Distribution of deformed Laplacian limit points

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 $\lambda>1$ and we present a convergence criterion based on Shearer's approach. Our main result is that for any fixed value $\lambda_0>1$ there exists a unique value $0<s^* <\sqrt{\lambda_0} -1$ such that, and for any $s \in (0,s^*)$ the interval $[\lambda_0, \; +\infty)$ 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.