In this Paper, we establish a new application of the Mittag-Lefier Function method that will enlarge the application to the non linear Riccati Differential equations with fractional order. This method provides results that converge promptly to the exact solution. The description of fractional derivatives is made in the Caputo sense. To emphasize the consistency of the approach, few illustrations are presented to support the outcomes. The outcomes declare that the procedure is very constructive and relavent for determining non linear Ricati differential equations of fractional order.

One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding problem. In this paper, we embed complete multipartite graphs into certain trees, such as $k$-rooted complete binary trees and $k$-rooted sibling trees.

In this paper we compute the $P_3$-forcing number of honeycomb network. A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ whose components are all paths of length 3, then $S$ is called a $P_3$-forcing set of $G$. A Ps-forcing set of $G$ of minimum cardinality is called the $P_3$-forcing number of G denoted by $Z{P}_3(G)$.

In this paper, we introduced a new concept called nonsplit monophonic set and its relative parameter nonsplit monophonic number $m_{ns}(G)$. Some certain properties of nonsplit monophonic sets are discussed. The nonsplit monophonic number of standard graphs are investigated. Some existence theorems on nonsplit monophonic number are established.

In graph theory and network analysis, centrality measures identify the most important vertices within a graph. In a connected graph, closeness centrality of a node is a measure of centrality, calculated as the reciprocal of the sum of the lengths of the shortest paths between the node and all other nodes in the graph. In this paper, we compute closeness centrality for a class of neural networks and the sibling trees, classified as a family of interconnection networks.

Let $G=(V,E)$ be a graph. A total dominating set of $G$ which intersects every minimum total dominating set in $G$ is called a transversal total dominating set. The minimum cardinality of a transversal total dominating set is called the transversal total domination number of G, denoted by ${\gamma }_{tt}(G)$. In this paper, we begin to study this parameter. We calculate ${\gamma }_{tt}(G)$ for some families of graphs. Further some bounds and relations with other domination parameters are obtained for ${\gamma }_{tt}(G)$.

Let $G$ be any graph. The concept of paired domination was introduced having gaurd backup concept in mind. We introduce pendant domination concept, for which at least one guard is assigned a backup, A dominating set $S$ in $G$ is called a pendant dominating set if $(S)$ contains at least one pendant vertex. The least cardinality of a pendant dominating set is called the pendant domination number of G denoted by ${\gamma }_{pe}(G)$. In this article, we initiate the study of this parameter. The exact value of ${\gamma }_{pe}(G)$ for some families of standard graphs are obtained and some bounds are estimated. We also study the proprties of the parameter and interrelation with other invarients.

Transportation Problem (TP) is the exceptional case to obtain the minimum cost. A new hypothesis is discussed for getting minimal cost in transportation problem in this paper and also Vogel’s Approximation Method (VAM) and MODI method are analyzed with the proposed method. This approach is examined with various numerical illustrations.

This paper mainly surveys the literature on bulk queueing models and its applications. Distributed Different systems in the zone of queuing speculation merging mass queuing architecture. These mass queueing models are often related to confirm the clog issues. Through this diagram, associate degree challenge has been created to envision the paintings accomplished on mass strains, showing various wonders and also the goal is to present enough facts to inspectors, directors and enterprise those that are dependent on the usage of queueing hypothesis to counsel blockage troubles and need to find the needs of enthusiasm of applicable models near the appliance.

Frank Harary and Allen J. Schwenk have given a formula for counting the number of non-isomorphic caterpillars on $n$ vertices with $n\ge 3$. Inspired by the formula of Frank Harary and Allen
J. Schwenk, in this paper, we give a formula for counting the number of non-isomorphic caterpillars with the same degree sequence.

The line graph $L(G)$ of a connected graph G, has vertex set identical with the set of edges of $G$, and two vertices of $L(G)$ are adjacent if and only if the corresponding edges are adjacent in $G$. Ivan Gutman et al examined the dependency of certain physio-chemical properties of alkanes in boiling point, molar volume, and molar refraction, heat of vapourization, critical temperature, critical pressure and surface tension on the Bertz indices of $L^{\prime }(G)$ Dobrynin and Melnikov conjectured that there exists no nontrivial tree $T$ and $i\ge 3$, such that $W(L^{\prime }(T))=W(T)$. In this paper we study Wiener and Zagreb indices for line graphs of Complete graph, Complete bipartite graph and wheel graph.

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. The power domination number of G is the minimum cardinality of a power dominating set of G. In this paper, we solve the power domination number for certain nanotori such as H-Naphtelanic, $C_5{C}_6{C}_7[m,n]$ nanotori and $C_4{C}_6{C}_8[m,n]$ nanotori.

Let $G_k,(k\ge 0)$ be the family of graphs that have exactly k cycles. For $0\le k\le 3$, we compute the Hadwiger number for graphs in $G_k$ and further deduce that the Hadwiger Conjecture is true for such families of graphs.

Split domination number of a graph is the cardinality of a minimum dominating set whose removal disconnects the graph. In this paper, we define a special family of Halin graphs and determine the split domination number of those graphs. We show that the construction yield non-isomorphic families of Halin graphs but with same split domination numbers.

A graph $G(v,E)$ with $n$ vertices is said to have modular multiplicative divisor bijection $f:V(G)\to 1,2,..,n$ and the induced function $f\ast :E(G)\to 0,1,2,\dots ,n–1$ where $f\ast (uv)=f(u)f(v)(modn)$ for all $uv\in E(G)$ such that $n$ divides the sum of all edge labels of $G$. This paper studies MMD labeling of an even arbitrary supersubdivision (EASS) of corona related graphs.

In this paper, the distance and degree based topological indices for double silicate chain graph are obtained.

In this paper, we introduce a new form of fuzzy number named as Icosikaitetragonal fuzzy number with its membership function. It includes some basic arithmetic operations like addition, subtraction, multiplication and scalar multiplication by means of $\alpha$-cut with numerical illustrations.

In this paper, we determine the wirelength of embedding complete bipartite graphs $K_{2^{n-1},2^{n-1}}$ into 1-rooted sibling tree $S{T}_n^1$, and Cartesian product of 1-rooted sibling trees and paths.

A dominator coloring is a proper vertex coloring of a graph $G$ such that each vertex is adjacent to all the vertices of at least one color class or either alone in its color class. The minimum cardinality among all dominator coloring of $G$ is a dominator chromatic number of $G$, denoted by $X_d(G)$. On removal of a vertex the dominator chromatic number may increase or decrease or remain unaltered. In this paper, we have characterized nontrivial trees for which dominator chromatic number is stable.

If every induced sub graph $H$ of a graph $G$ contains a minimal dominating set that intersects every maximal cliques of $H$, then $G$ is SSP (super strongly perfect). This paper presents a cyclic structure of some circulant graphs and later investigates their SSP properties, while also giving attention to find the SSP parameters like colourability, cardinality of minimal dominating set and number of maximal cliques of circulant graphs.

A set $S$ of vertices in a graph $G$ is said to be a dominating set if every vertex in $V(G)\mathrm{\S }$ 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 $V(G)\mathrm{\S }$ 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 $q\equiv 3(mod4)$ 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.