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For a simple undirected connected graph G= (V, E), a maximal Roman dominating function (MRDF) of Gis a function f: V(G) → {0, 1, 2} with the following properties: (i) For every vertex v∈ {v∈ V|f(v) = 0}, there exists a vertex u∈ N(v) such that f(u) = 2. (ii) The set {v∈ V|f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v∈ {v∈ V|f(v) ≠ 0} such that N(v) ∩ {u∈ V|f(u) = 0} ∅. The weight of an MRDF of Gis the sum of its function values over all vertices, denoted as f(G) = ∑v∈V(G)f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

Based on the uncertainty theory, Liu [B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 2009, 1, 3–10] introduced an uncertain integral for applying uncertain differential equation, finance, control, filtering and dynamical systems. Since uncertain integrals are the important content of uncertainty theory, this paper explores an approach of the relationship between uncertain integrals by the well-known Chebyshev-type inequality. Also, we propose the concept of an uncertain fractional integral which is generalized version of an uncertain integral. The definition of a strong comonotonic uncertain process and some new properties of the uncertain integral were presented in [C. You and N. Xiang, Some properties of uncertain integral, Iran. J. Fuzzy Syst. 15 2018, 2, 133–142]. Based on the strong comonotonic uncertain process, as an application, we provide Chebyshev’s inequality for a fractional uncertain integral and an uncertain integral.

A Roman dominating function (RD-function) on a graph G= (V, E) is a function f: V→ {0, 1, 2} satisfying the condition that every vertex ufor which f(u) = 0 is adjacent to at least one vertex vfor which f(v) = 2. An Roman dominating function fin a graph Gis perfect Roman dominating function (PRD-function) if every vertex uwith f(u) = 0 is adjacent to exactly one vertex vfor which f(v) = 2. The (perfect) Roman domination number γR(G) (γpR(G)) is the minimum weight of an (perfect) Roman dominating function on G. We say that γpR(G) strongly equals γR(G), denoted by γpR(G) ≡ γR(G), if every RD-function on Gof minimum weight is a PRD-function. In this paper we show that for a given graph G, it is NP-hard to decide whether γpR(G) = γR(G) and also we provide a constructive characterization of trees Twith γpR(T) ≡ γR(T).

AbstractA quasi-total Roman dominating function (QTRD-function) on is a function such that (i) every vertex xfor which f(x) = 0 is adjacent to at least one vertex vfor which f(v) = 2, and (ii) if xis an isolated vertex in the subgraph induced by the set of vertices with non-zero values, then f(x) = 1. The weight of a QTRD-function is the sum of its function values over the whole set of vertices, and the quasi-total Roman domination number is the minimum weight of a QTRD-function on G. The quasi-total Roman bondage number of a graph Gis the minimum number of edges that have to be deleted to Gin order to increase the quasi-total Roman domination number. In this paper, we start the study of quasi-total Roman bondage in graphs. We first show that the decision problem associated with the quasi-total Roman bondage problem is NP-hard even when restricted to bipartite graphs. Then basic properties of the quasi-total Roman bondage number are provided. Finally, some sharp bounds for are also presented.

Discrimination between two probabilities is determined by divergences in information theory. In this paper, by using Fejér inequality, we introduce some extensions of Fejér-divergences and link these concepts with some well-known information divergences such as HH f-divergences, Riemann–Liouville fractional HH f-divergences, Hadamard fractional HH f-divergences and Conformable fractional HH f-divergences.

In a typical wireless telecommunication network, a large number of communication links is established with a limited number of available frequencies. The problem that addresses assigning available frequencies to transmitters such that interference is avoided as far as possible is called the frequency assignment problem. The problem is usually modeled as a graph coloring (labeling) problem. We study in this paper the (s, t)-relaxed L(2, 1)-labeling of a graph which considers the situation where transceivers that are very close receive frequencies that differ by at least two while transceivers that are close receive frequencies that differ by at least one. In addition, the model allows at most s(resp. t) anomalies at distance one (resp. two). The objective of the model is to minimize the span of frequencies in a corresponding network. We show that it is NP-complete to decide whether the minimal span of a (1, 0)-relaxed L(2, 1)-labeling of a graph is at most k. We also prove that the minimal span of this labeling for two classes of graphs is bounded above with the the square of the largest degree in the graph of interest. These results confirm Conjecture 6 and partially confirm Conjecture 3 stated in Lin [J. Comb. Optim.31(2016) 1–22].

