The spectral radius of graphs with no $K_{2,t}$ minor

Let $t\geq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $\mu\left( G\right)$ satisfies $$\mu\left( G\right) \leq\frac{t-1}{2}+\sqrt{n+\frac{t^{2}-2t-3}{4}},$$ with equality if and only if $n\equiv1$ $(\operatorname{mod}$ $t)$ and $G=K_{1}\vee\left\lfloor n/t\right\rfloor K_{t}$. For $t=3$ the maximum $\mu\left( G\right)$ is found exactly for any $n>40000$.