Graph theory
Graph theory is the investigation of graphs, which shows the relationships between every pair of objects while only showing each object once within the same diagram. It is often used in the context of voting theory to discuss Condorcet methods, most of which use pairwise counting to determine which candidates or groups of candidates are better than others.
Pairwise counting
A graph can be used to show which candidates (or winner sets, when discussing proportional or non-proportional multi-winner methods) are pairwise preferred over others. Within such a graph, each candidate is considered to be a node (a point), and when an edge (a line) with an arrow points from one candidate's node towards another candidate's node, that indicates that the former candidate is preferred by more voters than the latter candidate in their pairwise matchup. If the arrow points both ways, this means that both candidates are in a pairwise tie.
Some common concepts from Condorcet methods interpreted through graph theory:
Condorcet winner - a candidate who has arrows pointing away from them towards every other candidate, and no arrows from any other candidate pointing back to them.
Smith set - the smallest group of candidates such that arrows point away from any of the candidates in the group towards any candidate not in the group, and no arrows from any candidate not in the group point back to any candidate in the group.
Schwartz set - The same as the Smith set except arrows may point back from any candidate not in the group towards any candidate in the group.
Condorcet loser - The opposite of the Condorcet winner: a candidate who has arrows pointing towards them from every other candidate, and no arrows pointing away from them towards any other candidate.
Weak Condorcet winner - The same as the Condorcet winner except arrows may point back from other candidates towards them.
Weak Condorcet loser - The same as the Condorcet loser except arrows may point away from them towards other candidates.
The Schulze method in particular can be thought of in terms of beatpaths, which are defined for an ordered set of candidates such that on a graph representing pairwise preferences, an arrow points from the first candidate to the second, the second to the third, etc. all the way until the last candidate. A beat-or-tie path is the same except an arrow may point both ways between any pair of candidates in the beatpath.