In this paper, I propose an algorithm capable of solving the problem of isomorphic graphs in polynomial time. First, I define a pseudo tree that allows us to define for each vertex a label or a label. Secondly, I apply the pseudo tree for the first graph then I calculate the labels of each vertex of the first graph, then I do the same for the second graph. Thirdly, I look for for each graph vertex1 the graph vertices2 which have the same label, if or less a first graph vertex its label is not in the second graph vertices we deduce that the two pseudo trees are not isomorphic. In other cases I generate solutions and check them in polynomial time... This algorithm therefore allows isomorphic graphs to calculate the image of each vertex in polynomial time.
1. The document discusses strategies for students to score 10/10 grade in mathematics. It recommends practicing previous year question papers to understand concepts better.
2. It provides two sample question papers containing math problems like addition, subtraction, multiplication and division of integers, fractions and decimals. It notes that there are extra marks questions in both papers.
3. Studying the question papers carefully and understanding the logic behind extra marks questions will help students solve similar questions correctly and score full marks.
This document provides information about number patterns and sequences. It discusses arithmetic sequences where the difference between consecutive terms is constant, and quadratic patterns where the second differences are constant. Examples are provided to demonstrate how to determine the general term of a sequence from a given pattern. Practice problems with solutions allow the reader to apply the concepts to find missing terms, general terms, and values of specific terms.
This document provides an introduction to solving inequalities. It begins with essential questions and key vocabulary related to inequalities, including set-builder and interval notation. It then shows four examples of solving different types of one-step and multi-step inequalities. Each example solves the inequality, expresses the solution set using set-builder and interval notation, and graphs it on a number line. The examples demonstrate how to isolate the variable, determine if the inequality sign flips, and write the solution set in the two different notations.
Cs6702 graph theory and applications Anna University question paper apr may 2...appasami
This document contains a past exam paper for the subject "Graph Theory and Applications". It includes 15 multiple choice and long answer questions covering topics such as Euler graphs, planar graphs, spanning trees, matchings, counting principles, and graph algorithms. The key provided contains concise definitions and explanations for the concepts and steps to solve problems in 3 sentences or less.
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
3.complex numbers Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
This document introduces complex numbers and their algebra. It discusses how quadratic equations can lead to complex number solutions and how to represent complex numbers in the forms a + bi and rcis(θ). It then covers the basic arithmetic operations of addition, subtraction, multiplication and division of complex numbers. It provides examples of solving equations with complex number solutions. The key points are:
- Complex numbers allow solutions to quadratic equations that have no real number solutions.
- Complex numbers can be represented as a + bi or rcis(θ).
- Operations on complex numbers follow the same rules as real numbers but use i2 = -1.
- Equations with complex number variables can be solved using the same methods as real numbers
This document discusses surds, indices, and logarithms. It begins by defining radicals, surds, and irrational numbers. Some general rules for operations with surds like multiplication, division, and simplification are provided. The document then covers rules and operations for indices like exponentiation, roots, and properties like distributing exponents. Examples are given to demonstrate applying these index rules. The document concludes by defining logarithms as the inverse of exponentiation and provides an example equation.
This document provides information about slopes and equations of lines. It begins by defining key vocabulary like slope and different forms of line equations. It then gives examples of using the slope formula to find the slopes of lines between points and identifying whether lines are parallel, perpendicular, or neither based on their slopes. It concludes by stating the postulates for identifying parallel and perpendicular lines based on the relationship between their slopes.
1. The document discusses strategies for students to score 10/10 grade in mathematics. It recommends practicing previous year question papers to understand concepts better.
2. It provides two sample question papers containing math problems like addition, subtraction, multiplication and division of integers, fractions and decimals. It notes that there are extra marks questions in both papers.
3. Studying the question papers carefully and understanding the logic behind extra marks questions will help students solve similar questions correctly and score full marks.
This document provides information about number patterns and sequences. It discusses arithmetic sequences where the difference between consecutive terms is constant, and quadratic patterns where the second differences are constant. Examples are provided to demonstrate how to determine the general term of a sequence from a given pattern. Practice problems with solutions allow the reader to apply the concepts to find missing terms, general terms, and values of specific terms.
This document provides an introduction to solving inequalities. It begins with essential questions and key vocabulary related to inequalities, including set-builder and interval notation. It then shows four examples of solving different types of one-step and multi-step inequalities. Each example solves the inequality, expresses the solution set using set-builder and interval notation, and graphs it on a number line. The examples demonstrate how to isolate the variable, determine if the inequality sign flips, and write the solution set in the two different notations.
Cs6702 graph theory and applications Anna University question paper apr may 2...appasami
This document contains a past exam paper for the subject "Graph Theory and Applications". It includes 15 multiple choice and long answer questions covering topics such as Euler graphs, planar graphs, spanning trees, matchings, counting principles, and graph algorithms. The key provided contains concise definitions and explanations for the concepts and steps to solve problems in 3 sentences or less.
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
3.complex numbers Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
This document introduces complex numbers and their algebra. It discusses how quadratic equations can lead to complex number solutions and how to represent complex numbers in the forms a + bi and rcis(θ). It then covers the basic arithmetic operations of addition, subtraction, multiplication and division of complex numbers. It provides examples of solving equations with complex number solutions. The key points are:
- Complex numbers allow solutions to quadratic equations that have no real number solutions.
