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The vertices of a graph are to be labelled with 0 or 1 such that each edge gets the label as the product of its end vertices. If the number of vertices labelled with 0’s and 1’s differ by at most one and if the number of edges labelled with 0’s and 1’s differ by at most by one, then the labelling is called product cordial labelling. Complete characterizations of product cordial dragon graphs is given. We also characterize dragon graphs whose line graphs are product cordial. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) and edge set E(G). Denote by dG(u) the degree of a vertex u ∈ V (G). The Sombor index of G is defined as SO(G) = Σuv∈E(G) √d²u + d²v , whereas, the inverse sum indeg (ISI) index is defined as ISI(G) = Σuv∈E(G) dudv/du+dv . In this paper, we compute the bounds in terms of maximum degree, minimum degree, order and size of the original graphs G and H for Sombor and ISI indices of several graph operations like corona product, cartesian product, strong product, composition and join of graphs. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a simple graph. A set D ⊆ V (G) is a strong dominating set of G, if for every vertex x ∈ V (G) \ D there is a vertex y ∈ D with xy ∈ E(G) and deg(x) ≤ deg(y). The strong domination number γst(G) is defined as the minimum cardinality of a strong dominating set. In this paper, we examine the effects on γst(G) when G is modified by operations on edge (or edges) of G. [ABSTRACT FROM AUTHOR]

A signed graph is a graph whose edges are labeled either as positive or negative. The concepts of vector valued switching and balancing dimension of signed graphs were introduced by S. Hameed et al. In this paper, we deal with the balancing dimension of various products of signed graphs, namely the Cartesian product, the lexicographic product, the tensor product, and the strong product. [ABSTRACT FROM AUTHOR]

Topological indices play a significant role in crisp, fuzzy graphs and their real-life application. But to avoid the vagueness in the final result, these indices should be dealt with in the neutrosophic environment since it consolidates the uncertain quantity or values of an event in the name of "indeterminacy membership". Except for the wiener index, no other indices are introduced under the neutrosophic graphical setting. In this article, we consider the forgotten topological index (ToI) and the Edge forgotten index in the 3-valued logic neutrosophic graph and came up with some important theorem results and applications. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple connected graph. A perfect matching (or Kekulé structure in chemical literature) of G is a set of disjoint edges which covers all vertices of G. The anti-forcing number of G is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by af(G). For every m ∈ N, the mth power of G, denoted by Gm, is a graph with the same vertex set as G such that two vertices are adjacent in Gm if and only if their distance is at most m in G. In this paper, we study the anti-forcing number of the powers of some graphs. [ABSTRACT FROM AUTHOR]

This paper introduces certain relations between 1-edge contraction and the total vertex stress and the confluence number of a graph. A main result states that if a graph G with ζ(G) = k ≥ 2 has an edge vivj and a ζ-set CG such that vi, vj ∈ CG then, ζ(G/vivj ) = k − 1. In general, either S(G/ei) ≤ S(G/ej ) or S(G/ej ) ≤ S(G/ei) is true. This observation leads to an investigation into the question: for which edge(s) ei will S(G/ei) = max{S(G/ej ) : ej ∈ E(G)} and for which edge(s) will S(G/ej ) = min{S(G/el) : el∈ E(G)}? [ABSTRACT FROM AUTHOR]

A signed graph (G, σ) is called a parity signed graph if there exists a bijective mapping f : V (G) → {1, . . ., |V (G)|} such that for each edge uv in G, f(u) and f(v) have same parity if σ(uv) = +1, and opposite parity if σ(uv) = −1. The rna number σ−(G) of G is the least number of negative edges among all possible parity signed graphs over G. Equivalently, σ−(G) is the least size of an edge-cut of G that has nearly equal sides. In this paper, we show that for the generalized Petersen graph Pn,k, σ−(Pn,k) lies between 3 and n. Moreover, we determine the exact value of σ−(Pn,k) for k ∈ {1, 2}. The rna numbers of some famous generalized Petersen graphs, namely, Petersen graph, Dürer graph, Möbius-Kantor graph, Dodecahedron, Desargues graph and Nauru graph are also computed. Recently, Acharya, Kureethara and Zaslavsky characterized the structure of those graphs whose rna number is 1. We use this characterization to show that the smallest order of a (4n + 1)-regular graph having rna number 1 is 8n + 6. We also prove the smallest order of (4n − 1)-regular graphs having rna number 1 is bounded above by 12n − 2. In particular, we show that the smallest order of a cubic graph having rna number 1 is 10. [ABSTRACT FROM AUTHOR]

