Let $G$ be a graph of order $n$ and average degree $d$. Let $A$ be the adjacency matrix of $G$ and let $x^{n}+c_{1} x^{n-1}+c_{2} x^{n-2}+\cdots+c_{n}$ be its characteristic polynomial. Show that $c_{1}=0$ and $c_{2}=-n d / 2$. Show also that $-c_{3}$ is twice the number of triangles in $G$.

The eigenvalues of $A$ are $\lambda_{1} \geqslant \lambda_{2} \geqslant \cdots \geqslant \lambda_{n}$. Prove that $\lambda_{1} \geqslant d$.

Evaluate $\lambda_{1}+\cdots+\lambda_{n}$. Show that $\lambda_{1}^{2}+\cdots+\lambda_{n}^{2}=n d$ and infer that $\lambda_{1} \leqslant \sqrt{d(n-1)}$. Does there exist, for each $n$, a graph $G$ with $d>0$ for which $\lambda_{2}=\cdots=\lambda_{n}$ ?