The crossing number of a graph G is the minimum number of pairwise intersections of edges among all drawings of G. In this paper, we study the crossing number of Kn,n − nK2, Kn × P2, Kn × P3 and Kn × C4.

In this paper, we prove that the weighted BMO space $${\rm{BM}}{{\rm{O}}^p}(\omega ) = \left\{ {f \in L_{{\rm{loc}}}^1:\mathop {\sup }\limits_Q \left\| {{\chi _Q}} \right\|_{{L^p}(\omega )}^{ - 1}{{\left\| {(f {f_Q}){\omega ^{ 1}}{\chi \right\|}_{{L^p}(\omega )}} < \infty } \right\}$$ is independent of scale p ∈ (0, ∞) in sense norm when ω A1. Moreover, can replace Lp(ω) by Lp,∞(ω). As an appli...

Pigou's problem has many applications in real life scenarios like traffic networks, graph theory, data transfer internet etc. The two player classical network an unique Nash equilibrium with the Price of Stability and Anarchy agreeing each other. situation changes for $k-$person $n$ being total number people. If we fix behaviour $(n-2)$ people assume that $k-$persons take path $P_2$ where $k<(n...

Abstract Let $$ (P_n)_{n\ge 0}$$ ( P n ) ? 0 be the sequence of Perrin numbers defined by ternary relation P_0=3 = 3 , P_1=0 1 P...

We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic Artin $N$-stacks $Y$. In our setting, a deformation quantisation consists curved $A_{\infty}$ the structure sheaf $\mathcal{O}_{Y}$, equipped with morphism to ring differential operators $\mathcal{L}$; for smooth Lagrangian subvarieties algebraic varieties, this simplifies deforming...

We propose a new formula for the nonlinear viscous numerical flux and extend direct discontinuous Galerkin method with interface correction (DDGIC) of Liu Yan (H. Liu, J. Yan, The (DDG) diffusion corrections, Communications in Computational Physics 8 (3) (2010) 541) to compressible Navier-Stokes equations. DDGIC framework is based on observation that can be represented as sum multiple individua...

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n})\in \mathbb{S}^{n-1}$ and $d\sigma$ denote the normalised Lebesgue measure on $\mathbb{S}^{n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined $\R$ consider multilinear operator given by $$T(f_{1},f_{2},\dots,f_{n})(x)=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-\sigma_j)d\sigma, ~x\in \R.$$ In this paper we obtain necessary sufficien...

Let \(G=(V,E)\) be a finite graph. We consider the existence of solutions to generalized Chern–Simons–Higgs equation $$\begin{aligned} \Delta u=-\lambda e^{g(u)}\left( e^{g(u)}-1\right) ^2+4\pi \sum \limits _{j=1}^{N}\delta _{p_j} \end{aligned}$$on G, where \(\lambda \) is positive constant; g(u) inverse function \(u=f(\upsilon )=1+\upsilon -e^{\upsilon }\) on \((-\infty , 0]\); N integer; \(p_...