4 problems

Let $S$ be a set of $n$ points in the plane. Let $m_1(S)$ and $m_2(S)$ be the numbers of occurrences of the smallest and second smallest distances in $S$, respectively, and let…

Let $C_m$ be the cycle graph on $m$ vertices, $K_2$ and $C_3$ the indicated factors, and let $G$ be the Petersen graph. For nonnegative exponents, the displayed Cartesian products…

Let $P_{n_1}$ denote the path on $n_1$ vertices and $K_{n_2}$ the complete graph on $n_2$ vertices. For nonnegative integers $d_1,d_2$ and integers $n_1,n_2\geq 2$, write…

For $0<\alpha,\beta\leq\frac{1}{2}$, let $\underline{\Gamma}_{\mathrm{even,S}}(\alpha,\beta)$ and $\overline{\Gamma}_{\mathrm{even,S}}(\alpha,\beta)$ denote the lower and upper bou…