Let G = (V, E) be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph G is a function f: V → {0, 1, 2} such that each vertex v ∈ V with f(v) = 0 is adjacent to at least one vertex u ∈ V with f(u) = 2. A total Roman dominating function (TRDF) of G is a function f: V → {0, 1, 2} such that (i) it is a Roman dominating function, and (ii) the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, f(V ) = P u∈V f(u), called the total Roman domination number (γtR(G)). Similarly, a subset S ⊆ V is said to be a total dominating set (TDS) on the graph G if (i) S is a dominating set of G, and (ii) the induced subgraph G[S] does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general Graphs. In this paper, we propose a simple 10. 5-factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is O(|V | log k), where k is the number of vertices with weights 2. It is an improvement over the best-known 12-factor approximation algorithm with running time O(|V | log k) available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from 8 to 7. 79. The running time of the proposed algorithm is O(|V | + |E|).

This paper focuses on a particular type of starlike univalent function, namely $(1-sz)(1+z)$ with $-1/3\leq s\leq 1/3$. This function maps the open Unit Disk $\mathbb{D}$ onto a Limaçon domain denoted by \[ \left[(u-1)^2+v^2-s^2\right]^2<(1-s)^2\left[\left(u-1-s\right)^2+v^2\right].\]    The primary investigation concerns on analytic function families where $zh'/h$ maps the Unit Disk to a subset of this domain. The paper provides the lower and upper bounds of the real and imaginary parts of these functions and highlights some distinguishing characteristics of these bounds. Furthermore, the study employed subordination theory to derive conclusions and corollaries for functions belonging to the aforementioned class.

We provide some new sharp embeddings for p-Carleson measures and some related measures in the Unit Disk of the complex plane.

Let R be a ring (not necessarily commutative) with nonzero identity. We define 􀀀 (R) to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exist Unit elements u; v of R such that x+uyv is a Unit of R. In this paper, basic properties of 􀀀 (R) are studied. We investigate connectivity and the girth of 􀀀 (R), where R is a left Artinian ring. We also determine when the graph 􀀀 (R) is a cycle graph. We prove that if 􀀀 (R)  = 􀀀 (Mn(F)) then R  = Mn(F), where R is a ring and F is a finite field. We show that if R is a finite commutative semisimple ring and S is a commutative ring such that 􀀀 (R)  = 􀀀 (S), then R  = S. Finally, we obtain the spectrum of 􀀀 (R), where R is a finite commutative ring.

In the present paper, we introduce and investigate three interesting superclasses SD, SD* and KD of analytic, normalized and univalent functions in the open Unit Disk D. For functions belonging to these classes  SD, SD* and  KD, we derive several properties including (for example) the coefficient bounds and growth theorems. The various results presented here would generalize many well known results. We also get a new univalent criterion and some interesting properties for univalent function,starlike function,convex function and close-to-convex function. Many superclasses which are already studied by various researchers are obtained as special cases of our two new superclasses.

We provide new sharp results on the action of Toeplitz operators from Triebel and Besov spaces to new BMOA-type function spaces on the Unit Disk.  In this paper, we consider $s\geq1$ case in previous papers $s<1$ was covered for $BMOA_{s,q}$ and $BMOA_s^p$ spaces. We modify a little our previously known proofs.

In this paper we define Gevrey polyanalytic classes of order N on the Unit Disk D and we obtain for these classes a characteristic expansion into N􀀀 analytic polynomials on suitable neighborhoods of D. As an application of our main theorem, we perform for the Gevrey polyanalytic classes of order N on the Unit Disk D, an analogue to E. M. Dyn’ kin’ s theorem. We also derive, for these classes, their characteristic degree of the best uniform approximation on D by N􀀀 analytic polynomials.

We determine the forbidden induced subGraphs for the intersection of the classes of chordal bipartite Graphs and line Graphs of acyclic directed Graphs. This is a first step towards finding the forbidden induced subGraphs for the class of line Graphs of directed Graphs.