AbstractA large number of medical experiments have confirmed that the features of drugs have a close correlation with their molecular structure. Drug properties can be obtained by studying the molecular structure of corresponding drugs. The calculation of the topological index of a drug structure enables scientists to have a better understanding of the physical chemistry and biological characteristics of drugs. In this paper, we focus on Hyaluronic Acid-Paclitaxel conjugates which are widely used in the manufacture of anticancer drugs. Several multiplicative topological indices of this molecular structure are determined by using of the edge-partition method.Communicated by Ramaswamy H. Sarma

In this paper, the fractional pantograph equations in quantum calculus are studied. We propose a q-fractional variant of the nonlinear pantograph equation with a given initial condition and a nonlinear neutral q-fractional pantograph equation. Based on the fixed-point theory, the main theorems provide some new results regarding the existence and uniqueness of solutions for q-fractional pantograph equations.

The star k-edge-coloring of graph Gis a proper edge coloring using kcolors such that no path or cycle of length four is bichromatic. The minimum number kfor which Gadmits a star k-edge-coloring is called the star chromatic index of G, denoted by χ′s(G). Let GCD(n, k) be the greatest common divisor of nand k. In this paper, we give a necessary and sufficient condition of χ′s(P(n, k)) = 4 for a generalized Petersen graph P(n, k) and show that “almost all” generalized Petersen graphs have a star 5-edge-colorings. Furthermore, for any two integers kand n(≥2k+ 1) such that GCD(n, k) ≥ 3, P(n, k) has a star 5-edge-coloring, with the exception of the case that GCD(n, k) = 3, k≠ GCD(n, k) and n3≡1(mod3){n \over 3} \equiv 1\left( {\bmod 3} \right).

An outer independent (double) Roman dominating function is a (double) Roman dominating function ffor which the set of vertices assigned 0 under fis independent. The outer independent (double) Roman domination number (γoidR(G)) γoiR(G)is the minimum weight taken over all outer independent (double) Roman dominating functions of G. A vertex cover number β(G)is the minimum size of any vertex cover sets of a graph G. In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We also prove that the decision problem associated with γoidR(G)is NP-complete even when restricted to planar graphs with maximum degree at most four. We moreover prove that 2β(T)+1≤γoidR(T)≤3β(T)for any tree T, and show that each integer between the lower and upper bounds is realizable. Finally, we give an exact formula for this parameter concerning the corona graphs.

A set Sof vertices of a connected graph Gis a doubly connected dominating set(DCDS) if every vertex not in Sis adjacent to some vertex in Sand the subgraphs induced by Sand V-Sare connected. The doubly connected domination numberγcc(G)is the minimum size of such a set. The doubly connected domination subdivision numbersdγcc(G)is the minimum number of edges that must be subdivided (each edge in Gcan be subdivided at most once) to increase the doubly connected domination number. It was conjectured (Karami et al. in Mat Vesnic 64:232–239, 2012) that the doubly connected domination subdivision number of a connected planar graph is at most two. In this paper, we disprove this conjecture by showing that the doubly connected domination subdivision number of the regular icosahedron graph is three.

AbstractConsidering the appointments that teachers plan to teach some courses for specific classes, the problem is to schedule the curriculum such that the time for each teacher is consecutive. In this work, we propose an integer linear programming model to solve consecutive interval edge-coloring of a graph. By using the proposed method, we give the interval edge colorability of some small complete multipartite graphs. Moreover, we find some new classes of complete multipartite graphs that have interval edge-colorings and disprove a conjecture proposed by Grzesik and Khachatrian (2014).

