‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...

Let$mathbb{C}^n$ be the space of $n$ complex variables. Let$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on$mathbb{C}^n$. The Minkowski functional on complete Reinhardt$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept ofspirallike mapping of type $beta$ and order $alpha$ is defined.So, the concept of the strong and almost spirallike mappings o...

This paper is devoted to the multivariable $$H^\infty $$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $$(A_1,\ldots , A_d)$$ such family, $$A_k$$ R-sectorial R-type $$\omega _k\in (0,\pi )$$ $$k=1,\ldots ,d$$ and admits bounded (\Sigma _{\theta _1}\times \cdots \times \Sigma _d})$$ joint for some $$\theta (\omega _k...

Let $C$ a non-empty, bounded, closed and convex subset of Banach space $X$, denote by $\ell\_{\infty}(C)$ the $\ell\_{\infty}$-sum $C$. In present paper, using degree nondensifiability (DND), we introduce class $r$-$\Delta$-DND-contraction maps $f: \ell\_{\infty}(C)\rightarrow X$ prove that if $f(\ell\_{\infty}(C))\subset C$ then there is some $x^{}\in with $f(x^{},x^{},\ldots,x^{},\ldots)=x^{\...

‎Let $G$ be a finite group‎. ‎By $MT(G)=(m_1,cdots,m_k)$ we denote the type of‎ ‎conjugacy classes of maximal subgroups of $G$‎, ‎which implies that $G$ has exactly $k$ conjugacy classes of‎ ‎maximal subgroups and $m_1,ldots,m_k$ are the numbers of conjugates‎ ‎of maximal subgroups of $G$‎, ‎where $m_1leqcdotsleq m_k$‎. ‎In this paper‎, ‎we‎ ‎give some new characterizations of finite groups by ...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...

we characterize strictly diagonal type of embeddings offinitary symmetric groups in terms of cardinality and the characteristic. namely, we prove thefollowing.let $kappa$ be an infinite cardinal. if$g=underset{i=1}{stackrel{infty}bigcup} g_i,$ where $ g_i=fsym(kappa n_i)$,$(h=underset{i=1}{stackrel{infty}bigcup}h_i, $ where $ h_i=alt(kappa n_i) ), $ is a group of strictly diagonal type and$xi=(...