Incidence matrix

An incidence matrix represents a graph by placing vertices in rows and edges in columns, with entries recording which vertices touch each edge.

Graph theory: The mathematical study of graphs, structures made of vertices connected by edges. Incidence matrices encode the vertices and edges that graph theory studies.
Oriented incidence matrix: An incidence matrix whose edge columns use opposite signs at each edge's tail and head. It gives a standard signed form of the graph incidence matrix.
Network flow: The movement of quantities through a network of nodes and directed edges under capacity constraints. The signed incidence matrix expresses flow conservation at each vertex.
Unsigned incidence matrix: A graph incidence matrix with a 1 wherever a vertex touches an edge and 0 otherwise. It contrasts with signed conventions that distinguish an edge's tail from its head.
Adjacency matrix: A square matrix whose entries record which pairs of vertices in a graph are adjacent. It records vertex-to-vertex adjacency, unlike an incidence matrix's vertex-to-edge relation.
Graph Laplacian: A matrix encoding vertex degrees and adjacency, commonly defined as degree matrix minus adjacency matrix. For an oriented incidence matrix B, the product BBᵀ is the graph Laplacian.
Kirchhoff's circuit laws: Rules stating that currents balance at circuit junctions and voltage changes sum to zero around loops. The incidence matrix expresses current balance at circuit nodes.
Signed graph: A graph whose edges are assigned positive or negative signs. Signed graph incidence matrices use endpoint signs beyond the ordinary unsigned marking.
Bipartite graph: A graph whose vertices divide into two sets, with every edge joining vertices from different sets. An incidence matrix is itself the adjacency matrix of a graph's vertex-edge bipartite graph.
Rank-nullity theorem: A theorem stating that a linear map's domain dimension equals its rank plus its nullity. It relates incidence-matrix rank to the dimension of its null space.

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