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Radio k - Coloring of graph G is one of the divergence of frequency assignment problem (FAP). A simply connected graph G with diam(G) and a positive integer k, 1 ≤ k ≤ diam(G), a radio k - coloring of G is an allocation φ of positive integers to the nodes of G such that ϕ (u) − ϕ (v) ≥ 1 + k − d (u , v) , where u and v are any two different nodes of G and d(u, v) is the interval betwixt u and v. The supreme color granted by φ is called span, denoted by rck (φ). The radio k - chromatic number rck (G) of G is the minimal {rck (φ) : φ is a radio k - coloring of G}. In this paper,we attain the exact value of the radio k - chromatic number of bistar graph Bn,m and an operations on middle graph, central graph, total graph, line graph and jump graph of bistar graph denoted by M(Bn,m),C(Bn,m), T (Bn,m), L(Bn,m) and J(Bn,m) respectively. [ABSTRACT FROM AUTHOR]

A radio k-coloring is a proper node coloring of a graph G is a function f : V → (G) { 1, 2, ..., n} and for an integer 1 ≤ k≤diam(G) gratifying the order f (x) − f (y) ≥ k + 1 − d (x , y) for any two diverse nodes x and y, where d(x, y) is the length between x and y in G. The largest color consigned by f to some node of G is called span of f, denoted by rck (f). The radio k-chromatic number is the least of spans of all possible radio k-coloring of G, denoted by rck (G). In this paper we explore the radio k-chromatic number for the Star graph, Double star graph and its Middle graph, Central graph, Total graph, Line graph are also determined. [ABSTRACT FROM AUTHOR]

Consider the graph S = (V, E), which is undirected and loopless. A function g : V → {1, 2, 3, ....., n} is said to be a proper vertex coloring if d (u) ≠ d (v) ∀ edge uv ∈ E (S). A graph's chromatic number is the smallest integer χ(S) for which a proper vertex coloring with χ(S) colors exists. The b-chromatic number of a graph S shows proper p coloring, in which each color class has at least a vertex next to a vertex in each of the other color classes. V(S) is partitioned into sets for each suitable vertex coloring Vj = {u ∈ V (S)/d(u) = j}, for every j ∈ {1, 2, 3, ....., b,} known as color classes of d. In this paper, we investigate the b-chromatic number of line, central, middle and total graphs of Crown graph and Armed Crown graph denoted by L[Crn], L[ACrn], M[Crn], M[ACrn], C[Crn], C[ACrn], T [Crn], T [ACrn] respectively. [ABSTRACT FROM AUTHOR]

Various distance and distance-degree-based topological indices were already computed in various families of graphs. In this paper, we consider multiplicative circulant networks of order 3h where h is a positive integer and compute the exact values of the following distance based and distance-degree based topological indices: Wiener index, hyper-Wiener index, Schultz molecular topological index, Harary index, additively weighted Harary index, and multiplicatively weighted Harary index on these networks. [ABSTRACT FROM AUTHOR]

In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the results of the paper we get the known properties of the Bell numbers (Touchard's Congruence for k = 0). [ABSTRACT FROM AUTHOR]

A graph G with v vertices and e edges has a vertex magic total labeling (VMTL) if there is a 1-1 map taking vertices and edges onto the integers 1,2,3, . . .e + v such that for every vertex x, λ (x) + ∑ λ (xy) = k for some constant k. In this paper, we study some of the basic properties of vertex magic total labeling and give some examples for the graphs which admits VMTL. In particular, we discuss about the existence of VMTL for complete and corona graphs. [ABSTRACT FROM AUTHOR]

As the industry of construction, architecture and engineering adopts the digital era, value-added sensor network monitoring, simulation, engineering system optimization, and semantic modelling impact the design, construction, and operation of constructed assets. Such information-intensive models not only increase the value chain's effectiveness but further reduce the effects on the life-cycle of buildings. The industry as a whole has a number of problems that stem from the use of outdated methods, such as pollution and wasteful energy consumption. The aim of this research is to evaluate the significance of automation in the building trade by utilising several cutting-edge technologies. The paper examines the various applications of BIM in construction industry and identifies its limitations and prerequisites, laying the groundwork for ideas like the Construction Digital Twin, automation in the construction sector, and the integration of AI with IoT. Then, an outline of this notion is provided in terms of the underlying issues of BIM, identification of the potential to integrate automation in construction industry. [ABSTRACT FROM AUTHOR]

