On a relationship between the eigenvectors of weighted graphs and their subgraphs

We consider the problem of finding connections between eigen-vectors and subgraphs of a weighted undirected graph $G$.
Let $G$ have $n$ vertices labelled $1,\ldots,n$, $\lambda$ be an eigen-value of the graph $G$ of multiplicity $t\ge 1$, and let $X^{(i)}=(x_1^{(i)},\ldots,x_n^{(i)})$, $i=1,\ldots,t$, be linearly independent eigen-vectors corresponding to this eigen-value. We obtain formulas representing the components $x_j^{(i)}$ of the eigen-vectors $X^{(i)}$ in terms of some characteristics of special subgraphs of the graph $G$, $i=1,\ldots,t$, $j=1,\ldots,n$. An illustrative example is given.