MathOverflow

Consider the following function
\begin{align*}Y(d)\triangleq\frac{d\bigg[2-\beta f(d^{2})+\sqrt{\{2-\beta f(d^{2})\}^2-f(d^{3})\{1-4\beta+\beta^2 f(d)\}}\bigg]^+}{(1-\beta) f(d^{3})},\end{align*}
where $\beta\in[0,1/2),~\gamma\in[0,1)$ and $d\in[0,1]$ and $f(x) = 4 - \gamma - (1 - \gamma)x$....

(I have asked the same question on Mathematics Stack Exchange. But, while I got some interesting comments, I did not get an answer to the question, which is about bibliography.)
I am looking for bibliography on knights and knaves on graphs.
More specifically, suppose we are given a graph $G=(V,E)$ with one agent in each vertex; all agents make the same statement: "my closed neighborhood contains exactly $k$ knights" (with $k$ being a parameter)....

Consider the following PDE:$$-\Delta u + \alpha u + \beta (x \cdot \nabla) u = 0.$$Is there any nonzero weak solution of this equation on $\Bbb R^n$, in $H^1$ or other function spaces, for some positive $\alpha < \beta$?

I would like to compute asymptotics of the the following integral:
$$\int_{-\infty}^{\infty}\frac{dx}{2\pi} \frac{e^{-2d\sqrt{-x^2+\alpha^2+i\epsilon}}}{(-x^2+\alpha^2+i\epsilon)^2}$$
where $d, \alpha, \epsilon > 0$ and $d \alpha >>1$ and $\epsilon << 1$ (I will keep to first order in $\epsilon$ throughout)....