Let H and G be finite graphs where every edge of G belongs to at least one subgraph of G that is isomorphic to H. An (a,d) – H – antimagic total labeling of a graph G is a bijection f: V(G) ∪ E(G) → {1,2,..., │V(G)│+│E(G)│} such that for all subgraphs H' isomorphic to H, the H' – weights, F(H')= Σv∈V(H')f(v) + Σe∈E(H')f(e), form an arithmetic progression {a, a + d,..., a + (k − 1)d}, where a is a positive integer, d is a nonnegative integer, and k is the number of subgraphs of G isomorphic to H. If the vertex set V(G) receives the minimum possible labels {1,2,..., │V(G)│}, then f is called a super (a,d) –H –antimagic total labeling. In this paper we study super (a,d) – C4 –antimagic total labeling of graph P4 × Pn. We find that the graph P4 × Pn has a super (34n + 4, 2) – C4 – antimagic total labeling when a = 34n + 4 and d = 2 or when a = 32n + 4 and d = 4. [ABSTRACT FROM AUTHOR]

Let G be a finite and simple graph. The distance of vertices u and v in G is a shortest u - v path in G, and it is denoted by d(u, v). A rainbow u - v geodesic is a u - v path of length d(u, v) and all its edges have different colors. If G contains a rainbow u - v geodesic for every two vertices u and v of G, then G is called strongly rainbow-connected. The minimum number of colors needed to make a graph G strongly rainbow-connected is called strong rainbow connection number of G and is denoted by src(G). Let m, n, and t be positive integers, m ≥ 3, n ≥ 2, and t ≥ 2m. A prism graph having t pendants, (Cm × Pn)t, is a graph obtained from a prism graph Cm × Pn by attaching t pendant edges, at least one pendant edge to each vertex of degree 3. In this paper we study src((Cm × P3)t) and find that src((Cm × P3)t) = t. [ABSTRACT FROM AUTHOR]

Given a group of matrices, integers modulo prime. This group has many properties in group theory. In this paper, we will investigate the graph representation of this group using prime numbers, i.e., p = 2 and p = 3. We start in the case of simple matrices with orders 2x2 and 3x3. We show that the directed graph can be described the matrices in this group. The loop occurred in some graphs of the matrix. The weight of the edged should be defined in some graph representation on some matrices. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n, with no component of order two. Define an edge l-labeling θ:E(G)→{1,2,...,1}. Let xϵV(G), the weight of x is the sum of the l-labels of all its incident edges, denoted by α(x)=Σ░⟦θ(xy)⟧. The edge l-labeling is said modular irregular l-labeling of G if there exists a bijective weight α-function from V(G) to the group of integers modulo n. The smallest positive integer l such that G has a modular irregular l-labeling is said the modular irregularity strength of G, denoted by ms(G). Write ms(G)=∞, if G has no modular irregular strength. In this paper, we find ms of some graph classes, i.e. sunlet graphs, cycle barbell graphs, and m-Harary cycle graphs. [ABSTRACT FROM AUTHOR]

A graph G with m points is said to admit prime labeling if its points can be labeled with distinct positive integers not exceeding m such that the greatest common divisor of labels of each pair of adjacent point is one. A graph G which admits prime labeling is called a prime graph. In this paper we prove that some classes of graphs like ℋ-super subdivision of graphs related to cycle HSS(Cm)@K2, HSS(Cm)@2K2, HSS(Cm)@(K3∪K2), HSS(Cm)@(K3∪2K2), HSS(Cm ⊙ K1) are prime graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with an edge set E(G). If G1, G2, G3, ... , Gr are connected edge disjoint subgraphs of G with E(G) = E(G1) ∪ E(G2) ∪ E(G3) ∪ ... ∪ E(Gr), then G1, G2, G3, ... , Gr is a decomposition of G. An (a, d) − Continuous Monotonic Decompositions, or (a, d) − CMD, of G is a decomposition of G into r subgraphs G1, G2, G3, ... , Gr such that every Gi is connected and |E(Gi)| = a + (i − 1)d, for every i = 1, 2, 3,.. , r. Many authors have studied decomposition, including (a, d) − CMD, of graphs. In this paper we study (a, d) − CMD of some other class of graphs. Let n and m be positive integers, n ≥ 3. The corona product of a cycle Cn and an empty graph K m ¯ , denoted C n ⊙ K m ¯ , is a graph formed from Cn and n copies of K m ¯ by joining each ith vertex of Cn, with an edge, to every vertex of the ith copy of K m ¯. A caterpillar is a tree in which the removal of all its end vertices results a path. In this paper we find an (a, d) − CMD of C n ⊙ K m ¯ into caterpillars. [ABSTRACT FROM AUTHOR]