The harmonic status index of a simple connected graph G is defined as the sum of the weights 2 sG(u)+sG(v) over all edges uv of G, where sG(u) denotes the status of the vertex u in G which is the sum of distances between u and all other vertices of G. In this paper, we present upper bounds on the harmonic status index of some families of graph products in terms of certain structural invariants such as the order, size, maximum degree, inverse status and harmonic status index of their components. The graph products considered here are sum, disjunction, symmetric difference, Indu-Bala product, corona product, Cartesian product, lexicographic product, and strong product. Some applications of the obtained results are also presented as corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks. [ABSTRACT FROM AUTHOR]

Let tp (G) denote the number of paths in a graph G and let f: E → Z+ be an edge labeling of G. The weight of a path P is the sum of the labels assigned to the edges of P. If the set of weights of the paths in G is {1, 2, 3,..., tp (G)}, then f is called a Leech labeling of G and a graph which admits a Leech labeling is called a Leech graph. In this paper, we prove that the complete bipartite graphs K2, n and K3, n are not Leech graphs and determine the maximum possible value that can be given to an edge in the Leech labeling of a cycle. [ABSTRACT FROM AUTHOR]

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g -Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

This article successfully attempts to introduce the notion of Inverse Neutrosophic Mixed Graphs (INMG) together with its applications. This novel approach highlights the network modeling of real physical situations with indeterminacy. Here, INMG are developed with both directed and undirected relationships between nodes that express the circumstances where the truth membership degrees of the edges are superior or equivalent to the least of the truth membership degrees of the associated vertices and where false membership and indeterminacy values of the edges are inferior or equivalent to the maximum of false membership and indeterminacy values of the corresponding vertices. Furthermore, some fundamental functions and algebraic characteristics of INMG are examined to attain a profound insight into the properties and applications. To illustrate the application of INMG, the article provides a numerical example centered around social networks. By employing INMG in this context, the article demonstrates how the model can effectively capture and represent the complex relationships within social networks, taking into account the inherent uncertainties and indeterminacies present in such systems. [ABSTRACT FROM AUTHOR]

Recently, a serious problem in communications is security. Hiding data is one of the most important security techniques. Steganography is the art and science of hiding information in a cover media. Texts are the most usual method of communication and so they are very suitable for cover objects. In this paper, we give a new technique for text steganography. There exist different models, such as RGB, HSL, HSV to determine a color. The main goal of the proposed algorithm is the fact that some different but very similar colors in RGB have the same code in HSL. We use this fact to hide data. For this purpose, first we find a color B′ in RGB, which is very similar to black in such a way that they have the same code in HSL. Then, by changing the color of each character of the text to black or B′, we conceal the information. We will show that the capacity of this method is better than some other methods of text steganography, and then we show that the invisibility of this algorithm is very high, which is the most prominent feature of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of Connes biprojectivity for a dual Banach algebra A with respect to its w∗-closed ideal I, say I-Connes biprojectivity. Some Lipschitz algebras Lipα(X) and some matrix algebras are studied under this new notion. Also, with some mild assumptions, the relation between IConnes biprojectivity and left ϕ-contractibility is given, where ϕ is a ω∗-continuous multiplicative linear functional on A. As an application, we characterize Connes biprojectivity of some Lipschitz algebras. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a simple graph. A set D ⊆ V is a strong dominating set of G, if for every vertex x ∈ V \ D there is a vertex y ∈ D with xy ∈ E(G) and deg(x) ≤ deg(y). The strong domination number γst(G) is defined as the minimum cardinality of a strong dominating set. In this paper, we study the effects on γst(G) when G is modified by operations on vertices and edges of G. [ABSTRACT FROM AUTHOR]

