I know graphs contain certain information and they are useful, there is no doubt in it but unfortunately your answer does not even address my question, hope you don't mind

I am thankful for the answer but not at all satisfied,the papers you linked are quite recent, whereas the process of associating graphs started way back, so it does not make much sense

@SamHopkins; i just quoted $\mathbb Z_n$ as an example, you can even consider any group like $D_n$, $S_n$ anything; why is getting a number theory question cheating , i am just trying to find the necessity of associating the graph to group

@LSpice; its already given in number $2$.

@LSpice; if you read my question carefully, whatever u said is already written in my question and that is what i asked, what are those open problems

@M.Winter; math.stackexchange.com/a/1769191/665065; check this

@vidyarthi; can you provide a more detailed answer if possibe

But my question is about the smallest eigenvalue

Is my question clear?

The result gives $0\le \theta_0\leq \lambda_1$ but I want to know how is $\lambda_0$ related to $\theta_0$