I am trying to prove or disprove the following:
Let $G$ be a non-bipartite AND connected $k$-regular graph where the size of the second-largest eigenvalue of the adjacency matrix $A_G$ of $G$ is $\lambda$ (which as $G$ is connected is always strictly less than $k$). Then every eigenvalue of $A_G$, except for one, has absolute value no larger than $|\lambda|$.
Of course, the shaded above would not be true if $G$ is allowed to be bipartite; if $G$ is $k$-regular bipartite the eigenvalues of $A_G$ are symmetric around 0, so $-k$ is an eigenvalue of $A_G$ for any $k$-regular bipartite $G$. I am wondering if there is a $k$-regular connected graph $G$ that is not bipartite that has smallest eigenvalue in the interval $(-k,-|\lambda|]$, where $\lambda$ is as above the 2nd-largest eigenvalue of $A_G$.
I have been looking around the internet for a solution either way, meanwhile my matrix analysis is rusty and I know we have some really smart people here. Anyone have any insight into this?
Thanks!
The eigenvalues of the Petersen graph are $3$, $1$ and $-2$.