A note on median eigenvalues of subcubic graphs

Let $G$ be an simple graph of order $n$ whose adjacency eigenvalues are ${\lambda }_1\ge \cdots \ge {\lambda }_n$. The HL–index of $G$ is defined to be $R(G)=\max \{\mathrm{\mid }{\lambda }_h\mathrm{\mid },\mathrm{\mid }{\lambda }_l\mathrm{\mid }\}$ with $h=\lfloor \frac{n+1}{2}\rfloor$ and $l=\lceil \frac{n+1}{2}\rceil \mathrm{.}$ Mohar conjectured that $R(G)\le 1$ for every planar subcubic graph $G$. In this note, we prove that Mohar’s Conjecture holds for every $K_4$-minor-free subcubic graph. Note that a $K_4$-minor-free graph is also called a series–parallel graph. In addition, $R(G)\le 1$ for every subcubic graph $G$ which contains a subgraph $K_{2,3}$.