Minimum eigenvalue of a graph

Question

Let $G$ be a graph on vertices $\{1,2,\ldots ,n\}$. Suppose the vertex $1$ is connected to all other vertices. Let $\lambda$ be the least eigen value of the graph (i.e., its adjacency matrix).

Is it true that $\lambda\geq -1$?

Or is there any kind of lower bound of $\lambda$ for such graphs?

Any comment/reference would be very helpful.

Thank you.

Answer 1

Suppose e.g. the only edges are $(1,i)$ for $i = 2 \ldots n$. Then there is an eigenvalue of $-\sqrt{n-1}$ for eigenvector $\pmatrix{\sqrt{n-1}\cr -1\cr \ldots \cr -1}$