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An outer independent (double) Roman dominating function is a (double) Roman dominating function f for which the set of vertices assigned 0 under f is independent. The outer independent (double) Roman domination number ( γ oidR (G) ) γ oiR (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 γ oidR (G) is NP-complete even when restricted to planar graphs with maximum degree at most four. We moreover prove that 2 β (T) + 1 ≤ γ oidR (T) ≤ 3 β (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. [ABSTRACT FROM AUTHOR]

A double Roman dominating function on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 , 3 } with the properties that if f (u) = 0 , then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f (u) = 1 , then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w (f) = ∑ v ∈ V f (v) . The double Roman domination number γ d R (G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γ d R (G) = 3 γ (G) , where γ (G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P (3 k , k) , and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P (3 k , k) . This implies that P (3 k , k) is a double Roman graph if and only if either k ≡ 0 (mod 3) or k ∈ { 1 , 4 } . [ABSTRACT FROM AUTHOR]

For a simple graph G = (V , E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f : V (G) → { 0 , 1 , 2 , 3 } having the property that (i) ∑ w ∈ N (v) f (w) ≥ 3 if f (v) = 0 ; (ii) ∑ w ∈ N (v) f (w) ≥ 2 if f (v) = 1 ; and (iii) every vertex v with f (v) ≠ 0 has a neighbor u with f (u) ≠ 0 for every vertex v ∈ V (G) . The weight of a TR3DF f is the sum f (V) = ∑ v ∈ V (G) f (v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γ t { R 3 } (G) . In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γ t { R 3 } for trees. [ABSTRACT FROM AUTHOR]

For a graph G with no isolated vertex, let γ p r (G) and sd γ p r (G) denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order n ≥ 4 different from a healthy spider (subdivided star), then sd γ p r (T) ≤ min { γ p r (T) 2 + 1 , n 2 } , improving the (n − 1) -upper bound that was recently proven. [ABSTRACT FROM AUTHOR]

In the current close-to-smart power systems, renewable energies are the most significant elements that need to be carefully addressed in power system studies. In this regard, microgrids (MGs) have been introduced recently to activate the large penetration of renewables. However, integration of such generation technologies that are associated with severe uncertainty in their power output would significantly impact the scheduling of energy resources in MGs. Thus, efficient energy management systems are required to be employed in MGs. Therefore, this paper presents a day-ahead scheduling framework for an MG equipped with a solar photovoltaic (PV) unit. In this respect, different climate conditions and their impacts on the power output of the PV unit and the optimal scheduling of the MG have been investigated in this paper. To this end, four different days from the four seasons have been used to extract the data of solar irradiance. The scheduling problem has been formulated in a single-objective optimization framework, where the objective function is defined as minimizing the total operating cost over the scheduling period. An effective optimization algorithm named "modified krill herd (MKH)" algorithm is proposed to solve the mentioned day-ahead scheduling problem, while there are renewable and non-renewable generating units, besides an energy storage system. Furthermore, a comprehensive comparison has been made between the MKH algorithm and some well-known optimization algorithms to verify the superior performance of the suggested method. [ABSTRACT FROM AUTHOR]

Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been acknowledged as being an applicable and well-organized tool to epitomize and solve many multifarious real-world problems in which vague data and information are essential. Owing to unpredictable and unspecified information being an integral component in real-life problems that are often uncertain, it is highly challenging for an expert to illustrate those problems through a fuzzy graph. Therefore, resolving the uncertainty accompanying the unpredictable and unspecified information of any real-world problem can be done by applying a vague incidence graph (VIG), based on which the FGs may not engender satisfactory results. Similarly, VIGs are outstandingly practical tools for analyzing different computer science domains such as networking, clustering, and also other issues such as medical sciences, and traffic planning. Dominating sets (DSs) enjoy practical interest in several areas. In wireless networking, DSs are being used to find efficient routes with ad-hoc mobile networks. They have also been employed in document summarization, and in secure systems designs for electrical grids; consequently, in this paper, we extend the concept of the FIG to the VIG, and show some of its important properties. In particular, we discuss the well-known problems of vague incidence dominating set, valid degree, isolated vertex, vague incidence irredundant set and their cardinalities related to the dominating, etc. Finally, a DS application for VIG to properly manage the COVID-19 testing facility is introduced. [ABSTRACT FROM AUTHOR]

