(G)$ to relate $Q(G) = R(G)R(G)^T$ and $R(G)^TR(G) = A(G^l) + 2I_m$.
• Applies block matrix determinant identities (Lemma 2.2) to compute characteristic polynomials of transformed graphs.
• Employs eigenvalue decomposition techniques for matrices of the form $P(M, J)$, where $M$ is a graph matrix and $J$ is a all-ones matrix.
• Uses spectral symmetry and complementation properties (Lemma 2.3) to derive spectra for $G^{---}$ from $G^{+++}$.
• Applies Lemma 2.1 to compute eigenvalues of compound matrices involving $Q(G)$, $R^TR$, and $J$-matrices.
• Performs block matrix row operations to simplify determinants of large block matrices representing $G^{xyz}$.

Experimental results

Research questions

• RQ1 How can the signless Laplacian characteristic polynomial of any $xyz$-transformation of a regular graph be expressed in terms of the original graph’s spectrum and parameters $n$, $m$, $r$?
• RQ2 What is the spectral relationship between $G^{xyz}$ and the original graph $G$ under the signless Laplacian matrix?
• RQ3 How do transformations involving complementation ($z = -$) or completeness ($z = 1$) affect the signless Laplacian spectrum?
• RQ4 Can the spectrum of $G^{1--}$, $G^{+--}$, and $G^{---}$ be derived uniformly using spectral identities and matrix decomposition?
• RQ5 What is the role of the all-ones matrix and matrix block structures in simplifying the characteristic polynomial of complex graph transformations?

Key findings

• The signless Laplacian characteristic polynomial of $G^{1--}$ is given by $f(\lambda, G^{1--}) = (2-n)^{-n} (\lambda - n - m + 2r)^{m-n} |B|$, where $B$ is a matrix whose eigenvalues are explicitly computed.
• For $G^{+--}$, the characteristic polynomial is $f(\lambda, G^{+--}) = [(λ - m - r)(λ - n - 2m + 4r) + (4-n)m - 2r] (\lambda - n - m + 2r)^{m-n} \prod_{i=1}^{n-1} [(λ - n - m + 2r + q_i)(λ + r - m - q_i) - q_i]$, with $q_i$ being the signless Laplacian eigenvalues of $G$.
• The spectrum of $G^{---}$ is derived via complementation: $f(\lambda, G^{---}) = (\lambda - 2n - 2m + 4r + 2)(\lambda + 3r - n - m)(\lambda + 2r - n - m)^{m-n} \prod_{i=1}^{n-1} [(λ - n - m + r + q_i + 2)(λ + 2r - n - m + q_i) - q_i]$, showing a symmetric spectral shift.
• The method successfully generalizes to all 16 $xyz$-transformations with $x,y,z \in \{0,1,+, -\}$, providing a complete spectral characterization for regular graphs.
• The derived formulas are valid for all $r$-regular graphs and depend only on $n$, $m$, $r$, and the eigenvalues $q_1, \dots, q_n$ of $G$, enabling efficient spectral computation.
• The use of block matrix identities and eigenvalue perturbation techniques allows exact derivation of the characteristic polynomials without requiring full matrix computation.