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  "body": "\n## Definition\n\nLet $G$ be a finite [[graph|0JD5C82QMB]] with vertices $v_1, \\dots, v_n$ and edges $e_1, \\dots, e_m$.\nThe incidence matrix of $G$ is the $n \\times m$ matrix $B$ with $B_{ij} = 1$ if $v_i$ is an end of $e_j$, and $B_{ij} = 0$ otherwise.\nSuppose that each edge $e_j$ is given an orientation, from a tail to a head.\nThe oriented incidence matrix is the $n \\times m$ matrix $N$ with $N_{ij} = 1$ if $v_i$ is the head of $e_j$.\nAlso $N_{ij} = -1$ if $v_i$ is the tail of $e_j$, and $N_{ij} = 0$ otherwise.\n\n## Remarks\n\nExample: for the triangle with edges $e_1 = v_1 v_2$, $e_2 = v_2 v_3$, $e_3 = v_1 v_3$,\n$$B = \\begin{pmatrix} 1 \u0026 0 \u0026 1 \\\\ 1 \u0026 1 \u0026 0 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix},$$\nand each column has exactly two entries $1$, at the ends of its edge.\nNon-example: a $0$–$1$ matrix with a column containing three entries $1$ is not the incidence matrix of a graph, since an edge has exactly two ends; such matrices describe hypergraphs.\nLet $D$ be the diagonal matrix of degrees and $A$ the [[adjacency matrix|0NZ8MDBN6Q]] of a simple graph.\nThen $B B^{T} = D + A$, and $N N^{T} = D - A$ is the [[Laplacian matrix|TN6M6DVZRP]] for every orientation.\nSome authors use the transpose, with rows indexed by edges, or the opposite sign convention for $N$.\nMathlib defines the unoriented incidence matrix as `SimpleGraph.incMatrix`.\n",
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