View the vertices as elements of $\mathbb{Z}^n$. If $a\in\mathbb{Z}^n$, define a function $f_a$ on the vertices by $$ f_a(x) = (-1)^{a^Tx}. $$ This function is an eigenvectors and if $a$ has weight $w$, the eigenvalue is $n-2w$. I can make this look more combinatorial by viewing vertices (and $a$) as subsets of $\{1,\ldots,n\}$ and noting that $f_a(x)$ is determined by the parity of $a \cap x$ (abusing notation).
The actual difficulty with this question is in deciding what you mean by a "graph theoretical proof". From where I write, linear algebra is a standard and fundamental tool in graph theory.