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Is there a procedure to find the eigenvalues of A? ‎

$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t & 0 & I &&&&&&&\\&&I & 0 & P &&&&&&&\\&&& P^t & 0 & \ddots &&&&&\\&&&& \ddots & \ddots & \ddots &&&&\\&&&&& \ddots & 0 & I &&&\\&&&&&& I & 0 & P &&\\&&&&&&& P^t & 0 & I &0_{k\times y-k}\\&&&&&&&& I &&\\0&&&&&&&& 0_{y-k\times k} && Y_{y\times y}\end{bmatrix}$$ (All rows and columns except the last are $k \times k$ blocks.)

where $A$ is adjacency matrix of cubic graphs and $X, P$ are circulant matrices of order k, Y is a matrix of order y, $k \not=y$ and $I$ is a identity matrix and $0$ is a zero matrix.

For example: How to find eigenvalues of following block matrices? enter image description here

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    $\begingroup$ X-posted: math.stackexchange.com/q/2823989/339790 $\endgroup$ – Rodrigo de Azevedo Jun 20 '18 at 7:56
  • $\begingroup$ You mean efficiently or analytically? What's the problem with the regular eig()? $\endgroup$ – percusse Jun 25 '18 at 16:15
  • $\begingroup$ I want eigenvalues or characteristic polynomial associated to a cubic (3-regular) graphs with adjacency matrix A. Since $y\not= k$, then A is not Jacobian three-diagonal matrix and I cannot find characteristic polynomial of A. $\endgroup$ – Maryam Hak Jun 25 '18 at 21:12

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