Median eigenvalues of sparse subcubic graphs

For an n-vertex graph G, let ${\lambda }_1\left(G\right)\ge {\lambda }_2\left(G\right)\ge \dots \ge {\lambda }_n\left(G\right)$ be the sequence of eigenvalues of its adjacency matrix. The HL-index of an n-vertex graph G, denoted by $R\left(G\right)$, is defined as $R\left(G\right)=\max \left\{\right|{\lambda }_{\lfloor \frac{n+1}{2}\rfloor }\left(G\right)|,|{\lambda }_{\lceil \frac{n+1}{2}\rceil }\left(G\right)\left|\right\}$. Mohar proved in 2015 that every subcubic graph G satisfies that $R\left(G\right)\le \sqrt{2}$ and conjectured that $R\left(G\right)\le 1$ when restricted to planar graphs. Bipartite subcubic graphs and $K_4$-minor-free subcubic graphs have been verified to satisfy this conjecture. In this paper, we prove that two classes of sparse subcubic graphs G satisfy that $R\left(G\right)\le 1$: Subcubic graphs with maximum average degree smaller than $\frac{44}{17}$ and subcubic planar graphs of girth at least 8.