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

Let $t\ge 3$ 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 $$\mu \left(G\right)\le \frac{t-1}{2}+\sqrt{n+\frac{t^2-2t-3}{4}},$$ with equality if and only if $n\equiv 1$ $(\mathrm{mod}$ $t)$ and $G=K_1\vee \lfloor n\mathrm{/}t\rfloor K_t$. For $t=3$ the maximum $\mu \left(G\right)$ is found exactly for any $n>40000$.