Conditional Independence (CI) graphs are a type of Probabilistic Graphical Models that are primarily used to gain insights about feature relationships. Each edge represents the partial correlation between the connected features which gives information about their direct dependence. In this survey, we list different methods and study the advances in techniques developed to recover CI graphs. We cover traditional optimization methods as well as recently developed deep learning architectures along with their recommended implementations. To facilitate wider adoption, we include preliminaries that consolidate associated operations, for example techniques to obtain covariance matrix for mixed datatypes. [ABSTRACT FROM AUTHOR]

As the interdependence between arbitrary objects in the real world grows, it becomes gradually important to use chain graphs containing directed and undirected edges to learn the structure among objects. However, independence among some variables corresponds to multiple structures and the direction of edges among variables cannot be uniquely determined. This limitation restricts existing chain graphs structure learning algorithms to only learning their Markov equivalence class. To alleviate this limitation, we define the block domain knowledge and propose a block domain knowledge-driven learning chain graphs structure algorithm (KDLCG). The KDLCG algorithm learns the adjacencies and spouses of all variables, which are utilized to directly construct the skeleton and orient the edges of the complexes, thereby learning the Markov equivalence class of the chain graphs. Subsequently, the KDLCG algorithm then updates some edges with Meek rules, guided by block domain knowledge. Finally, the KDLCG algorithm directs some edges by estimating causal effects between two variables, driven by block domain knowledge. Meanwhile, we conduct theoretical analysis to prove the correctness of our algorithm and compare it with the LCD algorithm and MBLWF algorithm on synthetic and real-world datasets. The experimental results validate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

An edge e of a matching covered graph G is removable if G − e is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick G different from K4 and C6 has at least ∆ − 2 removable edges, where ∆ is the maximum degree of G. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more. [ABSTRACT FROM AUTHOR]

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if G = C d 2k is the dth power of the cartesian product of the even cycle C2k with itself, and at most 2d−1 edges of G are precolored, then there is a proper 2d-edge coloring of G that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least 4 from each other. For odd cycles of length at least 5, we prove that if G = C d 2k+1 is the dth power of the cartesian product of the odd cycle C2k+1 with itself (k ≥ 2), and at most 2d edges of G are precolored, then there is a proper (2d+1)-edge coloring of G that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410–444]. [ABSTRACT FROM AUTHOR]

A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge have the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h(H) is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound h(H) = O( k√k k!m) on the harmonious number of hypergraphs of maximum degree ∆ with m edges. We use the local cut lemma of A. Bernshteyn. [ABSTRACT FROM AUTHOR]

Graph and digraph decompositions are a fundamental part of design theory. Probably the best known decompositions are related to decomposing the complete graph into 3-cycles (which correspond to Steiner triple systems), and decomposing the complete digraph into orientations of a 3-cycle (the two possible orientations of a 3-cycle correspond to directed triple systems and Mendelsohn triple systems). Decompositions of the λ -fold complete graph and the λ -fold complete digraph have been explored, giving generalizations of decompositions of complete simple graphs and digraphs. Decompositions of the complete mixed graph (which contains an edge and two distinct arcs between every two vertices) have also been explored in recent years. Since the complete mixed graph has twice as many arcs as edges, an isomorphic decomposition of a complete mixed graph into copies of a sub-mixed graph must involve a sub-mixed graph with twice as many arcs as edges. A partial orientation of a 6-star with two edges and four arcs is an example of such a mixed graph; there are five such mixed stars. In this paper, we give necessary and sufficient conditions for a decomposition of the λ -fold complete mixed graph into each of these five mixed stars for all λ > 1 . [ABSTRACT FROM AUTHOR]

