Let $G$ be an simple graph of order $n$ whose adjacency eigenvalues are $\lambda_1\ge\dots\ge\lambda_n$. The HL--index of $G$ is defined to be $R(G)= \max\{ |\lambda_{ h}|, |\lambda_{ l}|\}$ with $h=\left\lfloor\frac{ n+1}{ 2}\right\rfloor$ and $ l=\left\lceil\frac{ n+1}{ 2}\right\rceil.$ 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 $K_{ 2,3}$.
Categories: math.CO