N. K. Geetha ¹

Affiliations
1 Department of Mathematics, Saveetha School of Engineering, Saveetha University, Chennai - 600072, Tamil Nadu, IN

Abstract

Cubic graph, where all vertices have degree three can be associated as 3-regular graph and trivalent graph. Let a point P be the given point set, with a stipulation of n ≥ 4 points in the regular plane such that n is even in general situation. It has been probed that the problem whether there is a straight-line cubic plane graph on P. Algorithm is not known for polynomialtime of a problem. With research on the reduction to the existence of the convex hull of P along with the certain diagonals of the boundary cycle, the polynomial-time algorithm is proposed that checks for 2-connected cubic plane graphs. The algorithm is constructive and runs in time complexity of O (n3). It is also shown that which graph structure can be expected when there is a cubic plane graph on P; e. g., a cubic plane graph on P implies a connected cubic plane graph on P, and a 2-connected cubic plane graph on P implies a 2-connected cubic plane graph on P that contains the boundary cycle of P.

Keywords

Algorithm, Cubic Graph, Euler's Formula Plane Graph, Polynomial Time Algorithm, Trivalent Graph

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N. Sivakumar ¹, N. K. Geetha ², J. Venkatamuthukumar ¹, V. Jayakumar ³

Affiliations
1 Department of Physics, School of Engineering, Saveetha University, Thandalam, Chennai - 602105, Tamil Nadu, IN
2 Department of Mathematics, School of Engineering, Saveetha University, Thandalam, Chennai - 602105, Tamil Nadu, IN
3 Department of Mechanical Engineering, School of Engineering, Saveetha University, Thandalm, Chennai - 602 105, Tamil Nadu

Abstract

Background/Objectives: Mechanical properties of the selected materials play an important role in device fabrications. Thus the perfect cleavage (010) plane of doped and un-doped Potassium Hydrogen Phthalate (KHP) crystal is subjected to Vicker’s micro hardness study. Methods/Statistical Analysis: Slow evaporation solution growth method is employed for the growth of Fe³⁺ doped and un-doped Potassium Hydrogen Phthalate (KHP) crystals. Here, 0.1 mol% of FeCl₃ used as dopant. Saturated solution of both doped and un-doped KHP solutions are prepared according to the solubility data (12.5g/100ml at 30°C). Findings: A well developed (010) plane of pure and un-doped crystals have been subjected to hardness studies with various loads. It is found that Vickers Hardness numbers (H_V) for undoped and doped KHP crystals varied for different loads. Meyer’s index or work hardening co-efficient (n) values are found to be greater than 1.6 which reveals that the grown crystals are belongs to soft material’s category. i.e., for pure KHP the value of ‘n’ is 3.68 and for Fe doped KHP, it is 3.46. The minimum load indentation (W) to initiate the plastic deformation on the surface of the crystals are calculated based on Hays and Kendall’s theory. The value of W for KHP and doped KHP are found to be 1.965 and 4.368 respectively. The hardness related constants like materials constant (k₁) and load dependent constant (A₁) for the pure and un-doped KHP crystals have been estimated. The Elastic stiffness constant (C₁₁) are also calculated from Vickers micro hardness values. Applications/Improvements: Potassium Hydrogen Phthalate (KHP) is an efficient crystal analyzer material and is used in X-ray spectrometer. The improvement in hardness by dopant provides the use of materials for efficient optical device applications.

Keywords

Elastic Stiffness Constant, Load Dependent Constant, Material Constant, Meyer’s Index, Minimum Load Indentation, Vickers Hardness Number.

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N. K. Geetha ¹, N. Siva Kumar ², P. Sekar ³

Affiliations
1 Department of Mathematics, Saveetha School of Engineering, Saveetha University, Chennai - 602105, Tamil Nadu, IN
2 Department of Physics, Saveetha School of Engineering, Saveetha University, Chennai - 602105, Tamil Nadu, IN
3 Department of Mathematics, C. Kandaswami Naidu College for Men, Anna Nagar, Chennai – 600102, Tamil Nadu, IN

Abstract

Background/Objectives: Graph theory matrix approach is a logical and systematical approach originated from combinatorial mathematics. Graph theory matrix approach is adopted to find the optimal combination of operating parameters. Methods/Statistical Analysis: Graph theory matrix approach helps to analyze and understand the system as a whole by identifying system and sub-system up to the component level. Attributes digraph is developed to represent the inheritance and the interdependencies of the subsystems. Matrix method is adopted to convert the digraph into mathematical form. Permanent function is deduced to determine the parameter index to find the optimal combination of operating parameters on a diesel engine. Findings: The combination of 18 Ampere load, 270 BTDC Injection timing and 200 bar Injection pressure forms the optimal combination of operating parameters having the highest value of Permanent index. Applications/Improvements: Graph theory matrix approach offers simple, generic, easy and convenient computation. It finds applications in the fields of education, neural networks, automotive industry, manufacturing, electronic devices, total quality management, location of plants, supply chain management, information technology, human resource selection etc.

Keywords

Engine, Graph Theory, Matrix Approach, Operating Parameter, Permanent Index.

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N. K. Geetha ¹, P. Sekar ¹

Affiliations
1 Department of Mathematics, C. Kandaswami Naidu College for Men, Anna Nagar, Chennai - 600102, IN

Abstract

Background: Graph theory matrix approach as a decision making tool. Methods: Graph theory matrix approach for determining the permanent index in comparing and ranking the alternatives in decision making. Findings: The best combination of attributes that influence the system can be found out using this combinatorial approach. Application: Decision making can be done appropriately using this approach.

Keywords

Digraph, GTMA, Matrix, Permanent Function, Permanent Index

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