Modeling is an important area of software development. The most prevalent approach to modeling is the graphical approach, which means that models are edited through a graphical interface. In the case of highly complex and large models, however, editing a model through text can be more efficient than the graphical approach. In this paper, a method is presented that can be used to compare and merge models based on their textual representations. This approach helps optimizing teamwork due to its possible application in version control systems for textual representations. Our method is universal, thus it can be applied to any modeling environment and formal language. Differences between the presented method and other, already existing comparing and merging approaches are also described. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Grotzsch graph Gz has admits Vertex Prime Labeling. When the graph has Vertex Prime Labeling that graph is called Vertex Prime graph. A Graph G = (V, E) is said to have a Vertex Prime Labeling if its edges can be labeled with distinct integers from {1,2,3,⋯, |E|} such that for each vertex of degree at least two, the Greatest Common Divisor of the labels on its incident edges is 1. We also prove that Vertex Prime Labeling in the context of the operation of duplication. [ABSTRACT FROM AUTHOR]

Let [n] denote the set {1, 2, ..., n}, τ be a permutation of [4] and π=B1|B2| ... |Bk be a partition of [n], in the standard sense, where the blocks be arranged such that the first entries from each block be in increasing order, and entries in each block be also in increasing order. A flattened partition f of [n] is the permutation of [n] obtained by erasing the symbol which separates each block in π; f avoids the pattern τ, or f is τ-avoiding, if there is no subsequence of f which is order-isomorphic to τ. A run in f is a subsequence of the form fifi+1 ... fi+p with fii+1<...i+p, where fii-1 and fi+p>fi+p+1; fi is called the starting point of the run. Pattern avoidance in flattened partitions is an open and active area of research, and so far only few related works have been published. In this paper, we give a formula which counts the number of flattened partitions of [n] avoiding the pattern 1234 and 1243, for n≥1, by considering the number of runs in it and using a simple yet powerful principle, namely the pigeonhole principle. One of our results is related to one of the sequences in Online Encyclopedia of Integers Sequences. [ABSTRACT FROM AUTHOR]

This paper deals with the maximization of the synergy within a carrier's transportation network. This key concept of synergy is particularly important when participating in combinatorial auctions for the procurement in full-truckload transportation service. Our ideas derive from the advances achieved in the field of graph theory and are based on the technique of minimizing the distance between the booked and auctioneed lanes. We develop two optimization formulations that mainly differ in the objective function and that have been described by using an illustrative example that has taken from the literature and suitably adapted for the purpose of our application. [ABSTRACT FROM AUTHOR]

Incompletely defined Boolean functions are widely used in logic design because such forms are convenient and flexible in many practical applications. The problem of how to find efficient solution for incompletely defined Boolean functions as well systems of such functions in order to reduce the number of logic gates used for their implementation still attracts the attention of the researchers in the field of digital logic synthesis. In this paper the results of a comparative research of the efficiency of the previously proposed methodology and algorithmic procedure for special post-determination of incompletely defined logical functions using numeric tables, is presented. Verification of the methodology and algorithmic procedure is presented based on comparative analysis of 50 randomly generated systems of incompletely defined logical functions. The results show that in 68 percentages of them after applying the algorithmic procedure better circuit realization is obtained. The efficiency in realization for the systems of re-defined logical functions varies from 2 to 38 percentages. [ABSTRACT FROM AUTHOR]

The corona virus disease (COVID 19) was first identified and diagnosed in Wuhan, china in December 2019. It is tremendously communicable which noxiously transmitted to 212 countries and territories throughout the world. It stumbled on India through a student who was all the way from Wuhan on January 30, 2020. By 3rd may 2020 the virus strike more than 37000 people and is currently developing rapidly. It is significant to concenter the number of infections in the whole country. Since our country is consists thickly populated states it's quite important to analyze the data on the number of infected people across the country. The objective of this paper is to scheme the data of infected people state wise which would help the state government to keep track of the spread and monitor the requirement of health resources accordingly. It would not be appropriate and adequate to analyze with only one model. The analyzation has to be veracious by raising it through other models. Here three growth models namely the logistic, exponential and the susceptible models are established. The data driven model interpreting the results of exponential and logical model exhibits the maximum Daily Infection Rate (DIR) as the outcome. The above mentioned outcome is used to analyze the success of the lockdown. State wise DIR values are interpreted with the results of all the models to categorize the state as less, moderate or highly infected area. It is endowed that Maharashtra, Delhi, Gujarat, Madhya Pradesh, Andhra Pradesh, Uttar Pradesh, and west Bengal or in the highly affected category. Tamilnadu, Rajasthan, Punjab and Bihar are in the moderately affected category. Moreover Kerala, Haryana, Jammu & Kashmir, Karnataka and Telangana are in the controlled category. By establish the various models and interpreting their outcome we tabulate the predicted numbers in state wise manner. A Web application is also constructed to anticipate and fore take the recent updates regularly. [ABSTRACT FROM AUTHOR]

