Suppose that the automorphism group of the finite undirected graph X\equal{}(P, E) is isomorphic to the quaternion group (of order 8). Prove that the adjacency matrix of X has an eigenvalue of multiplicity at least 4.
( P\equal{} \{ 1,2,\ldots, n \} is the set of vertices of the graph X. The set of edges E is a subset of the set of all unordered pairs of elements of P. The group of automorphisms of X consists of those permutations of P that map edges to edges. The adjacency matrix M\equal{}[m_{ij}] is the n×n matrix defined by m_{ij}\equal{}1 if {i,j}∈E and m_{i,j}\equal{}0 otherwise.)
L. Babai linear algebramatrixsuperior algebrasuperior algebra unsolved