The concept of the Sombor indices of a graph was introduced by Gutman. A vertex-edge variant of the Sombor index of graphs is called the KG-Sombor index. Recently, the Sombor and KG-Sombor indices of Kragujevac trees were studied, and the extremal Kragujevac trees with respect to these indices were empirically determined. Here we give analytical proof of the results. [ABSTRACT FROM AUTHOR]

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order 12p, where p is a prime. [ABSTRACT FROM AUTHOR]

A defective vertex coloring of a graph is a coloring in which some adjacent vertices may have the same color. An edge whose adjacent vertices have the same color is called a bad edge. A defective coloring of a graph G with minimum possible number of bad edges in G is known as a near proper coloring of G. In this paper, we introduce the notion of equitable near proper coloring of graphs and determine the minimum number of bad edges obtained from an equitable near proper coloring of some graph classes. [ABSTRACT FROM AUTHOR]

Most of the photonic valley-Hall edge states were constructed by changing structures to break the spatial inversion symmetry, restricting the practical application potential. In this paper, we construct a tunable topological valley-Hall edge state based on the large optical Kerr effect. It is demonstrated that topological phase transition happens by engineering the intensity of the injected pump and that a valley-Hall edge state can be generated at the interface between two regions with different topological properties. In addition, eigenfrequency and transmission characteristics of the edge state as a function of applied pump intensity are investigated. The topological protected valley-dependent transmission is studied under non-uniform distributed pump intensity. This work may open a new path toward designing reconfigurable all-optical metadevices. [ABSTRACT FROM AUTHOR]

The agglomeration of thin films on substrates is driven by minimization of the free surface and film/substrate interface energies and has been studied extensively for single component metal films. Only a few studies have investigated the agglomeration of kinetically constrained metal bilayer films, for which unusual long-range edge retraction was recently reported. This study has explored the agglomeration of kinetically constrained thin films of Au and Ni that were subsequently deposited on SiO2/Si substrates and annealed under high vacuum conditions at 545, 675, and 730 °C. Long-range edge retraction of the metal bilayer films revealed seven regions across the receding edge that are microstructurally distinct. The absolute and relative widths of the regions depend on the deposition sequence of the two metal films and annealing temperature. Arrhenius analysis of growth rates for different regions was used to identify energy barriers for mass transport mechanisms. The presence of native nickel oxide was found to have a significant effect on the kinetics of long-range edge retraction. The experimental results suggest that the formation of multiple regions across the receding edge is part of the kinetic evolution of long-range edge retraction of metal bilayer films. [ABSTRACT FROM AUTHOR]

Single functionality and fixed operating bands maintain the key drawbacks in existing acoustic topological insulators. Here, we report an acoustic system to realize the frequency band-selected one-way topological edge state transmission. The system is combined with a double-layer sonic crystal and a lossy acoustic metasurface. The topological insulators enable the frequency band-selected effect by separating and nesting the two layers of the sonic crystals. The sound one-way transmission effect is ensured by the metasurface. Consequently, the one-way topological edge state transmission is realized, and the operating band of the system can be shifted between two distinct ranges. Our work may have potential in the areas where multiple frequency bands are required, such as sound isolation, acoustic switch, mechanical imaging, acoustic split, and integrated acoustic communications. [ABSTRACT FROM AUTHOR]

We have investigated the magnetization dynamics in single and trilayer circular permalloy nanodots with a diameter of 120 nm using broadband ferromagnetic resonance spectroscopy. For single-layer nanodots, two well-separated modes near the saturation field, a high-frequency center mode due to excitations at the center of the nanodots and a low-frequency edge mode due to the inhomogeneous effective field near the edges, were observed. Both the center mode and the edge mode are found to be sensitive to the thickness of the nanodots. However, for trilayer nanodots, two center modes arise due to the in-phase and out-of-phase precession of spins in magneto-dynamically coupled layers. Our experimental results are substantiated by micromagnetic simulations, which are in good agreement. [ABSTRACT FROM AUTHOR]

