The answer to this question could be trivial.
The Graph Let $G$ be graph formed of two $d$-regular connected components. That is, $G= H_1\cup H_2$, where $H_1$, and $H_2$ are $d$-regular and disjoint. Let $x\in H_1$ and $y\in H_2$. Let $G'= G+xy$, then $G'$ is connected graph. My question is:
Question: What is the largest positive eigenvalue of $G'$ ? Or $\lambda_1 (G') = ??$
-It is obvious that $\lambda_1 (G) = d$ with multiplicity 2 ( since it is formed of 2 $d$-regular connected components). But I have no idea how to estimate $\lambda_1$ when a single edge is added between $H_1$, and $H_2$.
-The interlacing theorem could help in commutating bound for the maximal eigenvalue.
Any idea will be useful!