A k-rainbow dominating function(kRDF) of Gis a function f:V(G)→P({1,2,…,k})for which f(v)=∅we have ⋃u∈N(v)f(u)={1,2,…,k}. The weightw(f) of a function fis defined as w(f)=∑v∈V(G)f(v). The minimum weight of a kRDF of Gis called the k-rainbow domination numberof G, which is denoted by γrk(G). In this paper, we determine the exact values of the 2-rainbow domination numbers of C4□Cnand C8□Cn. It follows that γr2≠2γfor graphs C4□Cn(n≥4) and C8□Cn(n≥8), answering in part a question raised by Brešar.

Let Dbe a finite and simple digraph with vertex set V(D). A signed dominating function (SDF) of Dis a function f:V(D)⟶{-1,1}such that f(N-[v])=∑x∈N-[v]f(x)≥1for every v∈V(D), where N-[v]consists of vand all vertices of Dfrom which arcs go into v. The weight of an SDF is the sum of its function values over all vertices, and the minimum weight of an SDF of Gis the signed domination number γs(D).In this paper, we investigate the signed domination number of the Cartesian product of two directed cycles by showing that γs(Cm□Cn)=⌈m3⌉nif n≡0(mod2m)or n≥mand m≡1(mod3),answering a conjecture posed in Shaheen (J Progress Res Math 6(2):770–777, 2016). Moreover, the exact value of γs(C8□Cn)is also provided.

Purpose: A comprehensive study on the fluid flow and heat transfer in a nanofluid channel is carried out. The configuration of the channel is as like as quarter channel. The channel is filled with CuO–water nanofluid. Design/methodology/approach: The Koo–Kleinstreuer–Li model is used to estimate the dynamic viscosity and consider the Brownian motion. On the other hand, the influence of nanoparticles' shapes on the heat transfer rate is considered in the simulations. The channel is included with the injection pipes which are modeled as active bodies with constant temperature in the 2D simulations. Findings: The Rayleigh number, nanoparticle concentration and the thermal arrangements of internal pipes are the governing parameters. The hydrothermal aspects of natural convection are investigation using different approaches such as average Nusselt number, total entropy generation, Bejan number, streamlines, temperature fields, local heat transfer irreversibility, local fluid friction irreversibility and heatlines. Originality/value: The originality of this work is investigation of fluid flow, heat transfer, entropy generation and heatline visualization within a nanofluid-filled channel using a finite volume method. [ABSTRACT FROM AUTHOR]

ABSTRACTAs a chiral precursor for the important anticoagulant Edoxaban, enantioselective synthesis of (S)-3-cyclohexene-1-carboxylic acid is of great significance. The complicated procedures and generation of massive solid waste discourage its chemical synthesis, and the alternative biocatalysis route calls for an enzyme capable of asymmetric hydrolysis of racemic methyl-3-cyclohexene-1-carboxylate. To this end, we engineered the E. coliesterase BioH for improved S-enantioselectivity via rational design. By combinatorial modulation of steric and aromatic interactions, a positive mutant Mu3 (L24A/W81A/L209A) with relatively high S-selectivity in hydrolyzing racemic methyl-3-cyclohexene-1-carboxylate was obtained, improving the enantiomeric excess from 32.3% (the wild type) to 70.9%. Molecular dynamics simulation was conducted for both (R)- or (S)- complexes of the wild type and Mu3 to provide hints for the mechanism behind the increased S-selectivity. Moreover, the reaction conditions of Mu3 in methyl-3-cyclohexene-1-carboxylate hydrolysis was optimized to improve the conversion rate to 2 folds.

Let G= (V, E) be a graph and let f: V(G) → {0, 1, 2} be a function. A vertex vis said to be protected with respect to f, if f(v) >0 or f(v) = 0 and vis adjacent to a vertex of positive weight. The function fis a co-Roman dominating functionif (i) every vertex in Vis protected, and (ii) each v∈ Vwith positive weight has a neighbor u∈ Vwith f(u) = 0 such that the function fuv: V→ {0, 1, 2}, defined by fuv(u) = 1, fuv(v) = f(v) − 1 and fuv(x) = f(x) for x∈ V\ {v, u}, has no unprotected vertex. The weightof fis ω(f) = ∑v∈V f(v). The co-Roman domination numberof a graph G, denoted by γcr(G), is the minimum weight of a co-Roman dominating function on G. In this paper, we give a characterization of graphs of order nfor which co-Roman domination number is 2n3${{2n} \over 3}$or n− 2, which settles two open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1–10]. Furthermore, we present some sharp bounds on the co-Roman domination number.

