let $mathfrak{f}$ be a formation and $g$ a finite group. a subgroup $h$ of $g$ is said to be weakly $mathfrak{f}_{s}$-quasinormal in $g$ if $g$ has an $s$-quasinormal subgroup $t$ such that $ht$ is $s$-quasinormal in $g$ and $(hcap t)h_{g}/h_{g}leq z_{mathfrak{f}}(g/h_{g})$, where $z_{mathfrak{f}}(g/h_{g})$ denotes the $mathfrak{f}$-hypercenter of $g/h_{g}$. in this paper, we study the structur...

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

The use o f retent ion indices in mice l la r electrokinet ic ch roma tog raphy ( M E K C ) is eva lua ted both f rom a theoretical and a practical point o f view. F undamen t a l equa t ions for the de te rmina t ion o f retent ion indices in M E K C are descr ibed, showing that re tent ion indices are independen t o f the surfac tant concentra t ion. Possibi l i t ies as well as l imita t ion...

We introduce the index i(v) = 1− χ(S−(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G = (V,E). Here S−(v) = {w ∈ S(v) | f(w) − f(v) < 0 } is the subgraph of the unit sphere S(v) generated by the vertices {w ∈ V | (v, w) ∈ E }. We prove that the sum ∑ v∈V i(v) of the indices always is equal to the Euler characteristic χ(G) of the graph G. This is...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...