Shadowgraph Research Articles

Improved Foucault method with double-shoot shadow grams in optical shop testing.

The lack of rapid, effective, and quantitative testing methods in the initial stages of optical processing has been identified as a significant challenge. In response, an innovative approach has been developed to enhance the Foucault knife-edge test method, integrating digital image processing and spatial geometry algorithms. This has resulted in the creation of a convenient and rapid optical testing method, which has been successfully implemented in practice. The method requires the collection of only two knife-edge shadow grams, which are then subjected to calculation. This enables the theoretical results of the knife-edge shadow gram at the focal point, and the reference results of the wavefront surface processing error of the mirror to be measured can be obtained. The method can be well applied to the testing work of optical workshops and provides a guiding solution with better performance for optical testing than the traditional knife-edge testing method.

Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, \cdots, d_n \rangle$, where $d_1 \ge d_2 \ge \cdots \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{\Delta(G)}a_kx^{k}$, where $a_k$ is the number of vertices of $G$ having degree $k$ for each $i = 1,2,\cdots n = \Delta(G)$. In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.

SDD Spectral Radii and SDD Energies of Graph Operations

Graph energies and spectral radii have application in molecular computing and computer sciences. In this paper, we concentrate on new findings about the Symmetric Division Deg, (SDD) energies, and (SDD) spectral radii of p splitting and p shadow graphs built on the foundation of any regular graph. In fact it becomes natural to ask how the SDD energies and spectral radii of newly formed graphs depend on the SDD energies and spectral radii of the given regular graph.

On D-distance (anti)magic labelings of shadow graph of some graphs

On D-distance (anti)magic labelings of shadow graph of some graphs

On Maximum Degree and Maximum Reverse Degree Energies of Splitting and Shadow graph of Complete graph

In this paper, the relations of maximum degree energy and maximum reserve degree energy of a complete graph after removing a vertex have been shown to be proportional to the energy of the complete graph. The results of splitting the graph and shadow graphs are also presented for the complete graph after removing a vertex.

Further results on edge irregularity strength of some graphs

The focal point of this paper is to precisely ascertain the edge irregularity strength of various finite, simple, and undirected captivating graphs, including splitting graph, shadow graph, jewel graph, jellyfish graph, and $m$ copies of 4-pan graph.

Characterization of Fractional Mixed Domination Number of Paths and Cycles

Let G′ be a simple, connected, and undirected (UD) graph with the vertex set M(G′) and an edge set N(G′). In this article, we define a function f:M∪N⟶0,1 as a fractional mixed dominating function (FMXDF) if it satisfies fRmx=∑yϵRmxfy≥1 for all x∈MG′∪NG′, where Rmx indicates the closed mixed neighbourhood of x, that is the set of all y∈MG′∪NG′ such that y is adjacent to x and y is incident with x and also x itself. Here, pf=∑x∈M∪Nfx is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by γfm∗G′ and is designated as the lowest poundage among all FMXDFs of G′. We compute the FMXDN of some common graphs such as paths, cycles, and star graphs, the middle graph of paths and cycles, and shadow graphs. Furthermore, we compute upper bounds for the sum of the two fractional dominating parameters, resulting in the inequality γf1′Τ+γfm∗Τ≤r+p−radΤ−α, where γf1′ and γfm∗ are the fractional edge domination number and FMXDN, respectively. Finally, we compare γfm∗ to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.

$J^2$-Hop Domination in Graphs: Properties and Connections with Other Parameters

A subset $T=\{v_1, v_2, \cdots, v_m\}$ of vertices of an undirected graph $G$ is called a $J^2$-set if$N_G^2[v_i]\setminus N_G^2[v_j]\neq \varnothing$ for every $i\neq j$, where $i,j\in\{1, 2, \ldots, m\}$.A $J^2$-set is called a $J^2$-hop dominating in $G$ if for every $a\in V(G)\s T$, there exists $b\in T$ such that $d_G(a,b)=2$. The $J^2$-hop domination number of $G$, denoted by $\gamma_{J^2h}(G)$, is the maximum cardinality among all $J^2$-hop dominating sets in $G$. In this paper, we introduce this new parameter and wedetermine its connections with other known parameters in graph theory. We derive its bounds with respect to the order of a graph and other known parameters on a generalized graph, join and corona of two graphs. Moreover,we obtain exact values of the parameter for some special graphs and shadow graphs using the characterization results that are formulated in this study.

Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

Closeness Centrality of Vertices in Graphs Under Some Operations

In this paper, we revisit the concept of (normalized) closeness centrality of a vertex in a graph and investigate it in some graphs under some operations. Specifically, we derive formulas that compute the closeness centrality of vertices in the shadow graph, complementary prism, edge corona, and disjunction of graphs.

ISI spectral radii and ISI energies of graph operations

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.

Prime Labeling of Nauru Graph

A graph G = (V(G), E(G)) is observed to admit prime labeling, if a graph that receives prime labeling is called prime graph .In this research article we investigate that the Nauru graph admits prime labeling. We construct the mirror graph and shadow graph of the Nauru graph. We also establish prime labeling using some graph operations such as duplication. switching and fusion with few ideas.