Given a positive integer tand a graph F, the goal is to assign a subset of the color set $$\{1,2,\ldots ,t\}$$ {1,2,…,t} to every vertex of Fsuch that every vertex with the empty set assigned has all tcolors in its neighborhood. Such an assignment is called the t-rainbow dominating function ($$t\mathrm{RDF}$$ tRDF ) of the graph F. A $$t\mathrm{RDF}$$ tRDF is independent ($$It\mathrm{RDF}$$ ItRDF ) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a $$t\mathrm{RDF}$$ tRDF gof a graph Fis the value $$w(g) =\sum _{v \in V(F)}|g(v)|$$ w(g)=∑v∈V(F)|g(v)| . The independent t-rainbow domination number $$i_{rt}(F)$$ irt(F) is the minimum weight over all $$It\mathrm{RDF}$$ ItRDF s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given.

AbstractA double Roman dominating function (DRDF) on a graph G= (V, E) is a function f: V(G) → {0, 1, 2, 3} having the property that if f(v) = 0, then vertex vmust have at least two neighbors assigned 2 under for one neighbor wwith f(w) = 3, and if f(v) = 1, then vertex vmust have at least one neighbor wwith f(w) ≥ 2. A DRDF fis called a global double Roman dominating function(GDRDF) if fis also a DRDF of the complement of G. The weight of a GDRDF is the sum of its function value over all vertices. The global double Roman domination number of G, denoted by γgdR(G), is the minimum weight of a GDRDF on G. In this paper, we initiate the study of the global double Roman domination number. We obtain some properties of global double Roman domination number.

A set $$S \subseteq V(G)$$ S⊆V(G) is a k-distance independent set of a graph Gif the distance between every two vertices of Sis greater than k. The k-distance independence number $$\alpha _k(G)$$ αk(G) of Gis the maximum cardinality over all k-distance independent sets in G. A k-distance coloring of Gis a function ffrom V(G) onto a set of positive integers (colors) such that for any two distinct vertices uand vat distance less than or equal to kwe have $$f(u) \not = f(v)$$ f(u)≠f(v) . The k-distance chromatic number of a graph Gis the smallest number of colors needed to have a k-distance coloring of G. The k-distance independence numbers and 2-distance chromatic numbers of Cartesian products of cycles are considered. A computer-aided method with the isomorphic rejection is used to determine the k-distance independence numbers of Cartesian products of cycles. By using these results, several lower and upper bounds on the maximal cardinality $$A^L_q(n, d)$$ AqL(n,d) of a q-ary Lee code of length nwith a minimum distance at least dare improved. It is also established that the 2-distance chromatic number of Gequals $$\left\lceil \frac{|V(G)|}{\alpha (G^2)} \right\rceil $$ |V(G)|α(G2) for $$G=C_m \Box C_n \Box C_k$$ G=Cm□Cn□Ck , whenever $$k \ge 3$$ k≥3 and $$(m,n)\in \{(3,3), (3,4), (3,5), (4,4)\}$$ (m,n)∈{(3,3),(3,4),(3,5),(4,4)} or k, mand nare all multiples of seven. Moreover, it is shown that the 2-distance chromatic number of the three-dimensional square lattice is equal to seven.

