Group Actions
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Chapter presentation project by Amanda Trommater for MAT 361 (Modern Algebra) at Franklin College, Fall 2009.
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Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
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A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
Bidimensionality
This document summarizes the concept of bidimensionality and how it can be used to design subexponential algorithms for graph problems on planar and other graph classes. It discusses how bidimensionality can be defined for parameters that are closed under minors or contractions by relating their behavior on grid graphs. It presents examples like vertex cover and dominating set that are bidimensional. It also discusses how bidimensionality can be extended to bounded genus graphs and H-minor free graphs using grid-minor/contraction theorems.
A common fixed point of integral type contraction in generalized metric spacess
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
Rainone - Groups St. Andrew 2013
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Graph Theory
A graph consists of a set of vertices and edges connecting pairs of vertices. Graph coloring assigns colors to vertices such that no adjacent vertices share the same color. The chromatic polynomial counts the number of valid colorings of a graph using a given number of colors. It was introduced to study the four color theorem and fundamental results were established in the early 20th century. The chromatic polynomial can be used to find the chromatic number of a graph.
Brian Covello: Research in Mathematical Group Representation Theory and Symmetry
Brian Covello's research review on group representation theory and symmetry. In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, a graphical method exists to determine their finite representations that associates with each representation a Young tableau (also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
Applications and Properties of Unique Coloring of Graphs
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
445 colouring0
1) This document discusses graph colouring and related concepts like chromatic number, independent sets, cliques, greedy algorithms, degeneracy, Mycielski's construction, critical graphs, and counting colourings.
2) Key results include Brooks' Theorem which gives an upper bound for chromatic number based on maximum degree, and Gallai-Roy-Vitaver Theorem relating chromatic number to longest directed path in an orientation.
3) Mycielski's construction is introduced as a way to increase the chromatic number of a graph by 1 while preserving being triangle-free.
Note on Character Theory-summer 2013
This document provides an overview of character theory for finite groups and its analogy to representation theory for the infinite compact group S1. It discusses several key concepts:
1) Character theory characterizes representations of a finite group G by their characters, which are class functions that map G to complex numbers. Characters determine representations up to isomorphism.
2) The irreducible representations of S1 are 1-dimensional and in bijection with integers n, where each representation maps z to zn.
3) By analogy to Fourier analysis, the characters of S1 form an orthonormal basis for L2(S1) and decompose representations into irreducibles in the same way as for finite groups.
April2012ART_01(1)
This document summarizes research on algebraic elements in group algebras. It begins by defining a group algebra k[G] over a commutative ring k. An element of k[G] is algebraic if it satisfies a non-constant polynomial. The document discusses tools for studying algebraic elements like partial augmentations corresponding to conjugacy classes. It also summarizes results on idempotents, including Kaplansky's theorem that the trace of an idempotent is real and rational. The author's past work on dimension subgroups is also briefly outlined.
ICAMS033-G.NITHYA.pptx
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
Ji2416271633
This document discusses weak isomorphism and isomorphism between fuzzy graphs. Some key points:
1. Weak isomorphism preserves the weights of nodes but not necessarily the weights of edges, while isomorphism preserves weights of both nodes and edges.
2. If two fuzzy graphs are isomorphic, they have the same order (number of nodes) and size (sum of edge weights).
3. Isomorphism between fuzzy graphs is proved to be an equivalence relation as it is reflexive, symmetric and transitive.
4. Weak isomorphism is proved to be a partial order relation as it satisfies reflexivity, anti-symmetry and transitivity.
5. Some properties of self-complement
1452 86301000013 m
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
Best Approximation in Real Linear 2-Normed Spaces
This pape r d e l i n e a t e s existence, characterizations and st rong unicity of best uniform
approximations in real linear 2-normed spaces.
AMS Su ject Classification: 41A50, 41A52, 41A99, 41A28.
The Total Strong Split Domination Number of Graphs
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
Suborbits and suborbital graphs of the symmetric group acting on ordered r ...
]
This document discusses suborbits and suborbital graphs of the symmetric group Sn acting on the set X of all ordered r-element subsets from a set X of size n. Some key points:
- Suborbits are orbits of the stabilizer Gx of a point x under the group action. Suborbital graphs are constructed from suborbits.
- Theorems characterize when a suborbit is self-paired or when two suborbits are paired in terms of properties of the permutations that define the suborbits.
- Formulas are derived for the number of self-paired suborbits in terms of the cycle structure of group elements and character values.
On the equality of the grundy numbers of a graph
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSES
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
On the Equality of the Grundy Numbers of a Graph
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
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Total Dominating Color Transversal Number of Graphs And Graph Operations
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