- Complex numbers can be represented as a + bi or rcis(θ).
- Operations on complex numbers follow the same rules as real numbers but use i2 = -1.
- Equations with complex number variables can be solved using the same methods as real numbers
This document discusses surds, indices, and logarithms. It begins by defining radicals, surds, and irrational numbers. Some general rules for operations with surds like multiplication, division, and simplification are provided. The document then covers rules and operations for indices like exponentiation, roots, and properties like distributing exponents. Examples are given to demonstrate applying these index rules. The document concludes by defining logarithms as the inverse of exponentiation and provides an example equation.
This document provides information about slopes and equations of lines. It begins by defining key vocabulary like slope and different forms of line equations. It then gives examples of using the slope formula to find the slopes of lines between points and identifying whether lines are parallel, perpendicular, or neither based on their slopes. It concludes by stating the postulates for identifying parallel and perpendicular lines based on the relationship between their slopes.
This document contains instructions and questions for a mathematics exam. It begins by providing spaces for the student to write their name, centre number, and candidate number. It then lists the total marks, time allowed, and materials permitted. The document contains 19 multiple choice and free response questions testing a variety of math skills like arithmetic, algebra, geometry, statistics, and graphing. It provides formulae for reference.
This document contains a mathematics exam for Secondary School students in Perak, Malaysia. It covers topics like sets, Venn diagrams, linear inequalities, simultaneous linear equations, quadratic equations and expressions, solid geometry, and mathematical reasoning. There are 10 multiple choice questions for each topic area testing students' understanding of key concepts and ability to solve related problems. The document is in Malay and contains diagrams to illustrate the questions.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
This document presents an algorithm for calculating the number of spanning trees in chained graphs. It begins by reviewing relevant graph theory concepts like planar graphs, spanning trees, and recursive formulas for counting spanning trees using deletion/contraction and splitting methods. It then derives explicit recursions for counting spanning trees in families of graphs like wheel graphs, fan graphs, and corn graphs. The main result is a theorem providing a system of equations to calculate the number of spanning trees in a chained graph based on splitting it into components and accounting for the connecting paths. Applications to counting spanning trees in chained wheel graphs and chained corn graphs are discussed.
Cs6702 graph theory and applications question bankappasami
This document contains questions and answers related to the course CS6702 Graph Theory and Applications. It is divided into 6 units which cover topics like introduction to graphs, trees, connectivity, planarity, graph colorings, directed graphs, permutations and combinations, and generating functions. For each unit, it lists short answer questions in Part A and longer proof or explanation questions in Part B. The questions test concepts fundamental to graph theory such as definitions of graphs, trees, connectivity, colorings, isomorphism, and counting methods applied to graph theory.
This document contains a mathematics exam paper with questions divided into multiple sections. Some key details:
- It is a 21⁄2 hour exam worth 50 marks total, divided into Part A and Part B.
- Part A contains 4 sections with various types of short and long answer questions on topics like real numbers, coordinate geometry, trigonometry, and mensuration.
- Part B contains shorter answer questions to be written directly on the question paper involving skills like interpreting logarithmic expressions and evaluating polynomials.
- The questions test a wide range of mathematics concepts and require calculations, proofs, formula applications, and reasoning about geometric shapes and algebraic expressions.
This document is the cover page and instructions for a 1 hour 45 minute GCSE Mathematics exam. It provides information such as the materials allowed, instructions for completing the exam, exam structure, and advice for students. The exam consists of 27 multiple choice and free response questions testing a variety of math skills, including algebra, geometry, statistics, and trigonometry. Students are advised to read questions carefully, watch the time, attempt all questions, and check their work. Calculators are not permitted.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
This document provides an overview of mathematical functions and relations through a series of lessons:
1. It defines key concepts like domains, ranges, and intervals used to describe functions and relations. Functions are defined as relations where no two ordered pairs have the same first element.
2. One-to-one functions are introduced, which satisfy both vertical and horizontal line tests. Only one-to-one functions can have inverse functions.
3. The process for finding the inverse of a function is described. The inverse is formed by swapping the inputs and outputs of the original function and solving for the new output. The domain of the original becomes the range of the inverse, and vice versa.
4
This document provides a review for algebra sections on various topics including:
1) Solving word problems involving equations with one unknown variable.
2) Finding sums, differences, and ratios of numbers.
3) Representing word problems using tables and equations.
4) Solving uniform motion problems using tables, diagrams, and the appropriate equation based on the type of motion described.
5) Graphing linear equations and finding slopes of lines from equations or two points.
6) Writing equations of lines in different forms given information like slopes, intercepts, or two points.
The document is an advertisement for Vedantu, an online education platform, promoting their free online admission test to win scholarships for classes 6-12, JEE, and NEET. It highlights success stories of students who scored well in board exams and engineering/medical entrance exams after taking online classes on Vedantu. It encourages students to register now for the admission test to secure limited seats and chance at 100% scholarship.
