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For an undirected graph, since the adjacency matrix will be symmetric, can we draw any relations between the principal eigenvector and the degree of nodes in the graph. Also can we do the same with the matrix $A{D^-}^1$, where $A$ is the adjacency matrix, and $D$ is a diagonal matrix with each diagonal entry being degree of each vertex

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Theres a whole field known as "Spectral graph theory", where the eigendecomposition of of certain matrices (typically the Laplacian matrix, $L=D-A$ where $D$ is the diagonal matrix with the sum of degrees of each vertex) is studied. This is a lot more interesting than looking at the spectrum of the adjacency matrix for many reasons (for example, you can make connections to harmonic function theory).

One can prove that the largest eigenvalue of the Adjacency matrix lies between $max(d_{avg},\sqrt{d_{max}})$ and $d_{max}$ (see these notes).

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