let $g$ be a connected graph. the multiplicative zagreb eccentricity indices of $g$ are defined respectively as ${bf pi}_1^*(g)=prod_{vin v(g)}varepsilon_g^2(v)$ and ${bf pi}_2^*(g)=prod_{uvin e(g)}varepsilon_g(u)varepsilon_g(v)$, where $varepsilon_g(v)$ is the eccentricity of vertex $v$ in graph $g$ and $varepsilon_g^2(v)=(varepsilon_g(v))^2$. in this paper, we present some bounds of the multi...

In this paper we introduce a definition of gradation of continuity ingraded fuzzy topological spaces and study its various characteristic properties.The impact of the grade of continuity of mappings over the N-compactnessgrade is examined. Concept of gradation is also introduced in openness, closed-ness, homeomorphic properties of mappings and T2 separation axiom. Effectof the grades interrelat...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

Let X be a Banach space and T bounded linear operator acting in lp(ℤc,X), 1 ≤ p ∞. The is called locally nuclear if it can represented the form $${(Tx)_k} = \sum\limits_{m \in {\mathbb{Z}^c}} {{b_{km}}} {x_{k - m}},\quad k {\mathbb{Z}^c},$$ where bkm: → are nuclear, $${\left\| \right\|_{{\mathfrak{S}_1}}} \le {\beta _m},\quad k,m $$\left\|\cdot\right\|{_{{\mathfrak{S}_1}}}$$ norm, β ∈ l1(ℤc,ℂ) ...