The document provides examples and explanations for solving linear equations. It begins by defining key vocabulary like open sentence, equation, and solution. It then shows how to translate between verbal and algebraic expressions. Various properties of equality like reflexive, symmetric, and transitive properties are explained. Finally, it demonstrates solving linear equations by isolating the variable using the inverse operations property of equality. Examples include solving equations with variables on both sides and checking solutions.
The document describes several numerical methods for finding the roots of functions including bisection, secant, and Newton's methods. It lists examples of functions and intervals that each method was applied to, including polynomials ranging from degrees 3 to 5 with various coefficients and constants. Muller's method is also mentioned as another root-finding algorithm examined.
Join Mr. D for a mini-SAT Boot Camp. Mr. D will be showing the tips and techniques from his SAT Boot Camps. This workshop will focus on math, the language behind the questions and show participants what they really need to know before taking the math section of the SAT.
Find out the words on the test that give tips for how to solve the questions as well learning what formulas you really need to know before the taking the SAT. Students and parents alike will learn how to unravel the questions being asked into something they can solve quickly and easily. These techniques can be used for other testing situations and subject areas as well.
This document provides an overview of the topics covered in an introductory mathematics analysis course for business, economics, and social sciences. It includes:
1) A review of key concepts like algebra, subsets of real numbers, properties of operations, and graphing numbers on a number line.
2) An outline of course structure with sections on algebra, algebraic expressions, fractions, and mathematical systems.
3) Examples of problems and their step-by-step solutions covering topics like simplifying expressions, factoring, addition/subtraction of fractions, and properties of real numbers.
The document is a mathematics exam paper consisting of 3 sections - Section A with 6 multiple choice questions worth a total of 40 marks, Section B with 4 questions to choose from worth a total of 40 marks, and Section C with 2 questions to choose from worth a total of 20 marks. The paper provides various formulae that may be helpful in answering the questions and covers topics such as algebra, calculus, geometry, statistics, and trigonometry.
This document contains an unsolved mathematics paper from 2007 containing multiple choice, reasoning, and matching questions. The paper has four sections, with the first being 9 straight multiple choice questions on topics like vectors, permutations, geometry, and probability. The second section contains 4 reasoning questions involving assertions and explanations. The third section has 3 linked comprehension questions about functions and roots. The last section involves matching mathematical conditions and expressions between two columns.
This document contains a multi-part math review assignment involving integers, number lines, coordinate planes, and geometry. Students are asked to represent word problems with integers, find absolute values, locate points and reflect them across axes on a coordinate plane, calculate distances and side lengths of line segments and shapes, and graph geometric figures. The assignment provides practice with key skills and concepts in pre-algebra and introductory algebra.
This document contains lecture notes on graph theory. It introduces basic graph theory concepts such as graphs, multigraphs, digraphs, isomorphism of graphs, and representations of graphs as plane figures. It defines graphs as pairs of vertices and edges, and introduces common graph theory notations and terminology like order, size, adjacency, and subgraphs. It also discusses coloring, labeling, and weighting of graphs. The document provides context and references for further reading on graph theory topics.
The document provides an introduction to graph theory concepts. It defines graphs as pairs of vertices and edges, and introduces basic graph terminology like order, size, adjacency, and isomorphism. Graphs can be represented geometrically by drawing vertices as points and edges as lines between them. Both simple graphs and multigraphs are discussed.
This document contains instructions and questions for a mathematics exam. It begins by providing spaces for the student to write their name, centre number, and candidate number. It then lists the total marks, time allowed, and materials permitted. The document contains 19 multiple choice and free response questions testing a variety of math skills like arithmetic, algebra, geometry, statistics, and graphing. It provides formulae for reference.
This document contains a mathematics exam for Secondary School students in Perak, Malaysia. It covers topics like sets, Venn diagrams, linear inequalities, simultaneous linear equations, quadratic equations and expressions, solid geometry, and mathematical reasoning. There are 10 multiple choice questions for each topic area testing students' understanding of key concepts and ability to solve related problems. The document is in Malay and contains diagrams to illustrate the questions.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
This document presents an algorithm for calculating the number of spanning trees in chained graphs. It begins by reviewing relevant graph theory concepts like planar graphs, spanning trees, and recursive formulas for counting spanning trees using deletion/contraction and splitting methods. It then derives explicit recursions for counting spanning trees in families of graphs like wheel graphs, fan graphs, and corn graphs. The main result is a theorem providing a system of equations to calculate the number of spanning trees in a chained graph based on splitting it into components and accounting for the connecting paths. Applications to counting spanning trees in chained wheel graphs and chained corn graphs are discussed.
Cs6702 graph theory and applications question bankappasami
This document contains questions and answers related to the course CS6702 Graph Theory and Applications. It is divided into 6 units which cover topics like introduction to graphs, trees, connectivity, planarity, graph colorings, directed graphs, permutations and combinations, and generating functions. For each unit, it lists short answer questions in Part A and longer proof or explanation questions in Part B. The questions test concepts fundamental to graph theory such as definitions of graphs, trees, connectivity, colorings, isomorphism, and counting methods applied to graph theory.
This document contains a mathematics exam paper with questions divided into multiple sections. Some key details:
- It is a 21⁄2 hour exam worth 50 marks total, divided into Part A and Part B.
