2025/06/06 by Krystal Guo, Gordon Royle, Guo, Krystal +1
Materials Science · Mathematics · #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2506.05861
openalex publication_date 2025/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a complete characterisation of the cubic graphs with no eigenvalues in the interval (-2,0). There is one thin infinite family consisting of a single graph on 6n vertices for each n \geqslant 2, and five ``sporadic'' graphs, namely the 3-prism K3 \mathbin\square K2, the complete bipartite graph K3,3, the Petersen graph, the dodecahedron and Tutte's 8-cage. The proof starts by observing that if a cubic graph has no eigenvalues in (-2,0) then its local structure around a girth-cycle is very constrained. Then a separate case analysis for each possible girth shows that these constraints can be satisfied only by the known examples. All but one of these case analyses can be completed by hand, but for girth five there are sufficiently many cases that it is necessary to use a computer for the analysis.