All limit points of the largest roots of matching polynomials are determined

The largest matching root $\mu (G)$ of a graph $G$ is that of its matching polynomial. In this paper, all limit points of the largest matching roots of graphs are determined. More precisely, we identify the limit points of the largest matching roots of graphs less than ${\tau }^{\frac{1}{2}}+{\tau }^{-\frac{1}{2}}$. For any $\gamma \ge {\tau }^{\frac{1}{2}}+{\tau }^{-\frac{1}{2}}$ with $\tau =\frac{\sqrt{5}+1}{2}$, there exists a graph sequence $\{G_i\mathrm{\mid }i\in \mathbb{N}\}$ such that $\underset{i\to \mathrm{\infty }}{\lim }\mu (G_i)=\gamma$.