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Discrete Math - Graph Theory Basics

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Path

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A sequence of vertices where each adjacent pair is connected by an edge.

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Adjacency Matrix

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A square matrix used to represent a finite graph, with rows and columns labeled by graph vertices.

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Forest

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A collection of trees.

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Directed Graph (Digraph)

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A graph in which the edges have a direction.

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Adjacent Vertices

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Two vertices are adjacent if they are connected by an edge.

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Weighted Graph

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A graph where edges have an associated weight or cost.

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Hamiltonian Path

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A path in a graph that visits each vertex exactly once.

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Adjacent Edges

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Two edges are adjacent if they share a common vertex.

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Graph

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A set of vertices connected by edges.

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Bipartite Graph

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A graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent.

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Cycle

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A path that starts and ends at the same vertex without traversing any other vertex more than once.

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Subgraph

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A graph formed from a subset of the vertices and edges of another graph.

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Planar Graph

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A graph that can be embedded in the plane, so no edges intersect except at their endpoints.

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Tree

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A connected graph with no cycles.

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Vertex Cover

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A set of vertices that includes at least one endpoint of every edge in the graph.

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Strongly Connected Component

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In a directed graph, a subset of vertices such that there is a directed path from each vertex in the subset to every other vertex.

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Incidence Matrix

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A matrix that shows the relationship between vertices and edges in a graph.

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Edge

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A connection between two vertices in a graph.

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Connected Graph

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A graph where there is a path between every pair of vertices.

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Graph Isomorphism

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Two graphs G1 and G2 are isomorphic if there is a bijection between their vertex sets that preserves adjacency.

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Independent Set

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A set of vertices in a graph, no two of which are adjacent.

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Degree of a vertex

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The number of edges incident to the vertex.

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Complete Graph

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A graph in which every pair of vertices is connected by an edge.

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Degree Sequence

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A list of degrees of the vertices in the graph, usually written in non-increasing order.

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Eulerian Path

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A path in a graph which visits every edge exactly once.

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