  The linear projection constant $\Pi(E)$ of a finite-dimensional real Banach
space $E$ is the smallest number $C\in [0,+\infty)$ such that $E$ is a
$C$-absolute retract in the category of real Banach spaces with bounded linear
maps. We denote by $\Pi_n$ the maximal linear projection constant amongst
$n$-dimensional Banach spaces. In this article, we prove that $\Pi_n$ may be
determined by computing eigenvalues of certain two-graphs. From this result we
obtain that the relative projection constants of codimension $n$ converge to
$1+\Pi_n$. Furthermore, using the classification of $K_4$-free two-graphs, we
give an alternative proof of $\Pi_2=\frac{4}{3}$. We also show by means of
elementary functional analysis that for each integer $n\geq 1$ there exists a
polyhedral $n$-dimensional Banach space $F_n$ such that $\Pi(F_n)=\Pi_n$.
