For a fixed graph $H$, what is the smallest number of colors $C$ such that there proper edge-coloring complete $K_n$ with containing no two vertex-disjoint color-isomor...

Neighborhood Lights Out is a game played on graphs. Begin with graph and vertex labeling of the from set $\{0,1,2,\dots, \ell-1\}$ for $\ell \in \mathbb{N}$. The by toggling vertices: when toggled, that each its neighbors has label increased $1$ (modulo $\ell$). won every 0. For any $n\in\mathbb{N}$ it clear one cannot win $K_n$ unless initial assigns all vertices same label. Given maximum numb...

Let $G$ and $H$ be graphs. The tensor product $Gotimes H$ of $G$ and $H$ has vertex set $V(Gotimes H)=V(G)times V(H)$ and edge set $E(Gotimes H)={(a,b)(c,d)| acin E(G):: and:: bdin E(H)}$. In this paper, some results on this product are obtained by which it is possible to compute the Wiener and Hyper Wiener indices of $K_n otimes G$.

We prove that the homogeneously polyanalytic functions of total order m, defined by system equations $$\overline{D}^{(k_1,\ldots ,k_n)} f=0$$ with $$k_1+\cdots +k_n=m$$ , can be written as polynomials degree $$<m$$ in variables $$\overline{z_1},\ldots ,\overline{z_n}$$ some analytic coefficients. establish a weighted mean value property for such functions, using reproducing Jacobi polynomials. ...

We study weak asymptotic behaviour of the Christoffel--Darboux kernel on main diagonal corresponding to multiple orthogonal polynomials. show that under some hypotheses limit $\tfrac{1}{n} K_n(x,x)\, d\mu$ is same as normalized zero counting measure type II also an extension Nevai's operators our context.

We consider intersections of $n$ diagonal forms degrees $k_1 \lt \cdots k_n$, and we prove an asymptotic formula for the number rational points bounded height on these varieties. The proof uses Hardy–Littlewood method recent breakthro

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, we write $\mathfrak{p}$ for either of the primes $K$ above $2$. $K_\infty$ unique $\mathbb{Z}_2$-extension unramified outside $\mathfrak{p}$. For certain quadratic biquadratic extensions $\mathfrak{F}/K$, prove simple exact formula $\lambda$-invariant Galois group maximal abelian 2-extension ...