Incidence Matrix
The (more correctly, "an") incidence matrix of a signed graph with n vertices and m edges is an n × m matrix, with a row for each vertex and a column for each edge. It is obtained by orienting the signed graph in any way. Then its entry ηij is +1 if edge j is oriented into vertex i, −1 if edge j is oriented out of vertex i, and 0 if vertex i and edge j are not incident. This rule applies to a link, whose column will have two nonzero entries with absolute value 1, a half-edge, whose column has a single nonzero entry +1 or −1, and a loose edge, whose column has only zeroes. The column of a loop, however, is all zero if the loop is positive, and if the loop is negative it has entry ±2 in the row corresponding to its incident vertex.
Any two incidence matrices are related by negating some subset of the columns. Thus, for most purposes it makes no difference which orientation we use to define the incidence matrix, and we may speak of the incidence matrix of Σ without worrying about exactly which one it is.
Negating a row of the incidence matrix corresponds to switching the corresponding vertex.
Read more about this topic: Signed Graph
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