The surface of a three-dimensional ice crystal naturally has a quasi-liquid layer (QLL) at temperatures below its bulk melting point, due to a phenomenon called surface premelting. Here, we show that the edges of a two-dimensional (2D) bilayer hexagonal ice adsorbed on solid surfaces undergo premelting as well, resulting in the formation of quasi-liquid bands (QLBs) at the edges. Our extensive molecular dynamics simulations show that the QLB exhibits structure and dynamics indistinguishable from the bilayer liquid phase, acting as a lower-dimensional analog of the QLL on the bulk ice. We further find that at low temperatures, the width of the QLBs at armchair-type edges of the 2D ice is almost identical to that at zigzag-type edges but becomes far greater than the latter at temperatures near the melting point. The chirality-dependent edge premelting of 2D ices should add an important new ingredient to the heterogeneity of premelting. [ABSTRACT FROM AUTHOR]

We prove that for k = 4 v n regular resolution requires length nO(k) to establish that an Erdős-Rényi graph with appropriately chosen edge density does not contain a k-clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional nO(k) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs. [ABSTRACT FROM AUTHOR]

Let S be a collection of elements in a vertex set V. If every vertex in a graph G falls on a geodesic connecting two vertices from S, then that graph is said to be a geodesic graph. g(G) is the smallest cardinality of the geodesic subset of a graph G and is known as the geodetic number. This study investigates how the removal of an edge affects the geodetic number of some unique families. [ABSTRACT FROM AUTHOR]

We present a design to achieve antichiral edge states in acoustic systems where edge states on the two parallel edges of a lattice with a strip geometry propagate in the same direction. This peculiar phenomenon is realized by using a honeycomb lattice consisting of acoustic resonators with staggered air flow; i.e., the air flow takes opposite directions in resonators belonging to different sublattices. The existence of antichiral edge states is revealed through full-wave simulations of the band structure and acoustic fields excited by a point source. Furthermore, we compare these antichiral edge states with conventional chiral edge states. It is found that the antichiral edge states are less robust than the chiral ones. Our work offers new possibilities for dispersion engineering and wave manipulations in acoustics. [ABSTRACT FROM AUTHOR]

The edge elasticity and its effect on flexoelectric response of the Janus MoSSe nanoribbons are systematically explored by means of density functional theory based first-principles calculations. We report edge stresses, edge elastic moduli, and structural deformations of the Janus MoSSe nanoribbons with various widths. It is shown that both armchair and zigzag terminated edges of the MoSSe nanoribbons are essentially subjected to tension, due to the existence of the edge stresses. The magnitude of average zigzag edge stresses is much larger than that of the average armchair ones. Furthermore, our results show that both misfit strain induced by asymmetric chalcogen atomic layers, and the edge stresses cause the spontaneous bending deformation of such Janus nanoribbons. More importantly, flexoelectronic properties of semiconducting armchair MoSSe nanoribbons are carefully evaluated and compared with those of armchair MoS 2 and MoSTe nanoribbons. In particular, it is found that the out-of-plane flexoelectronic coefficients strongly depend on their widths. Additionally, the flexoelectric response resulting from spontaneous bending is weaker than that from the opposite one. The implicit mechanisms on deformations and flexoelectronic properties of such Janus nanoribbons have been carefully explored. The results in this work provide useful insights into their potential applications in nanoscale electromechanical systems. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a finite, simple, and undirected graph without an isolated vertex. We define a dominating D of V (G) as a total pitchfork dominating set if 1 ≤ |N(t) ∩ V - D| ≤ 2 for every t ∈ D such that G[D] has no isolated vertex. In this paper, the effects of adding or removing an edge and removing a vertex from a graph are studied on the order of minimum total pitchfork dominating set γt pf (G) and the order of minimum inverse total pitchfork dominating set γ-t pf (G). Where γt pf (G) is proved here to be increasing by adding an edge and decreasing by removing an edge, which are impossible cases in the ordinary total domination number. [ABSTRACT FROM AUTHOR]

