Let $G = (V,E)$ be a simple graph with adjacency matrix $A$. It is well known that the largest eigenvalue $\lambda_{1}$ of $A$ is contained within the interval $[d, D]$ where $d$ is the average degree of $G$ and $D$ is the maximum degree of $G$. I'm aware of the result that $\lambda_{1} = D$ iff $G$ is $D$-regular.
I'm wondering about the tightness of the lower bound $\lambda_{1} \geq d$. In particular, how large can $C > 1$ be such that $\lambda_{1} \geq Cd$. For example, we known that $C \leq D/d$, but do we know of any better bounds than this. Can we construct examples of graphs where $C = 2$ for example?
