Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

Recently, Mandal and Gong [23] refined and analyzed the recursive method by Lempel and Mykkeltveit et al. for generating de Bruijn sequences, where the recursive feedback function is the sum of a feedback function with k-th order composition and a sum of (k + 1) product-of-sum terms. In this paper we first determine the linear complexity of a composited de Bruijn sequence. We then conduct a pro...

Let G be a graph. Assume that l and k are two natural numbers. An l-sum flow on a graph G is an assignment of non-zero real numbers to the edges of G such that for every vertex v of G the sum of values of all edges incident with v equals l. An l-sum k-flow is an l-sum flow with values from the set {±1, . . . ,±(k − 1)}. Recently, it was proved that for every r, r > 3, r 6= 5, every r-regular gr...

A subset A of a given finite abelian group G is called (k, l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size λk,l(G) of a (k, l)-sum-free subset in G. A (2, 1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group Z...

Let G = (V,E) be a simple undirected graph. For a given set L ⊂ R, a function ω : E −→ L is called an L-flow. Given a vector γ ∈ R , we say that ω is a γ-L-flow if for each v ∈ V , the sum of the values on the edges incident to v is γ(v). If γ(v) = c, for all v ∈ V , then the γ-L-flow is called a c-sum L-flow. In this paper we study the existence of γ-L-flows for various choices of sets L of re...

In the paper, we introduce and study a massive class of continuous functions defined on interval $(0;1)$ using special encoding (representation) argument with an alphabet $ \mathbb{Z}=\{0,\pm 1, \pm 2,...\}$ base $\tau=\frac{\sqrt{5}-1}{2}$: $\displaystyle x=b_{\alpha_1}+\sum\limits_{k=2}^{m}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_m(...

Let $ \mu be a positive Borel measure on the interval [0, 1) $. The Hankel matrix {\mathcal H}_\mu = (\mu_{n+k})_{n, k\geq 0} with entries \mu_{n, k} \mu_{n+k} induces operator H}_\mu(f)(z) \sum\limits_{n 0}^\infty\left(\sum\limits_{k 0}^\infty\mu_{n,k}a_k\right)z^n space of all analytic functions f(z) \sum^\infty_{n 0}a_nz^n in unit disk {\mathbb{D}} In this paper, we characterize boundedness ...

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sum P₁ + ⋯+P k with P₁,…, P k be idempotent (k > 3) are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extension...

Let k be a positive integer and let G be a graph of order n ≥ k. It is proved that the sum of k largest eigenvalues of G is at most 1 2 ( √ k+1)n. This bound is shown to be best possible in the sense that for every k there exist graphs whose sum is 12 ( √ k + 12 )n− o(k−2/5)n. A generalization to arbitrary symmetric matrices is given.

If k is a positive integer we say that a set S of real numbers is k-sum-free if there do not exist x, y, x is S such that x + y = kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of (0, 1] of maximum size.