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 $1$.