- Part A contains 4 sections with various types of short and long answer questions on topics like real numbers, coordinate geometry, trigonometry, and mensuration.
- Part B contains shorter answer questions to be written directly on the question paper involving skills like interpreting logarithmic expressions and evaluating polynomials.
- The questions test a wide range of mathematics concepts and require calculations, proofs, formula applications, and reasoning about geometric shapes and algebraic expressions.
This document is the cover page and instructions for a 1 hour 45 minute GCSE Mathematics exam. It provides information such as the materials allowed, instructions for completing the exam, exam structure, and advice for students. The exam consists of 27 multiple choice and free response questions testing a variety of math skills, including algebra, geometry, statistics, and trigonometry. Students are advised to read questions carefully, watch the time, attempt all questions, and check their work. Calculators are not permitted.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
This document provides an overview of mathematical functions and relations through a series of lessons:
1. It defines key concepts like domains, ranges, and intervals used to describe functions and relations. Functions are defined as relations where no two ordered pairs have the same first element.
2. One-to-one functions are introduced, which satisfy both vertical and horizontal line tests. Only one-to-one functions can have inverse functions.
3. The process for finding the inverse of a function is described. The inverse is formed by swapping the inputs and outputs of the original function and solving for the new output. The domain of the original becomes the range of the inverse, and vice versa.
4
This document provides a review for algebra sections on various topics including:
1) Solving word problems involving equations with one unknown variable.
2) Finding sums, differences, and ratios of numbers.
3) Representing word problems using tables and equations.
4) Solving uniform motion problems using tables, diagrams, and the appropriate equation based on the type of motion described.
5) Graphing linear equations and finding slopes of lines from equations or two points.
6) Writing equations of lines in different forms given information like slopes, intercepts, or two points.
The document is an advertisement for Vedantu, an online education platform, promoting their free online admission test to win scholarships for classes 6-12, JEE, and NEET. It highlights success stories of students who scored well in board exams and engineering/medical entrance exams after taking online classes on Vedantu. It encourages students to register now for the admission test to secure limited seats and chance at 100% scholarship.
The document provides examples and explanations for solving linear equations. It begins by defining key vocabulary like open sentence, equation, and solution. It then shows how to translate between verbal and algebraic expressions. Various properties of equality like reflexive, symmetric, and transitive properties are explained. Finally, it demonstrates solving linear equations by isolating the variable using the inverse operations property of equality. Examples include solving equations with variables on both sides and checking solutions.
The document describes several numerical methods for finding the roots of functions including bisection, secant, and Newton's methods. It lists examples of functions and intervals that each method was applied to, including polynomials ranging from degrees 3 to 5 with various coefficients and constants. Muller's method is also mentioned as another root-finding algorithm examined.
Join Mr. D for a mini-SAT Boot Camp. Mr. D will be showing the tips and techniques from his SAT Boot Camps. This workshop will focus on math, the language behind the questions and show participants what they really need to know before taking the math section of the SAT.
Find out the words on the test that give tips for how to solve the questions as well learning what formulas you really need to know before the taking the SAT. Students and parents alike will learn how to unravel the questions being asked into something they can solve quickly and easily. These techniques can be used for other testing situations and subject areas as well.
This document provides an overview of the topics covered in an introductory mathematics analysis course for business, economics, and social sciences. It includes:
1) A review of key concepts like algebra, subsets of real numbers, properties of operations, and graphing numbers on a number line.
2) An outline of course structure with sections on algebra, algebraic expressions, fractions, and mathematical systems.
3) Examples of problems and their step-by-step solutions covering topics like simplifying expressions, factoring, addition/subtraction of fractions, and properties of real numbers.
The document is a mathematics exam paper consisting of 3 sections - Section A with 6 multiple choice questions worth a total of 40 marks, Section B with 4 questions to choose from worth a total of 40 marks, and Section C with 2 questions to choose from worth a total of 20 marks. The paper provides various formulae that may be helpful in answering the questions and covers topics such as algebra, calculus, geometry, statistics, and trigonometry.
This document contains an unsolved mathematics paper from 2007 containing multiple choice, reasoning, and matching questions. The paper has four sections, with the first being 9 straight multiple choice questions on topics like vectors, permutations, geometry, and probability. The second section contains 4 reasoning questions involving assertions and explanations. The third section has 3 linked comprehension questions about functions and roots. The last section involves matching mathematical conditions and expressions between two columns.
This document contains a multi-part math review assignment involving integers, number lines, coordinate planes, and geometry. Students are asked to represent word problems with integers, find absolute values, locate points and reflect them across axes on a coordinate plane, calculate distances and side lengths of line segments and shapes, and graph geometric figures. The assignment provides practice with key skills and concepts in pre-algebra and introductory algebra.
This document contains lecture notes on graph theory. It introduces basic graph theory concepts such as graphs, multigraphs, digraphs, isomorphism of graphs, and representations of graphs as plane figures. It defines graphs as pairs of vertices and edges, and introduces common graph theory notations and terminology like order, size, adjacency, and subgraphs. It also discusses coloring, labeling, and weighting of graphs. The document provides context and references for further reading on graph theory topics.
