Median eigenvalues of subcubic graphs

TL;DR

The paper resolves whether the median eigenvalues of connected subcubic graphs lie in $[-1,1]$, showing this holds for all graphs except the Heawood graph and strengthening the claim with a positive-fraction bound near the medians. The authors develop a framework built around Cauchy interlacing, tail estimates $t_G^+$ and $t_G^-$, and a maximum-cut strategy, augmented by tail reducers, cut preservers, and cut enhancers to propagate local reductions to the entire graph. This yields a strong, structural verification of the conjectures for chemical graphs and provides a mechanism to bound a positive fraction of eigenvalues around the median, with potential extensions to broader graph families. The results have implications for spectral graph theory and the Hückel model of molecular orbitals, linking graph structure to HOMO-LUMO energy considerations in chemistry.

Abstract

We show that the median eigenvalues of every connected graph of maximum degree at most three, except for the Heawood graph, are at most $1$ in absolute value, resolving open problems posed by Fowler and Pisanski, and by Mohar.

Figures (3)

• Figure 1: The Heawood graph.
• Figure 2: A Venn diagram for \ref{['lem:mod-cut-preserver']}, where white represents set $A$, and gray represents set $B$.
• Figure 3: Cut enhancers.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Remark
• Definition 2.1: Tails and their estimates
• Lemma 2.2
• proof
• Proposition 2.3
• Definition 2.4: Decrease tail estimates
• Remark
• Lemma 2.5: Lemma 3.1 of Mohar M16