A set \(S\) of vertices in a graph \(G\) is said to be a dominating set if every vertex in \(V(G)\S\) is adjacent to some vertex in \(S\). A dominating set \(S\) is called a total dominating set if each vertex of \(V(G)\) is adjacent to some vertex in \(S\). Molecules arranging themselves into predictable patterns on silicon chips could lead to microprocessors with much smaller circuit elements. Mathematically, assembling in predictable patterns is equivalent to packing in graphs. In this pa-per, we determine the total domination number for certain nanotori using packing as a tool.

Among the varius coloring of graphs, the concept of equitable total coloring of graph \(G\) is the coloring of all its vertices and edges in which the number of elements in any two color classes differ by atmost one. The minimum number of colors required is called its equitable total chromatic number. In this paper, we obtained an equitable total chromatic number of middle graph of path, middle graph of cycle, total graph of path and total graph of cycle.

Making use of \(q\)-derivative operator, in this paper, we introduce new subclasses of the function class & of normalized analytic and bi-starlike functions defined in the open disk \(\mathbb{U}\). Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients \(|a_2|\) and \(|a_3|\). Moreover, we obtain Fekete-Szegö inequalities for the new function classes.

A set \(S\) of vertices in a graph \(G\) is called a dominating set of \(G\) if every vertex in \(V(G)\S\) is adjacent to some vertex in \(S\). A set S is said to be a power dominating set of \(G\) if every vertex in the system is monitored by the set \(S\) following a set of rules for power system monitoring. A zero forcing set of \(G\) is a subset of vertices B such that if the vertices in \(B\) are colored blue and the remaining vertices are colored white initially, repeated application of the color change rule can color all vertices of \(G\) blue. The power domination number and the zero forcing number of G are the minimum cardinality of a power dominating set and the minimum cardinality of a zero forcing set respectively of \(G\). In this paper, we obtain the power domination number, total power domination number, zero forcing number and total forcing number for m-rooted sibling trees, l-sibling trees and I-binary trees. We also solve power domination number for circular ladder, Möbius ladder, and extended cycle-of-ladder.

A proper vertex coloring of a graph where every node of the graph dominates all nodes of some color class is called the dominator coloring of the graph. The least number of colors used in the dominator coloring of a graph is called the dominator coloring number denoted by \(X_d(G)\). The dominator coloring number and domination number of central, middle, total and line graph of quadrilateral snake graph are derived and the relation between them are expressed in this paper.

A digraph G is finite and is denoted as \(G(V,E)\) with \(V\) as set of nodes and E as set of directed arcs which is exact. If \((u, v)\) is an arc in a digraph \(D\), we say vertex u dominates vertex v. A special digraph arises in round robin tournaments. Tournaments with a special quality \(Q(n, k)\) have been studied by Ananchuen and Caccetta. Graham and Spencer defined tournament with \(q\) vertices
where \(q \equiv 3 (mod 4)\) is a prime. It was named suitably as Paley digraphs in respect deceased Raymond Paley, he was the person used quadratic residues to construct Hadamard matrices more than 50 years earlier. This article is based on a special class of graph called Paley digraph which admits odd edge graceful, super edge graceful and strong edge graceful labeling.

Molecular graphs are models of molecules in which atoms are represented by vertices and chemical bonds by edges of a graph. Graph invariant numbers reflecting certain structural features of a molecule that are derived from its molecular graph are known as topological indices. A topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph. One of the most widely known topological descriptor is the Wiener index. The Weiner index \(w(G)\) of a graph G is defined as the half of the sum of the distances between every pair of vertices of \(G\). The construction and investigation of topological is one of the important directions in mathematical chemistry. The common neighborhood graph of G is denoted by con(\(G\)) has the same vertex set as G, and two vertices of con(\(G\)) are adjacent if they have a common neighbor in \(G\). In this paper we investigate the Wiener index of \(Y-tree,\, X-tree,\, con(Y-tree)\) and \(con(X-tree)\).

In the field of membrane computing. P system is a versatile model of computing, introduced by Paun [6], based on a combination of (i) the biological features of functioning of living cells and the members structure and (ii) the theoretical  concepts and results related to formal language theory. Among different Application areas of the model of P system, Ceterchi et al. [2] proposed an array-rewriting P system generating picture arrays based on the well-established notions in the area of array rewriting grammars and iso-array grammar have also been introduced. In this paper we consider structures in the two-dimensional plane called equi-triangular array composed of equilateral triangular array grammar and a corresponding P system, in the order to generate such structures. We Also examine the generative power of these new models of picture generation.