Let $G$ be a undirected $d$-regular graph, that is, a graph whose all vertices have the same degree $d$. It is known that the eigenvalues $\sigma\_i$, $i=1,\cdots,n$, of the adjacency matrix are real and they satisfy
$$
d = \sigma\_1 \geq \sigma\_2\geq\cdots\geq\sigma\_n\geq-d.
$$
Moreover, if $G$ is a bipartite graph then $\sigma\_n = -d$.


My question is: Does the opposite holds, that is, if $\sigma\_n = -d$ then $G$ is bipartite? If not, what would be the counterexample?



Spectra of undirected $d$-regular graphs
No that is not true. Take the disjoint union of a $d$-regular graph and a non-bipartite $d$-regular graph. You obtain a graph that is not bipartite and clearly the least eigenvalue is $-d$. 


What is true however is that a graph is bipartite iff its spectrum is symmetric over 0.


