The linear projection constant $Π(E)$ 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 $Π_n$ the maximal linear projection constant amongst $n$-dimensional Banach spaces. In this article, we prove that $Π_n$ 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 $1+Π_n$. Furthermore, using the classification of $K_4$-free two-graphs, we give an alternative proof of $Π_2=\frac{4}{3}$. 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 $Π(F_n)=Π_n$.