Fuzzy graph models enjoy the ubiquity of being in natural and human-made structures, namely dynamic process in physical, biological and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems which are often uncertain, it is highly difficult for an expert to model those problems based on a fuzzy graph (FG). Vague graph structure (VGS) can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. Likewise, VGSs are very useful tools for the study of different domains of computer science such as networking, capturing the image, clustering, and also other issues like bioscience, medical science, and traffic plan. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGSs. Operations are conveniently used in many combinatorial applications. In various situations, they present a suitable construction means; therefore, in this research, three new operations on VGSs, namely, maximal product, rejection, residue product were presented, and some results concerning their degrees and total degrees were introduced. Irregularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology and economy; so special concepts of irregular VGSs with several key properties were explained. Today one of the most important applications of decision making is in medical science for diagnosing the patient's disease. Hence, we recommend an application of VGS in medical diagnosis. [ABSTRACT FROM AUTHOR]

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital. [ABSTRACT FROM AUTHOR]

In recent years, the concept of domination has been the backbone of research activities in graph theory. The application of graphic domination has become widespread in different areas to solve human-life issues, including social media theories, radio channels, commuter train transportation, earth measurement, internet transportation systems, and pharmacy. The purpose of this paper was to generalize the idea of bondage set (BS) and non-bondage set (NBS), bondage number α (G) , and non-bondage number α k (G) , respectively, in the intuitionistic fuzzy graph (IFG). The BS is based on a strong arc (SA) in the fuzzy graph (FG). In this research, a new definition of SA in connection with the strength of connectivity in IFGs was applied. Additionally, the BS, α (G) , NBS, and α k (G) concepts were presented in IFGs. Three different examples were described to show the informative development procedure by applying the idea to IFGs. Considering the examples, some results were developed. Also, the applications were utilized in water supply systems. The present study was conducted to make daily life more useful and productive. [ABSTRACT FROM AUTHOR]

A dominating set in a graph G is a set of vertices S ⊆ V (G) such that any vertex of V − S is adjacent to at least one vertex of S. A dominating set S of G is said to be a perfect dominating set if each vertex in V − S is adjacent to exactly one vertex in S. The minimum cardinality of a perfect dominating set is the perfect domination number γ p (G). A function f : V (G) → { 0 , 1 , 2 } is a perfect Roman dominating function (PRDF) on G if every vertex u ∈ V for which f (u) = 0 is adjacent to exactly one vertex v for which f (v) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p (G). In this paper, we prove that for any nontrivial tree T, γ R p (T) ≥ γ p (T) + 1 and we characterize all trees attaining this bound. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) with vertex set V = V (G) and edge set E = E (G) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V (G) → { 0 , 1 , 2 , 3 } having the property that ∑ u ∈ N G (v) f (u) ≥ 3 , if f (v) = 0 , and ∑ u ∈ N G (v) f (u) ≥ 2 , if f (v) = 1 for any vertex v ∈ V (G) . The weight of a Roman { 3 } -dominating function f is the sum f (V) = ∑ v ∈ V (G) f (v) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } (G) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f (v) ≠ 0 has a neighbor w with f (w) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } (G) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs. [ABSTRACT FROM AUTHOR]

The concept of a metric dimension was proposed to model robot navigation where the places of navigating agents can change among nodes. The metric dimension m d (G) of a graph G is the smallest number k for which G contains a vertex set W, such that | W | = k and every pair of vertices of G possess different distances to at least one vertex in W. In this paper, we demonstrate that m d (H D N 1 (n)) = 4 for n ≥ 2 . This indicates that in these types of hex derived sensor networks, the least number of nodes needed for locating any other node is four. [ABSTRACT FROM AUTHOR]

The concept of almost interior Γ -hyperideals (A − I − Γ -hyperideals) in ordered Γ -semihypergroups is a generalization of the concept of interior Γ -hyperideals (I − Γ -hyperideals). In this study, the connections between I − Γ -hyperideals and A − I − Γ -hyperideals in ordered Γ -semihypergroups were presented. Also, we define the notion of m , n - Γ -hyperfilters of ordered Γ -semihypergroups and provide useful results on it. Moreover, we use the weak pseudoorders to construct quotient ordered Γ -semihypergroups by using ordered regular equivalence relations. These concepts lead us to a new research direction in ordered Γ -semihypergroups. [ABSTRACT FROM AUTHOR]

In this work, a new model has been developed to examine and present a bidding method and a suitable strategy for large consumers. The proposed model is consists of different energy suppliers as: wind micro turbines, energy storage systems, renewable energy sources (wind turbine and solar system) and bilateral contracts. To solve the mentioned problem, a hybrid robust-stochastic method in which the load uncertainty, temperature and wind speed, the amount of solar radiation and the cost of purchasing energy are considered in the proposed model. In this work, load uncertainty and other uncertainties are modeled. Also, two types of weibull and normal distribution are utilized, that the first distribution is used to create wind speed scenarios and second distribution to create solar radiation, temperature and electricity market prices scenarios. Bidding and offering curves have been obtained via a mixed-integer linear programming model and this model has a significant effect on uncertainty in the demand of the large consumers. Effectiveness of proposed method is applied on standard system through different scenarios. Obtained results shows the efficiency of proposed method in large consumers and the presence of load uncertainty. [ABSTRACT FROM AUTHOR]