Scientists, in the public imagination, are intrepid explorers. They stand at the edge of human knowledge and carve out new territory, break down barriers and rewrite the rules. Nature has no political ideology and carries no passport. Science, at its best, also espouses such cosmopolitan ideals. That data is neutral, and science is apolitical, makes for an alluring narrative. By clinging to it, the scientist appears assured, almost noble, rising above the messy and the mundane by sheer force of intellect. But reality does not conform to such convenient self-delusion. In this essay, the author argues that pretending to be above and beyond politics is by itself a political position; in adopting it, one has aligned with the state and sided with the powerful. [ABSTRACT FROM AUTHOR]

We demonstrate the influence of edge effects on the photothermal-induced phase shift measured by a homodyne quadrature laser interferometer and compare the experiments with rigorous theoretical descriptions of thermoelastic surface displacement of metals. The finite geometry of the samples is crucial in determining how the temperature is distributed across the material and how this affects the interferometer phase shift measurements. The optical path change due to the surface thermoelastic deformation and thermal lens in the surrounding air is decoded from the interferometric signal using analytical and numerical tools. The boundary/edge effects are found to be relevant to properly describe the interferometric signals. The tools developed in this study provide a framework for the study of finite size effects in heat transport in opaque materials and are applicable to describe not only the phase shift sensed by the interferometer but also to contribute to the photothermal-based technologies employing similar detection mechanisms. [ABSTRACT FROM AUTHOR]

The physical characteristics of the fundamental absorption edge of semi-insulating GaAs and unintentionally doped GaSb, InAs, and InSb are examined using spectroscopic ellipsometry. A five parameter model is developed to describe the key characteristics of the absorption edge. Among these parameters are the bandgap energy, the characteristic energy of the Urbach tail, and the absorption coefficient at the bandgap energy. The results indicate that the Coulomb interaction strongly influences the shape of the band edge with progressively less influence as the bandgap energy decreases. The energy dependence of the optical transition strength is observed to be nearly constant in narrow bandgap InSb. [ABSTRACT FROM AUTHOR]

Apertureless scanning near-field optical microscopy mapping of a slightly doped 4H-SiC epitaxial layer grown on a heavily doped 4H-SiC substrate was performed in a cleaved edge geometry. Surface phonon polariton waves excited by an external coherent light were observed on a sample surface that contains such an epilayer-defined strip near its edge. The light frequency was tuned close to the lattice resonance. Due to a low doping level in an epilayer, its electromagnetic response is determined mainly by the SiC lattice resonance. The rest of the sample surface corresponds to a substrate whose electromagnetic response is determined mainly by the free carriers so that phonon polariton phenomena get suppressed. Such an epilayer-defined strip (vanishing at 895 cm − 1 frequency) becomes more pronounced at 920 cm − 1 and, finally, the excited state gets completely confined within such a strip (938 cm − 1 ) due to the differences in the electromagnetic properties of doped and undoped SiC. [ABSTRACT FROM AUTHOR]

A facial total-coloring of a plane graph G is a coloring of the vertices and edges such that no facially adjacent edges, no adjacent vertices, and no edge and its endvertices are assigned the same color. A facial total-coloring of G is odd if for every face f and every color c, either no element or an odd number of elements incident with f is colored by c. In this note we prove that every cactus forest with an outerplane embedding admits an odd facial total-coloring with at most 16 colors. Moreover, this bound is tight. [ABSTRACT FROM AUTHOR]

A type (1; 1; 1) face-magic labeling of a planar graph G = (V;E; F) is a bijection from V [E [F to the set of labels f1; 2;: :: ; jV j + jEj + jFjg such that the weight of every n-sided face of G is equal to the same fixed constant. The weight of a face F 2 F is equal to the sum of the labels of the vertices, edges, and face that determine F. It is known that the grid graph Pm-Pn admits a type (1; 1; 1) face-magic labeling, but the proof in the literature is quite lengthy. We give a simple proof of this result and show two more infinite families of gridded graphs admit type (1; 1; 1) face-magic labelings. [ABSTRACT FROM AUTHOR]

