Extremal spectral radius of K3,3/K2,4-minor free graphs

Abstract Let ρ ⁎ ( s , t ) be the largest real root of the quadratic equation: ( x − s + 2 ) ( x − t + 1 ) − ( n − s + 1 ) ( s − 1 ) = 0 and F s , t ( n ) : = K s − 1 ∨ ( p K t + K q ) , where 2 ≤ s ≤ t , n − s + 1 = p t + q and 0 ≤ q t . Nikiforov in 2017 showed that the spectral radius ρ ( G ) satisfies ρ ( G ) ≤ ρ ⁎ ( 2 , t ) for any K 2 , t -minor free graph G of order n large enough, with equality if and only if G ≅ F 2 , t ( n ) . Tait in 2019 extended Nikiforov's result as follows: for 2 ≤ s ≤ t , if G is a K s , t -minor free graph G of order n large enough, then ρ ( G ) ≤ ρ ⁎ ( s , t ) , with equality if and only if G ≅ F s , t ( n ) . Note that when t does not divide n − s + 1 , the equalities above are impossible. Tait proposed the following conjecture: If G is a K s , t -minor free graph of order n large enough, then ρ ( G ) ≤ ρ ( F s , t ( n ) ) , with equality if and only if G ≅ F s , t ( n ) . The previous results due to Nikiforov showed that the conjecture holds for s + t ≤ 5 . In this paper, we confirm the conjecture for s + t = 6 .