if $G$ is bipartite and k-regular graph,prove that we can calculate the length of smallest even cycle from its spectrum of adjacency matrix.
because it is bipartite it doesn't have any odd cycle,also because it is bipartite and k-regular so we have $k$ and $-k$ as its eigenvalues,I don't know what should I do to come to an answer,any hint or guidance or any reference that you think it is useful for me to study will be great,thanks a lot.
If $G$ is bipartite and regular on $n$ vertices and has girth $k$ and $r<k$, the number of closed walks of length $r$ in $G$ is $n$ times the number of closed walks of length $r$ in the infinite $k$-regular tree that start at the root. So the first $r$ for which this equality fails is your girth.