Deduce the characteristic polynomial of a graph with $n$ vertices whose non-isolated vertices form a star with $d$ branches with $1 \leq d \leq n - 1$. Then determine the eigenvalues and eigenvectors of an adjacency matrix of this graph.
Deduce the characteristic polynomial of a graph with $n$ vertices whose non-isolated vertices form a star with $d$ branches with $1 \leq d \leq n - 1$.\\
Then determine the eigenvalues and eigenvectors of an adjacency matrix of this graph.