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V , E), where V and E are the vertex set and the edge set of G , respectively. The order and size of G is the number of vertices and edges of G , respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G , deg (ν) = k , we say that G is a k -regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipartite graph is a graph G with and such that every vertex of the set (part) X is connected to every vertex of the set (part) Y. If , then this graph is denoted by Km,n. The complete bipartite graph K1,n is called the star graph which has n + 1 vertices. The distance between two vertices u and ν of G , denoted by d (u , ν), is defined as the minimum number of edges of the walks between them. The complement of graph G is denoted by and is a graph with the same vertices such that two distinct vertices of are adjacent if, and only if, they are not adjacent in G. For more information on graphs, refer to [1]. [ABSTRACT FROM AUTHOR]

A function f: E (G) {0,1,2} is an edge Italian dominating function (EIDF) if it satisfies the rule that every edge with weight 0 is either adjacent to an edge with weight 2 or adjacent to at least two edges with weight 1 each. The weight of an EIDF is S. The minimum is the edge Italian domination number (EIDN). The symbol is used to denote the EIDN. In this paper, we obtain the EIDN of some wheel related graphs like gear graph, helm graph, flower graph, web graph etc. [ABSTRACT FROM AUTHOR]

The Sigma colouring (σ - colouring) of a graph σ with G vertices is an injection from V(G) to {1,2,...} such that the colour sums (adding the colours of the neighbouring vertices) of any two neighbouring vertices are different. The smallest number of colours needed to colour a graph G is represented by its Sigma Chromatic number, σ(G). In this article we obtain the σ-colouring of some graphs such as Barbell Graph, Twig graph, Shell graph, Tadpole, Lollipop, Fusing all the vertices of cycle and duplication of every edge by a vertex in Cn. [ABSTRACT FROM AUTHOR]

Any line graph L(G), the vertices correspond to the edges of G(V, E) and two vertices in L(G) are adjacent if and only if the corresponding edges in G are adjacent". "If there are vertices u, v in x, y such that dg, G -[G,-] for any pair of vertices S, G in G, then the set G of vertices of f is said to be a double geodetic set of The lowest cardinality of a double geodetic set is represented by the double geodetic number. In this study, we determine double geodetic number of several line graphs. [ABSTRACT FROM AUTHOR]

The change of the ion transport is investigated with the variation of the focus ring property at the wafer edge of a capacitively coupled plasma under an intermediate pressure of a few Torr. The particle fluxes and the ion trajectories at different locations are investigated with the variations of the gap size between the wafer edge and the focus ring, the focus ring height, and the permittivity of the focus ring. The incident angle and the particle fluxes to the wafer edge increase with the gap size. Conversely, the particle fluxes to the wafer edge decrease with the increase in the focus ring height. The incident angle of ions still keeps normal to the surface at the wafer edge, but on the left side of the focus ring, it increases dramatically with the increase in the focus ring height. With the change of the permittivity of the focus ring, it is possible to control the ratio of the ion flux to the neutral flux on the focus ring surface by enhancing only the ion flux independently. [ABSTRACT FROM AUTHOR]

We propose a reconfigurable photonic crystal based on split-ring structures, which hosts tunable edge states by controlling the rotation angle of the split-rings. The split-ring structure breaks the inversion symmetry and introduces a nontrivial Dirac mass in the otherwise gapless Dirac photonic spectrum. The sign of the Dirac mass depends on the rotation angle that thus introduces two topologically distinct phases. It is shown that an interface between two split-ring photonic crystals with opposite rotation angles supports gapped edge states. Despite the topologically trivial nature of the split-ring photonic crystal, the dispersion of the edge states is tunable through the rotation angle of the split-ring, making it useful in frequency-selective beam splitters. Our study provides an alternative way for the controlling of edge states and thus can be useful for future integrated photonic circuits. [ABSTRACT FROM AUTHOR]