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An Eulerian Circuit is a closed path in a graph that visits every edge exactly once and returns to the starting vertex. A connected graph has an Eulerian Circuit if and only if every vertex has an even degree.
Graph theory is a branch of mathematics that studies the properties and applications of graphs, which are structures made up of nodes (vertices) connected by edges. It is fundamental in computer science, network analysis, and combinatorics for solving problems related to connectivity, flow, and optimization.
An Eulerian Path is a trail in a graph that visits every edge exactly once, and it exists if and only if the graph is connected and has exactly zero or two vertices of odd degree. If a graph has exactly two vertices of odd degree, any Eulerian Path will start at one of these vertices and end at the other.
The degree of a vertex in a graph is the number of edges incident to the vertex, which provides insight into the vertex's connectivity within the graph. In directed graphs, the degree is split into in-degree and out-degree, representing incoming and outgoing edges, respectively.
A connected graph is a type of graph in which there is a path between every pair of vertices, ensuring that all vertices are reachable from any other vertex. This property makes connected graphs fundamental in network analysis, ensuring the integrity and communication across the entire structure.
Graph connectivity is a fundamental property that determines whether there is a path between any two vertices in a graph, indicating how well the graph is 'held together'. Understanding connectivity is crucial for analyzing network robustness, optimizing routes, and ensuring efficient communication in various applications like computer networks and transportation systems.
Euler's theorem states that if n is a positive integer and a is an integer coprime to n, then a raised to the power of Euler's totient function φ(n) is congruent to 1 modulo n. This theorem generalizes Fermat's Little Theorem and is fundamental in number theory, particularly in the field of modular arithmetic and cryptography.
Path and cycle decomposition involves the partitioning of a graph's edges into disjoint paths and cycles. This decomposition is a fundamental technique used to analyze and solve problems related to graph structure, connectivity, and optimization.
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