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 $1$.

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 

and graph6 string

GQil^W

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 $1$.