Abstract An edge metric generator of a connected graph G is a vertex subset S for which every two distinct edges of G have distinct distance to some vertex of S , where the distance between a vertex v and an edge e is defined as the minimum of distances between v and the two endpoints of e in G. The smallest cardinality of an edge metric generator of G is the edge metric dimension, denoted by dim e (G). It follows that 1 ≤ dim e (G) ≤ n − 1 for any n -vertex graph G. A graph whose edge metric dimension achieves the upper bound is topful. In this paper, the structure of topful graphs is characterized, and many necessary and sufficient conditions for a graph to be topful are obtained. Using these results we design an O (n 3) time algorithm which determines whether a graph of order n is topful or not. Moreover, we describe and address an interesting class of topful graphs whose super graphs obtained by adding one edge are not topful. [ABSTRACT FROM AUTHOR]

In this paper we obtain some bounds and exact values of the total domination numbers and open packing numbers for pyrene torus, hexabenzocoronene, H-phenylenic nanotube and H-napthelenic nanotube. [ABSTRACT FROM AUTHOR]

A set D ⊆ V ( G ) of a graph G is a dominating set if every vertex v ∈ V ( G ) is either in D or adjacent to a vertex in D . The domination number γ ( G ) of a graph G is the minimum cardinality of a dominating set of G . Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n -vertex maximal outerplanar graph having t vertices of degree 2, then γ ( G ) ≤ n + t 4 . We improve their upper bound by showing γ ( G ) ≤ n + k 4 , where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ ( G ) ≤ 5 n 16 for a Hamiltonian maximal planar graph G with n ≥ 7 vertices. [ABSTRACT FROM AUTHOR]

Background: Influenza B virus can cause epidemics with high pathogenicity, so it poses a serious threat to public health. A feature representation algorithm is proposed in this paper to identify the pathogenicity phenotype of influenza B virus. Methods: The dataset included all 11 influenza virus proteins encoded in eight genome segments of 1724 strains. Two types of features were hierarchically used to build the prediction model. Amino acid features were directly delivered from 67 feature descriptors and input into the random forest classifier to output informative features about the class label and probabilistic prediction. The sequential forward search strategy was used to optimize the informative features. The final features for each strain had low dimensions and included knowledge from different perspectives, which were used to build the machine learning model for pathogenicity identification. Results: The 40 signature positions were achieved by entropy screening. Mutations at position 135 of the hemagglutinin protein had the highest entropy value (1.06). After the informative features were directly generated from the 67 random forest models, the dimensions for class and probabilistic features were optimized as 4 and 3, respectively. The optimal class features had a maximum accuracy of 94.2% and a maximum Matthews correlation coefficient of 88.4%, while the optimal probabilistic features had a maximum accuracy of 94.1% and a maximum Matthews correlation coefficient of 88.2%. The optimized features outperformed the original informative features and amino acid features from individual descriptors. The sequential forward search strategy had better performance than the classical ensemble method. Conclusions: The optimized informative features had the best performance and were used to build a predictive model so as to identify the phenotype of influenza B virus with high pathogenicity and provide early risk warning for disease control. [ABSTRACT FROM AUTHOR]

Graphs serve as one of the main tools for the mathematical modeling of various human problems. Fuzzy graphs have the ability to solve uncertain and ambiguous problems. The cubic graph, which has recently gained a position in the fuzzy graph family, has shown good capabilities when faced with problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Simultaneous application of fuzzy and interval-valued fuzzy membership indicates a high flexibility in modeling uncertainty issues. The vertex cover is a fundamental issue in graph theory that has wide application in the real world. The previous definition limitations in the vertex covering of fuzzy graphs has directed us to offer new classifications in terms of cubic graph. In this study, we introduced the strong vertex covering and independent vertex covering in a cubic graph with strong edges and described some of its properties. One of the motives of this research was to examine the changes in the strong vertex covering number of a cubic graph if one vertex is omitted. This issue can play a decisive role in covering the graph vertices. Since many of the problems ahead are of hybrid type, by reviewing some operations on the cubic graph we were able to determine the strong vertex covering number on the most important cubic product operations. Finally, two applications of strong vertex covering and strong vertex independence are presented. [ABSTRACT FROM AUTHOR]

