Vertices are the objects in a graph, and edges are the connections joining pairs of them. People often call vertices nodes and edges links; the words are interchangeable. A single edge always ties exactly two vertices together.
Two more terms describe local structure. The neighbors of a vertex are the vertices directly joined to it by an edge. The degree of a vertex is how many edges touch it, which equals its neighbor count in a simple graph.
Why care? Degree tells you how connected something is. A social account with degree ten million is a hub you must handle carefully. Neighbors are exactly what every traversal reads next, so representing them cheaply drives your whole performance. When someone calls a graph sparse, they mean average degree stays small, and most vertices have few neighbors.
Rewriting in plainer words…
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