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arXiv · 2311.01884

A note on median eigenvalues of subcubic graphs

Yuzhenni Wang·Xiao-Dong Zhang
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Abstract

Let $G$ be an simple graph of order $n$ whose adjacency eigenvalues are $λ_1\ge\dots\geλ_n$. The HL--index of $G$ is defined to be $R(G)= \max\{|λ_{h}|, |λ_{l}|\}$ 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 $K_{2,3}$.

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Yuzhenni Wang, Xiao-Dong Zhang. 2023-11-03. A note on median eigenvalues of subcubic graphs. https://arxiv.org/abs/2311.01884

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