ABSTRACT
A graph is completely determined by either its adjacencies or its incidences, with a given graph, adequately labeled, there are associated several matrices, including the adjacency matrix, incidence matrix, cycle matrix, and cocycle matrix. The classic theorem on graphs and matrices is the Matrix-Tree Theorem, which gives the number of spanning trees in any labeled graph. In general, because of the correspondence between graphs and matrices, any graph-theoretic concept is reflected in the adjacency matrix. Sometimes a labeling is irrelevant which interpret the entries of the powers of the adjacency matrix. The incidence matrix of a block is contained in its cocycle matrix. If the cycles and cocycles are chosen in a special way, then the reduced incidence, cycle, and cocycle matrices of a graph have particularly nice forms. In fact a matroid is both graphical and cographical if and only if it is the cycle matroid of some planar graph.
