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Several expressions for the $j$-th component $\left( x_{k}\right)_{j}$ of the$k$-th eigenvector $x_{k}$ of a symmetric matrix $A$ belonging to eigenvalue$\lambda_{k}$ and normalized as $x_{k}^{T}x_{k}=1$ are presented. Inparticular, the expression \[ \left(x_{k}\right)_{j}^{2}=-\frac{1}{c_{A}^{\prime}\left( \lambda_{k}\right)}\det\left( A_{\backslash\left\{ j\right\} }-\lambda_{k}I\right) \] where$c_{A}\left( \lambda\right) =\det\left( A-\lambda I\right) $ is thecharacteristic polynomial of $A$, $c_{A}^{\prime}\left( \lambda\right)=\frac{dc_{A}\left( \lambda\right) }{d\lambda}$ and $A_{\backslash\left\{j\right\} }$ is obtained from $A$ by removal of row $j$ and column $j$,suggests us to consider the square eigenvector component as a graph centralitymetric for node $j$ that reflects the impact of the removal of node $j$ fromthe graph at an eigenfrequency/eigenvalue $\lambda_{k}$ of a graph relatedmatrix (such as the adjacency or Laplacian matrix). Removal of nodes in a graphrelates to the robustness of a graph. The set of such nodal centrality metrics,the squared eigenvector components $\left( x_{k}\right)_{j}^{2}$ of theadjacency matrix over all eigenvalue $\lambda_{k}$ for each node $j$, is'ideal' in the sense of being complete, \emph{almost} uncorrelated andmathematically precisely defined and computable. Fundamental weights (columnsum of $X$) and dual fundamental weights (row sum of $X$) are introduced asspectral metrics that condense information embedded in the orthogonaleigenvector matrix $X$, with elements $X_{ij}=\left( x_{j}\right)_{i}$.  In addition to the criterion {\em If the algebraic connectivity is positive,then the graph is connected}, we found an alternative condition: {\em If$\min_{1\leq k\leq N}\left( \lambda_{k}^{2}(A)\right) =d_{\min}$, then thegraph is disconnected.}

Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks