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The edges of a graph define a [[symmetric relation]] on the vertices, called the ''adjacency relation''. Specifically, two vertices {{mvar|x}} and {{mvar|y}} are ''adjacent'' if {{math|{''x'', ''y''} }} is an edge. A graph is fully determined by its [[adjacency matrix]] {{mvar|A}}, which is an {{math|''n'' × ''n''}} square matrix, with {{mvar|A{{sub|ij}}}} specifying the number of connections from vertex {{mvar|i}} to vertex {{mvar|j}}. For a simple graph, {{math|''A{{sub|ij}}''}} is either 0, indicating disconnection, or 1, indicating connection; moreover {{math|1=''A{{sub|ii}}'' = 0}} because an edge in a simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all {{mvar|A{{sub|ii}}}} being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all {{mvar|A{{sub|ij}}}} being equal to a positive integer. Undirected graphs will have a [[symmetric matrix|symmetric]] adjacency matrix (meaning {{math|1=''A{{sub|ij}}'' = ''A{{sub|ji}}''}}).
The edges are interpreted to have double arrows in some application, where an undirected edge is regarded as two directed ones.
=== Directed graph ===
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