The document provides an introduction to graph theory concepts. It defines graphs as pairs of vertices and edges, and introduces basic graph terminology like order, size, adjacency, and isomorphism. Graphs can be represented geometrically by drawing vertices as points and edges as lines between them. Both simple graphs and multigraphs are discussed.
Graph Dynamical System on Graph ColouringClyde Shen
This document discusses using a graph dynamical system to model the vertex coloring problem. It proposes using a Graph-cellular Automaton (GA), which is an extension of cellular automata, to distribute the coloring process. In the GA model, each vertex independently chooses its optimal color based on the colors of its adjacent vertices. The document outlines testing the GA approach on different types of graphs and analyzing whether it can find colorings for them. It provides background on graph coloring, planar graphs, and introduces the graph dynamical system and GA models.
This document discusses methods for summarizing Lego-like sphere and torus maps. It begins by introducing the concept of ({a,b},k)-maps, which are k-valent maps with faces of size a or b. It then discusses several challenges in enumerating and drawing such maps, including enumerating all possible Lego decompositions. Specific enumeration methods are described, such as using exact covering problems or satisfiability problems. The document also discusses challenges in graph drawing representations, and suggests using primal-dual circle packings as a promising approach.
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
Graph Analytics - From the Whiteboard to Your Toolbox - Sam LermaPyData
This document provides an introduction to graph theory concepts and working with graph data in Python. It begins with basic graph definitions and real-world graph examples. Various graph concepts are then demonstrated visually, such as vertices, edges, paths, cycles, and graph properties. Finally, it discusses working with graph data structures and algorithms in the NetworkX library in Python, including graph generation, analysis, and visualization. The overall goal is to introduce readers to graph theory and spark their interest in further exploration.
Cs6660 compiler design november december 2016 Answer keyappasami
The document discusses topics related to compiler design, including:
1) The phases of a compiler include lexical analysis, syntax analysis, semantic analysis, intermediate code generation, code optimization, and code generation. Compiler construction tools help implement these phases.
2) Grouping compiler phases can improve efficiency. Errors can occur in all phases, from syntax errors to type errors.
3) Questions cover topics like symbol tables, finite automata in lexical analysis, parse trees, ambiguity, SLR parsing, syntax directed translations, code generation, and optimization techniques like loop detection.
Attributed Graph Matching of Planar GraphsRaül Arlàndez
Many fields such as computer vision, scene analysis, chemistry and molecular biology have
applications in which images have to be processed and some regions have to be searched for
and identified. When this processing is to be performed by a computer automatically without
the assistance of a human expert, a useful way of representing the knowledge is by using
attributed graphs. Attributed graphs have been proved as an effective way of representing
objects. When using graphs to represent objects or images, vertices usually represent regions
(or features) of the object or images, and edges between them represent the relations
between regions. Nonetheless planar graphs are graphs which can be drawn in the plane
without intersecting any edge between them. Most applications use planar graphs to
represent an image.
Graph matching (with attributes or not) represents an NP-complete problem, nevertheless
when we use planar graphs without attributes we can solve this problem in polynomial time
[1]. No algorithms have been presented that solve the attributed graph-matching problem and
use the planar-graphs properties. In this master thesis, we research about Attributed-Planar-
Graph matching. The aim is to find a fast algorithm through studying in depth the properties
and restrictions imposed by planar graphs.
Bounded Approaches in Radio Labeling Square Grids -- Dev AnandaDev Ananda
This thesis presents research on determining the radio number of square grid graphs through bounded approaches. Section 1 provides background, defining key terms like radio labeling, span, and radio number. It also reviews previous work, including Jiang's proof of the radio number for all grids. Section 2 establishes an upper bound on the radio number for odd and even grids by constructing optimal radio labelings. Section 3 provides a lower bound through distance maximization and labeling techniques. Section 4 closes the gap between bounds, determining the radio number of even grids using bounded approaches. Future research directions are also discussed.
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,aijcsa
A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of
“opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops
counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called
Hexagonal trapezoid system Tb,a.
This document is a master's thesis that studies independence complexes constructed from independent sets of sequences of graphs. It examines generating functions, closed formulas, and homology groups of independence complexes for different graph sequences. The thesis provides formulas for the f-polynomials, Euler characteristics, and dimensions of homology groups for each independence complex studied. It also compares the results to known number sequences and establishes bijections to other problems when possible.
This document discusses algorithms for solving the feedback vertex set problem, which aims to find the minimum number of nodes that need to be removed from a graph to make it acyclic. It describes several algorithms including a naive algorithm, fixed parameter tractable algorithm, 2-approximation algorithm, disjoint feedback vertex set algorithm, and randomized algorithm. For each algorithm, it provides definitions, pseudocode, and an example to illustrate how it works. The document concludes that this problem remains an active area of research to develop more efficient algorithms.
This document provides a summary of concepts in number theory and finite fields that are important for cryptography. It discusses divisors and properties of divisibility, the division algorithm, greatest common divisors, modular arithmetic including operations and properties, the Euclidean algorithm for finding GCDs, groups, rings, fields including finite Galois fields GF(p) and GF(2n), and polynomial arithmetic over finite fields. It provides examples to illustrate key concepts like modular arithmetic, the extended Euclidean algorithm, and polynomial operations over finite fields.
