In 1982, B. Bollobas and Vega in the paper gave the configurational model to generate $r$-regular random graphs. They gave the following theorem (Theorem 1 in the paper). Theorem: Let $r\geq 3$ and $\epsilon> 0$ be fixed and define $d=d(n)$ as the least integer satisfying $$(r-1)^{d-1}\geq (2+\e...

Assume $G$ is a simple $k$-regular graph of order $n$ with adjacency matrix $A$ which is non-singular. Does anyone know some lower bounds for $\vert \det (A) \vert$ with respect to $n$, $k$ or both? Thanks in advance.

Let $p(x)$ be a positive measurable function on $(-1,1)$. Consider the Prandtl equation $$ u(x)-\frac{p(x)}\pi \int_{-1}^1 \frac{u'(t)}{t-x}dt=p(x)h_0(x),\quad u(1)=u(-1)=0.\quad\quad(\star) $$ What is state of the art of the existsence and regularity theory for $(\star)$, what is a standard refe...

Many years ago I found an inequality that directly controlled whether a point $\displaystyle c$ belongs or does not belong to the Mandelbrot set. Roughly, it was something like this: If $\displaystyle g(c) \leq 1/4$ then $\displaystyle c$ belongs to $\displaystyle M$ (ie to the Mandelbrot set), w...