Mac Lane's planarity criterion, also known as the "Mac Lane's formation", is a combinatorial condition used to determine whether a graph can be embedded in the plane without any edges crossing. Specifically, the criterion states that a graph is planar if and only if it does not contain a specific type of subgraph as a "minor.
The matching polynomial is a well-defined polynomial associated with a graph that encapsulates information about its matchings—sets of edges without shared vertices.
The minimum rank of a graph is a concept from algebraic graph theory that is associated with the graph's adjacency matrix or Laplacian matrix. Specifically, it refers to the smallest rank among all real symmetric matrices corresponding to the graph.
Modularity, in the context of networks, refers to the degree to which a network can be divided into smaller, disconnected sub-networks or communities. It is often used in network analysis to identify and measure the strength of division of a network into modules, which are groups of nodes that are more densely connected to each other than to nodes in other groups. ### Key Points about Modularity: 1. **Community Structure**: Modularity helps in detecting community structure within networks.
The Parry–Sullivan invariant is a concept in the field of dynamical systems and statistical mechanics, particularly related to the study of interval exchanges and translations. It is associated with the study of the dynamics of certain classes of transformations, particularly those that exhibit specific structural and statistical properties. The invariant itself is often connected to topological and measure-theoretic characteristics of systems that exhibit a certain type of symmetry or recurrence.
A Ramanujan graph is a type of expander graph named after the Indian mathematician Srinivasa Ramanujan, whose work in number theory inspired this concept. Ramanujan graphs are particularly characterized by their exceptional expansion properties and have applications in various areas of mathematics and computer science, including combinatorics, number theory, and network theory.
In graph theory, the term "rank" can have a couple of different meanings, depending on the context in which it is used. 1. **Rank of a Graph**: The rank of a graph can refer to the maximum number of edges that can be included in a spanning tree. In this context, it is often considered in relation to the concept of the graph's connectivity and the number of vertices (V) and edges (E).
A Seidel adjacency matrix is a type of matrix used in graph theory, particularly for the representation of certain types of graphs known as Seidel graphs. It is derived from the standard adjacency matrix of a graph but has a distinctive form.
A semi-symmetric graph is a type of graph that exhibits certain symmetrical properties but does not necessarily exhibit full symmetry. More formally, a semi-symmetric graph can be defined through its vertex and edge structure in relation to their automorphisms and symmetrical actions. In the context of graph theory, the properties that characterize a semi-symmetric graph can vary somewhat depending on the specific definition being used. Generally, however, a semi-symmetric graph maintains some degree of regularity and uniformity in its structure.
Sims' conjecture is a hypothesis in the field of algebraic topology and combinatorial group theory, specifically relating to the properties of certain types of groups. Named after mathematician Charles Sims, the conjecture primarily deals with the structure of finite groups and representation theory. While specific details or formulations may vary, Sims' conjecture is generally focused on establishing a relationship between the orders of groups and their representations or modules.
Spectral clustering is a technique used in machine learning and data analysis for grouping data points into clusters based on the properties of the dataset. It leverages the eigenvalues and eigenvectors of matrices derived from the data, particularly the similarity matrix, to identify clusters. Here’s an overview of the key steps and concepts involved in spectral clustering: 1. **Similarity Graph**: First, a similarity graph is constructed from the data points.
Spectral graph theory is a branch of mathematics that studies the properties of graphs through the eigenvalues and eigenvectors of matrices associated with them. These matrices include the adjacency matrix, the degree matrix, and the Laplacian matrix, among others. Spectral graph theory connects combinatorial properties of graphs with linear algebra and provides powerful tools for analyzing graphs in various contexts.
A strongly regular graph is a specific type of graph characterized by a regular structure that satisfies certain conditions regarding its vertices and edges. Formally, a strongly regular graph \( G \) is defined by three parameters \( (n, k, \lambda, \mu) \) where: - \( n \) is the total number of vertices in the graph.
The Tutte matrix is a mathematical construct used in the study of graph theory, particularly in the context of understanding the properties of bipartite graphs and the presence of perfect matchings. It is named after the mathematician W. T. Tutte.
A "two-graph" typically refers to a specific type of graph in the field of graph theory, but it might not be a widely standardized term. In general, graph theory involves studying structures made up of vertices (or nodes) connected by edges.
A **vertex-transitive graph** is a type of graph in which, for any two vertices, there is some automorphism of the graph that maps one vertex to the other. In simpler terms, this means that the graph looks the same from the perspective of any vertex; all vertices have a similar structural role within the graph. ### Key Properties: 1. **Automorphism:** An automorphism is a bijection (one-to-one correspondence) from the graph to itself that preserves the edges.
A **walk-regular graph** is a type of graph that has a uniform structure relative to walks of certain lengths. Specifically, a graph is called \( k \)-walk-regular if the number of walks of length \( k \) from any vertex \( u \) to any other vertex \( v \) depends only on the distance between \( u \) and \( v \), rather than on the specific choice of \( u \) and \( v \).