1. The document discusses various types of functions including: polynomial functions, constant functions, linear functions, quadratic functions, cubic functions, rational functions, and square root functions.
2. It provides the general forms and properties of each type of function, such as domains, ranges, graphs, vertices, and examples with step-by-step workings.
3. Key aspects like determining if a relation is a function using the vertical line test, finding maximum/minimum values of quadratic functions, and defining the domains of square root functions are explained.
A Hypocoloring Model for Batch Scheduling ProblemIOSR Journals
In this paper we define Scheduling, Batch Scheduling, Coloring, Subcoloring, Hypocoloring, Chromatic number, Subchromatic number, and Hypochromatic number for a given graph. A batch Scheduling problem has been obtained by using the above discussed concepts.An exponential algorithm has been developed for triangle free graphs. The solution is obtained by introducing COCA - ‘Contract or Connect” Principle. The Subchromatic and Hypochromatic number for shell graphs has been illustrated. A Hypocoloring model for personnel assignment problem has been briefly discussed.
This document introduces differential forms as an alternative approach to vector calculus. It provides a brief overview of 1-forms and 2-forms, including how to calculate line integrals and surface integrals of differential forms. The author explains that differential forms are similar to vector fields but written in a "funny notation" that is ultimately quite powerful. The document is intended as a supplement for teaching multivariable calculus using differential forms at the undergraduate level in a informal way without advanced linear algebra or manifolds.
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1. Polynomial algorithm for isomorphic graphs, all cases
Mohamed mimouni
20 street kadissia 60000 Oujda Morocco
mimouni.mohamed@gmail.com
April 12, 2019
Abstract
In this paper, I propose an algorithm capable of solving the problem of isomorphic graphs
in polynomial time. First, I define a pseudo tree that allows us to define for each vertex a
label or a label. Secondly, I apply the pseudo tree for the first graph then I calculate the
labels of each vertex of the first graph, then I do the same for the second graph. Thirdly, I
look for for each graph vertex1 the graph vertices2 which have the same label, if or less a first
graph vertex its label is not in the second graph vertices we deduce that the two pseudo trees
are not isomorphic. In other cases I generate solutions and check them in polynomial time...
This algorithm therefore allows isomorphic graphs to calculate the image of each vertex in
polynomial time.
MOTS-CLES : isomorphic graphs, isomorphism, automorphism, hypergraphs, isomorphs
1
3. 1 Definitions of the terms
1.1 A graph
A graph is a set of points called nodes (sometimes vertices or cells) connected by lines (segments)
or arrows called edges (or links or bows).
1.2 A hypergraph
A hypergraph is a graph whose edges connect one or two or more vertices.
1.3 An isomorphism
An isomorphism f between the graphs G and H is a bijection between the vertices of G and those
of H, such as a pair of vertices u, v of G is an edge of G if and only if (u), (v) is an edge of H.
2 Building a pseudo tree
Let’s say G a simple graph and s a vertex. The pseudo tree header s is done in the following way :
1. The vertex s in level 0.
2. The adjacents of s in level 1.
3. The adjacencies of each vertex of level 1 without repetition, which are not already in the
other levels, in the level 2.
4. Repeat step 3 to build the other levels.
5. End when there are no new adjacent ones.
If in the case (case of not connected graph) where there are still graph vertices G, which are not
in the pseudo-trees, other pseudo-trees must be generated.
3 Label
For each vertex of G a label is formed of four elements :
1. n the level it is at.
2. h the number of adjacent ones who are in the upper level.
3. m the number of adjacents who are in the same level.
4. b the number of adjacent ones who are in the lower level.
4 Algorithme
Let’s say G a graph {g1, g2, g3, , , gn} and H a graph {h1, h2, h3, , , hn}
The two graphs G and H are isomorphic if for the isomorphism function f(gi) = hi, we have the
label of gi equals the label of hi.
label(gi) = label(hi) = n − h − m − b
4.1 For the graph G
Construct one or more pseudo-trees folds so as to have a label for each vertex of the graph G.
4.2 For the graph H
Build pseudo trees and calculate labels for each vertex of H. Search in the graph G vertices that
have the same label. If at least one vertex of H has an unequal label of the labels of the vertices
of H, the header is modified and the labels of the vertices of H. If each vertex of H has a label
equal to graph labels G, we do a validation or verification of the solutions. If the solution is not
validated, the headers are changed. End if a solution is validated or check.
3
4. 5 Mathematical proof
5.1 principal theorem
Let’s say P(a) and Q(a ) two pseudo isomorphic trees.
If f(a) = a we have:
For each vertex s of P(a) : label(s) = label(s)
5.1.1 proof
Isomorphism keeps the distance, therefore:
1. n = d(a, s) = d(a , f(s)) so f(s) is also in level n.
2. for h, there are h vertices of level n-1 adjacent to s such as d(s, th) = d(s , th) and d(a, th) =
d(a , th) = n − 1, according to the definition there is also h vertices in Q(a’)
3. for m, there are m vertices of level n adjacent to s so d(s, tm) = d(s , tm) and d(a, tm) =
d(a, tm) = n, according to the definition there are also m vertices in Q(a’).