Open global shadow graph and it’s zero forcing number

Zero forcing number of a graph is the minimum cardinality of the zero forcing set. A zero forcing set is a set of black vertices of minimum cardinality that can colour the entire graph black using the color change rule: each vertex of G is coloured either white or black, and vertex v is a black vertex and can force a white neighbour only if it has one white neighbour. In this paper we identify a class of graph where the zero forcing number is equal to the minimum rank of the graph and call it as a new class of graph that is open global shadow graph”. Some of the basic properties of open global shadow graph are studied. The zero forcing number of open global shadow graph of a graph with upper and lower bound is obtained. Hence giving the upper and lower bound for the minimum rank of the graph.

Automatic Foucault test method based on transverse cutting at two different axial positions near the foci

Foucault knife-edge method is an irreplaceable qualitative testing method in the early stage of optical processing, but it needs to accurately place the knife-edge at the focus of the mirror surface to be measured for testing. In view of the complex and difficult calibration process of the knife-edge instrument and the inability to perform quantitative testing efficiently, here the Foucault knife-edge testing method is improved by digitizing and automating it. The knife-edge instrument is first placed on the motorized precision translation stage for two programmed cutting movement perpendicular to the optical axis with controlled axial spacing. And the normal vector of each point on the mirror surface to be measured is solved through the knife-edge transverse position relationship for the bright and dark transition of the corresponding point on a series of knife-edge shadow grams obtained by two cuts. Then the automation and quantification of knife-edge method can be realized. The experimental results of this method are compared with the interferometer testing results, which have good consistency and high accuracy. This method does not require high calibration accuracy for the knife-edge instrument and can perform rapid and efficient testing. It provides a testing method with good performance for automatic and quantitative testing in the early stage of optical processing.

On sombor index of m-shadow graph of some graphs

This new topological invariant is applied in the areas of chemical graph theory. In this paper we have obtained the Sombor index of m−shadow graphs of path graph Dm(Pn), cycle graph Dm(Cn), ladder graph Dm(Ln) and tadpole graph Dm(Tn,p). We have also concluded that, the Sombor index of m-shadow graph of any regular or finite connected graph G is SO(G)m3

On equitable near-proper coloring of some derived graph classes

An equitable near-proper coloring of a graph $G$ is a defective coloring in which the number of vertices in any two color classes differ by at most one and the bad edges obtained is minimised by restricting the number of color classes that can have adjacency among their own elements. This paper investigates the equitable near-proper coloring of some derived graph classes like Mycielski graphs, splitting graphs and shadow graphs.

ABC energies and spectral radii of some graph operations

The present article presents some new results relating to Atomic Bond Connectivity energies and Spectral radii of generalized splitting and generalized shadow graphs constructed on the basis of some fundamental families of cycle graph Cn, complete graph Kn and complete bipartite graph Kn,n referred as base graphs. In fact we relate the energies and Spectral radii of splitting and shadow graphs with the energies and Spectral radii of original graphs.

Quantum sets of the multicolored-graph approach to contextuality

The CHSH inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter (CSW) came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the CSW approach, one may think that the theta body of the CHSH graph, is equal to the quantum set of the CHSH Bell inequality, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while CSW approach only demands the exclusivity structure of a non-contextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like CHSH, the colored-graph approach was introduced. Does it make any difference to think about CHSH as a Bell scenario or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph and the NC CHSH inequality by a simple graph, which is the shadow of the colored graph. In general, we have that the theta body of the colored graph is a subset of the theta body of its shadow graph in the same way that the Lov\`asz number, which corresponds to the quantum bound, of the simple graph is greater than or equal to the Lov\`asz number of the colored graph. In the case of the CHSH inequality, we have that the Lov\`asz numbers are equal. Does this accident also hold for the corresponding quantum sets? Is it true that the theta bodies are equal, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negativ. This implies that there are quantum correlations which can not be obtained under Bell restrictions.

Lucas Antimagic Labeling of some Star Related Graphs

Objective: To identify a new family of Lucas antimagic graph. Methods: A (p;q) graph G is said to be a Lucas antimagic graph if there exists a bijection f : E(G) ! fL1;L2;


Lqg such that the induced injective function f  : V(G) ! f1;2; : : :åLqg given by f  (u) = åe2E(u) f (e) are all distinct (where E(u) is the set of edges incident to u). Findings: In this paper the Lucas Antimagic Labeling of Subdivision of star, Shadow graph of star, Splitting graph of star, Subdivision of Bistar, Shadow graph of Bistar, Splitting graph of Bistar are found. Novelty: It involves the mathematical formulation for labeling the edges of a given graph which in turn gives rise to a new type of labeling called the Lucas antimagic labeling. Keywords: Subdivision graph; Shadow graph; Splitting graph; Star; Bistar

Hop Independent Hop Domination in Graphs

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is called a hop independent hop dominating set of G if S is both hop independent and hop dominating set of G. The minimum cardinality of hop independent hop dominating set of G, denoted by γhih(G), is called the hop independent hop domination number of G. In this paper, we show that the hop independent hop domination number of a graph G lies between the hop domination number and the hop independence number of graph G. We characterize these types of sets in the shadow graph, join, corona, and lexicographic product of two graphs. Moreover, either exact values or bounds of the hop independent hop domination numbers of these graphs are given.