Let $G$ be a planar graph with $n$ vertices. Show that $\lambda_1(G) \leq −3 \lambda_n(G)$.
Let $\chi(G)$ denote the chromatic number of the graph $G$. Let $\lambda_1(G) , \lambda_n(G)$ denote the largest and smallest eigenvalue of $G$.
Then we have a result in hand: Let $G$ be a graph with $n$ vertices and with at least one edge. Then $$\chi(G) \geq 1-\frac{\lambda_1(G)}{\lambda_n(G)}$$
If $\chi(G) = 4$, then we have $\lambda_1(G) \leq −3 \lambda_n(G)$.
But is it always true that chromatic number of a planer graph is $4$?