# Consider the class of simple, connected unicyclic graphs on $n$ vertices (a graph on $n$ vertices is unicyclic, if it has $n$ edges).

Consider the class of simple, connected unicyclic graphs on $n$ vertices (a graph on $n$ vertices is unicyclic, if it has $n$ edges). Now from this collection, consider the collection $S$ of all those graph for which the corresponding adjacency matrices are singular (i.e, having determinant is zero). Now from this collection $S$, can we find a graph that satisfies the following property: if $\lambda$ is an non zero eigenvalue of the adjacency matrix iff $\dfrac{1}{\lambda}$ is also an eigenvalue of the adjacency matrix.

Please somebody help