A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...

Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called e...

We make some sharp estimates to obtain a Schwarz lemma for the symmetrized polydisc $${\mathbb {G}}_n$$ , family of domains naturally associated with spectral interpolation, defined by $$\begin{aligned} {\mathbb {G}}_n :=\left\{ \left( \sum _{1\le i\le n} z_i,\sum i<j\le n}z_iz_j \ldots \prod _{i=1}^n z_i \right) : \,|z_i|<1, i=1,\ldots ,n \right\} . \end{aligned}$$ first few extended $$\wideti...

The aim of this paper is to study the matrix discrepancy problem. Assume that $$\xi _1,\ldots ,\xi _n$$ are independent scalar random variables with finite support and $${\textbf{u}}_1,\ldots ,{\textbf{u}}_n\in {\mathbb C}^d$$ . Let $${\mathcal C}_0$$ be minimal constant for which following holds: $$\begin{aligned} \textrm{Disc}({\textbf{u}}_1{\textbf{u}}_1^*,\ldots ,{\textbf{u}}_n{\textbf{u}}_...

Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $...

We consider products of a i.i.d. sequence in set $\{f_1,\ldots,f_m\}$ preserving orientation diffeomorphisms the circle. we can naturally associate Lyapunov exponent $\lambda$. Under few assumptions, it is known that $\lambda\leq 0$ and equality holds if only $f_1,\ldots,f_m$ are simultaneously conjugated to rotations. In this paper, state quantitative version fact case where $C^k$ perturbation...