A note on median eigenvalues of subcubic graphs

Abstract

Let G be a simple graph of order n whose adjacency eigenvalues are λ1≥⋯≥λn. The HL-index of G is defined to be R(G)=max{|λh|,|λl|} with h=n+12 and l=n+12. Mohar conjectured that R(G)≤1 for every planar subcubic graph G. In this note, we prove that Mohar’s Conjecture holds for every K4-minor-free subcubic graph. In addition, R(G)≤1 for every subcubic graph G which contains a subgraph K2,3.