Graphs, trees, and basic graph properties and terminology from an intro discrete mathematics course.
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- What is a graph?
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A mathematical structure consisting of vertices (nodes) and edges (connections) between them.- What is a vertex in a graph?
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A fundamental unit of a graph, also called a node; the point where edges connect.- What is an edge in a graph?
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A connection between two vertices (or a vertex and itself).- Define a directed graph.
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A graph in which each edge has a direction, represented as an arrow pointing from one vertex to another.- Define an undirected graph.
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A graph in which edges have no direction; the connection between two vertices is mutual.- What is a weighted graph?
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A graph in which each edge has an associated numerical value, called its weight.- Define the degree of a vertex.
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The number of edges incident to (connected to) that vertex.- State the Handshaking Lemma.
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The sum of the degrees of all vertices equals twice the number of edges.- What is a simple graph?
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An undirected graph with no loops and no multiple edges between the same pair of vertices.- What is a multigraph?
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A graph in which multiple edges can connect the same pair of vertices.- Define a loop in a graph.
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An edge that connects a vertex to itself.- What is a connected graph?
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A graph in which there is a path between every pair of vertices.- What is a disconnected graph?
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A graph in which at least one pair of vertices has no path between them.- Define a connected component.
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A maximal connected subgraph of a graph; a subset where every vertex is reachable from every other vertex in that subset.- What is a walk in a graph?
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A sequence of vertices in which each consecutive pair is connected by an edge; vertices and edges may repeat.
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