coefficient of smallest power of $x$ in the adjacency characteristic polynomial of $g$ is $1$

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This shows connected graphs on 7 vertices with nonsingular adjacency matrix.

for g in graphs.nauty_geng('7 -c'):
    if g.adjacency_matrix().determinant() == 0:
        t = g.adjacency_matrix().charpoly()
        g.show()
        print(t)

Using the above code, how we can obtain only those graphs (if any) satisfying the following property: coefficient of smallest power of $x$ in the adjacency characteristic polynomial of $g$ is $1$.