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 Fs,t(n):=Ks−1∨(pKt+Kq), where 2≤s≤t, n−s+1=pt+q and 0≤q<t. Nikiforov in 2017 showed that the spectral radius ρ(G) satisfies ρ(G)≤ρ⁎(2,t) for any K2,t-minor free graph G of order n large enough, with equality if and only if G≅F2,t(n). Tait in 2019 extended Nikiforov's result as follows: for 2≤s≤t, if G is a Ks,t-minor free graph G of order n large enough, then ρ(G)≤ρ⁎(s,t), with equality if and only if G≅Fs,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 Ks,t-minor free graph of order n large enough, then ρ(G)≤ρ(Fs,t(n)), with equality if and only if G≅Fs,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.