Determinants, Permanents, and Bipartite Graphs.

Frank Harary

Expert InsightsPublished 1968

The combinatorial properties of a nonnegative matrix M are captured by that binary matrix A = A(M) in which the entries are 1 whenever those of M are positive. If A is a square matrix, then it can be regarded as the adjacency matrix of a directed graph (digraph). If A is rectangular, a bipartite graph (bigraph) can be associated with A, and this can also be done for A square. The determinant of the adjacency matrix of a graph or digraph has been expressed in terms of its structure, and so has the permanent. The objectives of this study are to express the permanent of a square or rectangular binary matrix in terms of the associated bigraph and to formulate the determinant of a square matrix in terms of its bigraph. 12 pp. Refs.

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Harary, Frank, Determinants, Permanents, and Bipartite Graphs. Santa Monica, CA: RAND Corporation, 1968. https://www.rand.org/pubs/papers/P3840.html.
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