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

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. In particular, 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 the characteristic 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 centrality metric for node $j$ that reflects the impact of the removal of node $j$ from the graph at an eigenfrequency/eigenvalue $\lambda{k}$ of a graph related matrix (such as the adjacency or Laplacian matrix). Removal of nodes in a graph relates to the robustness of a graph. The set of such nodal centrality metrics, the squared eigenvector components $\left( x{k}\right){j}^{2}$ of the adjacency matrix over all eigenvalue $\lambda{k}$ for each node $j$, is 'ideal' in the sense of being complete, \emph{almost} uncorrelated and mathematically precisely defined and computable. Fundamental weights (column sum of $X$) and dual fundamental weights (row sum of $X$) are introduced as spectral metrics that condense information embedded in the orthogonal eigenvector 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 the graph is disconnected.}