An interesting graph distance constrained labelling problem can model the frequency channel assignment problem as well as code assignment in computer networks. The frequency assignment problem asks for assigning frequencies to transmitters in a broadcasting network with the aim of avoiding undesired interference. One of the graph theoretical models of The frequency assignment problem is the concept of distance constrained labelling of graphs. Let uand vbe vertices of a graph G= (V(G),E(G)) and d(u, v) be the distance between uand vin G. For an integer d≥ 0, an L(d, 1)-labelling of Gis a function f: V(G) → {0, 1, · · · } such that for every u, vϵ V(G), |f(u) – f(v)| ≥ dif d(u, v) = 1 and |f(u) – f(v)| ≥ 1 if d(u, v) = 2. The span of fis the difference between the largest and the smallest numbers in f(V(G)). The λd,1-number of Gis the minimum span over all L(d, 1)-labellings of G. For natural numbers nand k, where n> 2k, a generalised Petersen graph P(n, k) is obtained by letting its vertex set be {u1, u2, · · · , un} υ {v1, v2, · · · , vn} and its edge set be the union of uiui+1, uivi, vivi+kover 1 ≤ i≤ n, where subscripts are reduced modulo n. In this paper, we show the λd,1-numbers of the generalised Petersen graphs P(n, k) for n≥ 5.

Sidon set and its related problems including modular Sidon set have many applications in information theory and communication system. They have been widely studied in radio frequency selection, radio antennae placement and error correcting codes. In this paper, we establish a backtracking algorithm with pruning techniques to construct modular Sidon sets. By using master-slave parallel model, 50 new optimal modular Sidon sets are constructed.

Membrane algorithms (MAs), which inherit from P systems, constitute a new parallel and distribute framework for approximate computation. In the paper, a membrane algorithm is proposed with the improvement that the involved parameters can be adaptively chosen. In the algorithm, some membranes can evolve dynamically during the computing process to specify the values of the requested parameters. The new algorithm is tested on a well-known combinatorial optimization problem, the travelling salesman problem. The empirical evidence suggests that the proposed approach is efficient and reliable when dealing with 11 benchmark instances, particularly obtaining the best of the known solutions in eight instances. Compared with the genetic algorithm, simulated annealing algorithm, neural network and a fine-tuned non-adaptive membrane algorithm, our algorithm performs better than them. In practice, to design the airline network that minimize the total routing cost on the CAB data with twenty-five US cities, we can quickly obtain high quality solutions using our algorithm.

Abstract: Given a graph G and a set of t colors, assume that we assign an arbitrary subset of these colors to each vertex of G. If we require that each vertex to which an empty set is assigned has in its neighborhood all t colors, then this assignment is called a t-rainbow dominating function of the graph G. The corresponding invariant γ rt (G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the t-rainbow domination number of G. In this paper, bounds for the t-rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given. The 3-rainbow domination numbers of several classes of graphs such as paths, cycles and the generalized Petersen graphs P(n, k), are investigated. The 3-rainbow domination number of P(n,1) is determined and the upper bounds for P(n,2) and P(n,3) are provided. By computer search, we obtain that the upper bounds for P(n,2) match their exact values for n ⩽17. [Copyright &y& Elsevier]

In this note, a dynamic programming-like method is used to detect $$k$$-term semi-progression efficiently. By using this approach, we obtain some exact values and new lower bounds on semi-progression van der Waerden numbers.

It is known that Intuitionistic fuzzy models give more precision, flexibility and compatibility to the system as compared to the classic and fuzzy models. Intuitionistic fuzzy tree has an important role in neural networks, computer networks, and clustering. In the design of a network, it is important to analyze connections between the levels. In addition, the intuitionistic fuzzy tree is becoming increasingly significant as it is applied to different areas in real life. The study proposes the novel concepts of intuitionistic fuzzy graph (IFG) and some basic definitions. We investigate the types of arcs, for example, αμ-strong, βμ-strong, and δμ-arc in an intuitionistic fuzzy graph, and introduce some of their properties. In particular, the present work develops the concepts of intuitionistic fuzzy bridge (IFB), intuitionistic fuzzy cut nodes (IFCN) and some important properties of an intuitionistic fuzzy bridge. Next, we define an intuitionistic fuzzy cycle (IFC) and an intuitionistic fuzzy tree (IFT). Likewise, we discuss some properties of the IFT and the relationship between an intuitionistic fuzzy tree and an intuitionistic fuzzy cycle. Finally, an application of intuitionistic fuzzy tree is illustrated in other sciences.