4. for b, there are b vertices of level n+1 adjacent to s so d(s, tb) = d(s , tb) and d(a, tb) =
d(a , tb) = n + 1, according to the definition there is also b vertices in Q(a’) .
which shows that label(s) = label(s)
5.2 The algorithm
According to principal theorem we have two cases:
Case 1 label(s) = label(s ) =⇒ f(s) = s .
Case 2 label(s) = label(s ), it is necessary to check f(s)=s’ (We can’t say that f(s) = s )
5.2.1 The verification
So after finding for each vertex of G one or more images of isomorphism f, we must do one last
step: solution verification, this verification and polynomial.
5.2.2 The complexity
All the steps of the algorithm are polynomial, so the algorithm proposed in this paper and Poly-
nomial, so the isomorphism is in P.
6 Algorithm complexity
Each vertex must be in a pseudo-tree, so the complexity is O(n2
).
For the second graph the complexity can reach O(n3
).
The calculation of the labels of each vertex is O(n).
The solution validation is to O(n3
).
So this algorithm is polynomial.
4
5. 7 Examples
7.1 connected graphs
Either the two graphs G and H, present by lists of the following adjacents:
Graph G Graph H
a:b,d,e,h 1:4,6,7,8,9
b:a,c,e,g,i 2:6,7,8,9
c:b,f,g,h,i 3:5,6
d:a,e,f,i 4:1,5,6,9
e:a,b,d,h,i 5:3,4,7,9
f:c,d 6:1,2,3,4,8
g:b,c,h 7:1,2,5,9
h:a,c,e,g 8:1,2,6
i:b,c,d,e 9:1,2,4,5,7
7.1.1 Pseudo tree of head a for graph G
Graph G levels
a level 0
b,d,e,h level 1
f,i,c,g level 2
7.1.2 Label of each graph vertex G
vertex adjacent level top even bottom label
a b,d,e,h 0 empty a b,d,e,h 0-0-0-4
b a,c,e,g,i 1 a b,d,e,h f,i,c,g 1-1-1-3
d a,e,f,i 1 a b,d,e,h f,i,c,g 1-1-1-2
e a,b,d,h,i 1 a b,d,e,h f,i,c,g 1-1-3-1
h a,c,e,g 1 a b,d,e,h f,i,c,g 1-1-1-2
f c,d 2 b,d,e,h f,i,c,g empty 2-1-1-0
i b,c,d,e 2 b,d,e,h f,i,c,g empty 2-3-1-0
c b,f,g,h,i 2 b,d,e,h f,i,c,g empty 2-2-3-0
g b,c,h 2 b,d,e,h f,i,c,g empty 2-2-1-0
explanations After finishing for graph G, we move on to graph H : Label (a)=0-0-0-0-4 so for
graph H, we test f(a)=2, f(a)=4, f(a)=5 and f(a)=7.
7.1.3 Pseudo tree of head 2,4,5 and 7 for graph H
2 4 5 7 level 0 a
6,7,8,9 1,5,6,9 3,4,7,9 1,2,5,9 level 1 b,d,e,h
3,1,4,5 3,8,2,7 6,1,2 6,8,3,4 level 2 f,i,c,g
8 level 3
We are forward to vertex 5 four levels so f(a) = 5.
7.1.4 The labels
head 2
vertex adjacent level top same bottom label images
2 6,7,8,9 0 empty 2 6,7,8,9 0-0-0-4 a
6 1,2,3,4,8 1 2 6,7,8,9 3,1,4,5 1-1-1-2 d,h
7 1,2,5,9 1 2 6,7,8,9 3,1,4,5 1-1-1-2 d,h
8 1,2,6 1 2 6,7,8,9 3,1,4,5 1-1-1-1 empty
At the vertex 8, the label (1-1-1-1-1) does not exist in graph labels G, so f(a) = 2.
head 4
vertex adjacent level top same bottom label images
4 1,5,6,9 0 empty 4 1,5,6,9 0-0-0-4 a
1 4,6,7,8,9 1 4 1,5,6,9 3,8,2,7 1-1-2-2 empty
For vertex 1 the label (1-1-2-2) does not exist in the graph labels G, so f(a) = 4.
