Yes, the graph with adjacency matrix

     0     0     1     0     1     1     1     1
     0     0     0     1     0     0     0     1
     1     0     0     0     1     1     1     1
     0     1     0     0     0     0     0     0
     1     0     1     0     0     1     1     1
     1     0     1     0     1     0     1     1
     1     0     1     0     1     1     0     0
     1     1     1     0     1     1     0     0 

is such a graph, as the characteristic polynomial is $x(x+2)(x+1)^4(x^2-6x+6)$, and the two roots of $(x^2-6x+6)$ are both larger than $1$.