Distribution of deformed Laplacian limit points

TL;DR

This work investigates the limit points of the deformed Laplacian spectrum $\rho(M_G(s))$ across a one-parameter family $M_G(s)=I - sA + s^2(D-I)$. It first shows that for a simple family of trees, every $\lambda\ge1$ is an $s$-deformed limit point for some $s$, establishing a baseline distribution of limit points. It then leverages Shearer’s caterpillar sequences to prove a stronger, parametric result: for each fixed $\lambda>1$ there is a unique $0<s^*(\lambda)<1$ such that, for all $s$ in $(0,s^*)$, the interval $[\lambda,\infty)$ consists entirely of $s$-deformed limit points. The paper also provides a practical approximation framework and numerical data, including an efficient Diagonalize-based algorithm to compute limiting radii even for large, sparse graphs, highlighting the potential for a unified spectral theory bridging Laplacian and signless Laplacian regimes.

Abstract

This paper investigates limit points of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a quadractic one-parameter family of matrices. First, we show that any value greater or equal to 1 is a deformed Laplacian limit point (for different values of the parameter $s$) using a simple family of trees. Second, we define $(T_k)_{k \in \mathbb{N}}$ the Shearer's sequence of caterpillars for $λ>1$ and we present a convergence criterion based on Shearer's approach. Our main result is that for any fixed value $λ_0>1$ there exists a unique value $0<s^* <\sqrt{λ_0} -1$ such that, and for any $s \in (0,s^*)$ the interval $[λ_0, \; +\infty)$ is entirely formed by $s$-deformed Laplacian limit points (for the same value of $s$). Finally, we provide some numerical data exploring the limit properties.

Figures (11)

• Figure 1: Starlike tree $T_{1,n,n}$.
• Figure 2: Algorithm $\hbox{Diagonalize}(M, x)$.
• Figure 3: Initialization (left) and output (right) of $\hbox{Diag}(M_{T_{1,n,n}}(s), -\lambda_n)$ for $T_{1,n,n}$.
• Figure 4: Rational function $\varphi(t)=-3 -\frac{2}{t}, \; t \neq 0$.
• Figure 5: Graph of $f$ (left) and $g$(right).

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Theorem 2
• Example 1
• Definition 4
• Theorem 3
• Definition 5
• Lemma 1: DOT2025