5
6. head 7
vertex adjacent level top same bottom label images
7 1,2,5,9 0 empty 7 1,2,5,9 0-0-0-4 a
1 4,6,7,8,9 1 7 1,2,5,9 6,8,3,4 1-1-1-3 b
2 6,7,8,9 1 7 1,2,5,9 6,8,3,4 1-1-1-2 d,h
5 3,4,7,9 1 7 1,2,5,9 6,8,3,4 1-1-1-2 d,h
9 1,2,4,5,7 1 7 1,2,5,9 6,8,3,4 1-1-3-1 e
6 1,2,3,4,8 2 1,2,5,9 6,8,3,4 empty 2-2-3-0 c
8 1,2,6 2 1,2,5,9 6,8,3,4 empty 2-2-1-0 g
3 5,6 2 1,2,5,9 6,8,3,4 empty 2-1-1-0 f
4 1,5,6,9 2 1,2,5,9 6,8,3,4 empty 2-3-1-0 i
7.1.5 Check the solutions
vertex adjacent images images adjacent vertex adjacent state
7 1,2,5,9 a (b),(d,h),(d,h),(e) a b,d,e,h ok
1 4,6,7,8,9 b (i),(c),(a),(g),(e) b a,c,e,g,i ok
2 6,7,8,9 d (c),(a),(g),(e) d a,e,f,i no
2 6,7,8,9 h (c),(a),(g),(e) h a,c,e,g ok
5 3,4,7,9 d (f),(i),(a),(e) d a,e,f,i ok
5 3,4,7,9 h (f),(i),(a),(e) h a,c,e,g no
9 1,2,4,5,7 e (b),(d,h),(i),(d,h),(a) e a,b,d,h,i ok
6 1,2,3,4,8 c (b),(d,h),(f),(i),(g) c b,f,g,h,i ok
8 1,2,6 g (b),(d,h),(c) g b,c,h ok
3 5,6 f (d,h),(c) f c,d ok
4 1,5,6,9 i (b), (d,h),(c),(e) i b,c,d,e ok
explanations In this table we have f(2) = h and not d. and f(5) = d and not h.
The check to give 2 errors, we must then do another check
vertex adjacent images images adjacent vertex adjacent state
7 1,2,5,9 a (b),(,h),(d),(e) a b,d,e„h ok
1 4,6,7,8,9 b (i),(c),(a),(g),(e) b a,c,e,g,i ok
2 6,7,8,9 h (c),(a),(g),(e) h a,c,e,g ok
5 3,4,7,9 d (f),(i),(a),(e) d a,e,f,i ok
9 1,2,4,5,7 e (b),(,h),(i),(d),(a) e a,b,d„h,i ok
6 1,2,3,4,8 c (b),(,h),(f),(i),(g) c b,f,g„h,i ok
8 1,2,6 g (b),(,h),(c) g b,c„h ok
3 5,6 f (d),(c) f c,d ok
4 1,5,6,9 i (b),(d),(c),(e) i b,c,d,e ok
explanations Now with no errors so both graphs G and H are isomorphic and here is the final
solution:
vertex Image
a 7
b 1
c 6
d 5
e 9
f 3
g 8
h 2
i 4
6
7. 8 pseudo codes
8.1 pseudo-arbres
Here is a pseudo code that generates a pseudo tree.
Algorithm 1: generate a pseudo tree P(a) head a for a graph G
Data: A graph G and a vertex a of G
Result: A pseudo tree P(a)
1 initialization; the vertex a in the level 0;
2 the adjacent from a in level 1;
3 the vertices of G in array graph[];
4 delete the vertices of the level 0 and 1 array graph[];
5 niv ← 1 ;
6 adjacents ← adjacent to a ;
7 while graph[] is not empty do
8 initialize array list[] empty ;
9 niv ← niv + 1 ;
10 for every element of adjacent do
11 adjacents1 ← the adjacents of element ;
12 adjacents1 ← the intersections between adjacent1 and graph[] ;
13 liste[] ← list[]+adjacents1 ;
14 liste[] ← liste[] order and without duplication ;
15 graph[] ← the difference between graph[] and list[] ;
16 if graph[] is empty then
17 break;
18 end
19 end
20 if lise[] is empty then
21 break;
22 else
23 list in level niv ;
24 end
25 end
For unconnected graphs, it is necessary to generate other pseudo-trees for the other remaining
vertices.
8.2 The label
Here’s a pseudo code that calculates the label of a vertex in a pseudo tree.
Algorithm 2: Calculate the label of a vertex s in a pseudo tree
Data: A vertex s in a pseudo tree
Result: s label for a pseudo tree
1 n ← determine the level of s;
2 nivh ← determine the vertices that are in the level n+1;
3 nivm ← determine the vertices that are in the level n;
4 nivb ← determine the vertices that are in the level n-1;
5 h ← the number of adjacent vertices s that are in nivh;
6 m ← the number of adjacent vertices s that are in nivm;
7 b ← the number of adjacent vertices s that are in nivb;
8 label(s) ← "n-h-m-b";
7
8. 8.3 Validate a solution
Here’s a pseudo code to check out a proposed solution.
Algorithm 3: Check a solution
Data: A proposed solution
Result: validated or erroneous solution
1 initialization;
2 err ← 0;
3 for Each f(a)=a’ proposed do
4 adjacent(a) ← the adjacent vertices a;
5 adjacent(a ) ← the adjacent vertices a’;
6 for each adjacent to a do
7 x ← an adjacent vertex adjacent a;
8 x ← the images of x by isomorphism;
9 Intersection ← the intersection between the images of x by isomorphism and the
adjacent ones of a’;
10 if intersection is empty then
11 err ← err+1;
12 break;
13 end
14 end
15 if err = 0 then
16 break;
17 end
18 end
19 if err = 0 then
20 "The solution is validated";
21 else
22 "The solution is not validé";
23 end
8.4 comment
For the not connected graphs it is necessary of course to study the isomorph between the connected
components.
9 Conclusion
In this paper I presented a version of a polynomial algorithm, which gives us the images of each
vertices by isomorphic bijection, or to say that the two graphs are not isomorphic.
8