Let G(V, E) be a graph with order n no component of 2. An edge k-labeling α: E(G) →{1,2,…,k} is called modular irregular G if the corresponding weight function wt_ α:V(G) → Z_n defined by α(x) =Ʃ_(xyϵE(G)) α(xy) bijective. The value wt_α(x) vertex x. Minimum k such that has irregularity strength G. In this paper, we define labeling on C_n⊙mK_1. Furthermore, determine C_n⊙mK_1.Keywords: corona p...

We consider the frame property of the Gabor system G(g, α, β) = {eg(t − αm) : m,n ∈ Z} for the case of rational oversampling, i.e. α, β ∈ Q. A ’rational’ analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, α, β) does not generate a frame if αβ = n−1 n . Special attention is paid to the first Hermite function h1(t) = te −πt .

This work aimed to develop a whole-cell biotransformation process for the production of α-ketoisocaproate from L-leucine. A recombinant Escherichia coli strain was constructed by expressing an L-amino acid deaminase from Proteus vulgaris. To enhance α-ketoisocaproate production, the reaction conditions were optimized as follows: whole-cell biocatalyst 0.8 g/L, leucine concentration 13.1 g/L, te...

Let G be a simple, undirected, and connected graph on n vertices with eigenvalues λ1 ≤ ... ≤ λn. Moreover, let m, δ, and α denote the size, the minimum degree, and the independence number of G, respectively. W.H. Haemers proved α ≤ −λ1λn δ2−λ1λnn and, if η is the largest Laplacian eigenvalue of G, then α ≤ η−δ η n is shown by C.D. Godsil and M.W. Newman. We prove α ≤ 2σ−2 σδ m for the largest n...

Probably the best-known among the remaining unsolved problems in graph theory is Hadwiger’s Conjecture: If χ(G) denotes the chromatic number of graph G, then G has the complete graph Kχ(G) as a minor. The following would immediately follow from the truth of Hadwiger’s Conjecture: Conjecture: If G has n vertices and if α(G) denotes the independence number of G, then G has Kdn/α(G)e as a minor. I...

A. (Forgive my choice of) Notation. Let G be a complex, connected, semisimple (or reductive) Lie group, T a maximal torus in G, and K the identity component of the fixed points of an involution in G. Equivalently, K is the complexification of a maximal compact subgroup of the identity component of a real form of G. Of course we assume T ⊂ K. We let ΛG denote the lattice of T -characters (the we...

For a family of graphs F , a graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F . F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph G = (V,∪i=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colo...

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that there exists k-vertex coloring in which any two vertices receiving color i are at distance least i+1. In this short note, we present upper and lower bound for lexicographic product G∘H graphs H. Both bounds coincide many cases. particular, happens if |V(H)|−α(H)≥diam(G)−1, where α(H) denotes independence

The stability number of the graph G, denoted by α(G), is the cardinality of a maximum stable set of G. In this paper we characterize the square-stable graphs, i.e., the graphs enjoying the property α(G) = α(G), where G is the graph with the same vertex set as in G, and an edge of G is joining two distinct vertices, whenever the distance between them in G is at most 2. We show that every square-...

where an, bn > 0 and ∑ n an = ∞. Then there exist a threshold α, such that f( ) = ∞ for < α, while f( ) < ∞ for > α. The so-called precise asymptotics problem consists in finding, under appropriate moment conditions, an elementary function g( ) > 0, > α, such that lim ↓α g( ) = 0 and lim ↓α g( )f( ) = l = 0,∞, i.e., in establishing that f( ) ∼ l/g( ) as ↓ α. For almost exhaustive references on ...