This paper is a study of rings and ideal structures in the multiset environment. What are the conditions to be imposed on a multiset derived from a ring to become a multiset ring and multiset ideal have been mentioned. We delved into properties, examples and inter relationships of different types of multiset ideals. Characterization of multiset prime ideals derived from the ting of integers Z, and Multiset Factor rings are explained. [ABSTRACT FROM AUTHOR]

Let G =(V, E) be finite and simple graphs with vertex set V(G) and edge set E(G). A graph G is called super edge-magic if there exists a bijection f: V(G) ∪ E(G) → {1, 2, ⋯, |V(G)| + |E(G)|} and f(V(G)) = {1, 2, ⋯, |V(G)|} such that f(x) + f(xy) + f(y) is a constant for every edge xy ∈ E(G). A graph G with isolated vertices is called pseudo super edge-magic if there exists a bijection f: V(G) → {1, 2, ⋯, |V(G)|} such that the set {f(x) + f(y) ∶ xy ∈ E(G)} ∪ {2f(x) ∶ deg(x) = 0} consist of |E(G)| + |{x ∈ V(G) ∶ deg(x) = 0}| consecutive integers. In this paper, we construct (pseudo) super edge-magic 2-regular graphs from a super edge-magic cycle by using normalized Kotzig arrays. We also show that the graph C3 ∪ Cn ∪ K1 is pseudo super edge-magic for n ≡ 1(mod 4). By this result, we obtain some new classes of super edge-magic 2-regular graphs. In addition, we show that union of cycles and paths are super edge-magic. [ABSTRACT FROM AUTHOR]

An injective function f from the set of vertices in a graph G to a set {0,1,...,n} is called graceful labeling if the function f induced the edge function f* from the set of edges of G to a set of positive integers {1,2,...,n} with f*(xy) = |f(x) − f(y)| for every edge xy ∈ E(G) such that every edge has a different label. A graph G that can be labeled with graceful labeling is called a graceful graph. The multiple-fan graph with k pendants F(n1,n2,...,nr),k is a graph obtained from r fan graphs Fni, with i = 1,2,...,r, which share the center vertex and then we add k pendants to the center vertex of the multiple-fan graph. In this paper, we show that the multiple-fan graph with k pendants is graceful, with ni ≥ 2, i = 1,2,..., r and k ≥ r − 1. [ABSTRACT FROM AUTHOR]

The paper investigates a system of discrete equations Δ² x(k) + B² x(k – m) = Hu(k), k = 0, 1,… where x is an n-dimensional unknown vector, B is an n × n regular matrix, m ≥ 0 is an integer, H is an n × p matrix, and u is an input (control) function. The problem of relative controllability is formulated and considered with the results illustrated by an example. [ABSTRACT FROM AUTHOR]

In this paper, we propose an optimal diagonal algorithm under a generalized interval arithmetic for solving the problem of complete interval integer transport without converting the crisp equivalent problem. The proposed algorithm is simple to implement and it offers a numerical illustration to demonstrate the efficacy of the proposed process. [ABSTRACT FROM AUTHOR]

In this paper, we consider the sequence of integers ri, si, ti Ⲉ ℤ of regular continued fraction (RCF) expansions generated from the extended Euclidean algorithm. These sequences always satisfy ri = sia + tib where ri is a remainder whereas si and ti arising from the extended Euclidean algorithm are equal, up to sign, to the convergent of the continued fraction expansion of a/b. We discuss the behavior of these sequences and provide their full proof in detail with their simple calculation of sample. Throughout this work, we deal with the concept of Euclidean algorithm and extended Euclidean algorithm together with continued fraction algorithm. These algorithms are involved in the improvement of computational efficiency of the elliptic curve cryptography (ECC). Henceforth from that, we tend to associate these sequences in ECC. Last but not least, we found that the value of integers (ri, si, ti) satisfy various properties in RCF which then used to solve the shortest vector problem in representing point multiplications in ECC, namely the Gallant, Lambert & Vanstone (GLV) integer decomposition method and the integer sub decomposition (ISD) method. [ABSTRACT FROM AUTHOR]

The t−fold bristled graph(or t−thorny graph) is defined by attaching t pendant vertex to each vertex of a given graph, for any integer t ≥ 1. In this paper, we have shown that, the following equipartite graphs Kn ∗K...m,Kn − I ∗ K...m, Kn + I ∗ K...m, K∗m,n ∗ K...t and if G be a complete multipartite, G ∗ K...t−1 to be decomposed into t− fold bristled graphs. [ABSTRACT FROM AUTHOR]