Let G be a molecular graph with n vertices, m edges and G(L) be a line graph. Both G and L can be represented by their adjacency matrices A and E, respectively. The eigenvalues of G and L are denoted by, λ 1 ⩾ λ 2 ⩾ ... ⩾ λ n and γ 1 ⩾ γ 2 ⩾ ... ⩾ γ n , respectively. The Estrada index A and L are defined as E E (G) = ∑ i = 1 n e λ i and E E (L) = ∑ i = 1 n e γ i . In this article, we present some new lower and upper bounds for the Estrada index molecular and line graphs of phenylenes. [ABSTRACT FROM AUTHOR]

During the past years, hybrid solar-wind power systems containing photovoltaic (PV) and wind generators are used to minimize the intermittency problem of renewable power generation units. The improved modeling and control schemes for a grid-tied hybrid PV–wind system is presented in the current research work. The maximum power point tracking namely "MPPT" algorithm together with controlling the pitch angle are used, respectively, for the PV system and wind power generation to attain the maximum power for any given external weather conditions. A radial basis function network sliding mode known as the RBFNSM method is used to control the pitch angle in the wind energy system, while the PV system uses a proportional–integral–derivative controller equipped with the fuzzy gain scheduling in order to enhance the transient state and mitigate the settling time to ensure the stability of the mentioned system. To test the suggested control scheme's effectiveness, MATLAB simulations are carried out under various scenarios of the wind speed as well as solar irradiation. The obtained results show the efficiency of the adaptive MPPT method to harness the highest power under very challenging scenarios. The merits of the developed schemes are quickly and precisely tracking the maximum power output of the hybrid PV–wind system. Besides, the power flowing between the utility grid and the hybrid source with a fast transient response and improved stability performance is effectively controlled using the offered schemes. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V (G). A signed total Roman k -dominating function (STR k DF) on a graph G is a function f : V (G) → { − 1 , 1 , 2 } such that (i) every vertex v with f (v) = − 1 is adjacent to at least one vertex u with f (u) = 2 and (ii)  ∑ u ∈ N (v) f (u) ≥ k holds for any vertex v. The weight of an STR k DF f is ∑ u ∈ V (G) f (u) , and the minimum weight of an STR k DF is the signed total Roman k -domination number γ stR k (G) of G. In this paper, we establish some sharp bounds on the signed total Roman 2-domination number. [ABSTRACT FROM AUTHOR]

A molecular network can be characterized by several different ways, like a matrix, a polynomial, a drawing or a topological descriptor. A topological descriptor is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. Analyzing and determining the topological indices and structural properties of a network or a graph have been a worthy studied topic in the field of chemistry, networks analysis, etc. In this paper, we consider several types of the generalized Sierpiński networks and investigate the explicit expressions of some well-known valency-based topological indices. Taking into account the other structural property of the generalized Sierpiński networks, the average degree is determined. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , an independent Roman dominating function (IRDF) is a function f : V → { 0 , 1 , 2 } having the property that: (1) every vertex assigned a value of 0 is adjacent to at least one vertex assigned a value of 2, (2) there are no two adjacent vertices with positive assignments. The weight of an IRDF ( w (f) ) is the sum of assignments for all vertices. The minimum weight of an independent Roman dominating function on graph G is the independent Roman domination number, denoted by i R (G) . In this paper, we prove that the decision problem of minimum IRDF is N P -complete for chordal bipartite graphs. Then, we research the difference in complexity between the decision problem of RDF and IRDF. Finally, we propose a linear-time algorithm for computing the minimum weight of an independent Roman dominating function in trees. [ABSTRACT FROM AUTHOR]

Abstract: Mesh is an important and popular interconnection network topology for large parallel computer systems. A mesh can be divided into submeshes to obtain the upper bounds on the connection probability for the mesh. Combinatorial techniques are used to get closer upper bounds on the connection probability for 2-D meshes compared with the existing upper bounds we have known. Simulation results of meshes of various sizes show that our upper bounds are close to the exact connection probability. The combinatorial methods and tools used in this paper can be used to study the connection probabilities for other networks. [Copyright &y& Elsevier]

A 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.

An L(2,1)-labeling of a graph G = (V, E) is a function f from the vertex set V(G) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two and the labels on vertices at distance two differ by at least one. The span of f is the difference between the largest and the smallest numbers in f(V). The λ-number of G, denoted by λ(G), is the minimum span over all L(2,1)-labelings of G. We consider the λ-number of Pn⊠C m and for n ≤ 11 the λ-number of Cn⊠Cm. We determine λ-numbers of graphs of interest with the exception of a finite number of graphs and we improve the bounds on the λ-number of Cn⊠Cm, m ≥ 24 and n ≥ 26.