Discoveries / combinatorics

Boots-Royle/Cao-Vince Conjecture on Planar Spectral Radius

Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for every $n \ge 9$. Tait and Tobin proved it for sufficiently large $n$ in 2017; the conjecture now holds for all $n \ge 9$.