Cyclic matrices of weighted digraphs

Abstract

We address in this paper several properties of so-called augmented cyclic matrices of weighted digraphs. These matrices arise in different applications of digraph theory to electrical circuit analysis, and can be seen as an enlargement of basic cyclic matrices of the form $B W \rsp B^T$, where $B$ is a cycle matrix and $W$ is a diagonal matrix of weights. By using certain matrix factorizations and some properties of cycle bases, we characterize the determinant of augmented cyclic matrices in terms of Cauchy-Binet expansions and, eventually, in terms of so-called proper cotrees. In the simpler context defined by basic cyclic matrices, we obtain a dual result of Maxwell's determinantal expansion for weighted Laplacian (nodal) matrices. Additional relations with nodal matrices are also discussed. Finally, we apply this framework to the characterization of the differential-algebraic circuit models arising from loop analysis, and also to the analysis of branch-oriented models of circuits including charge-controlled memristors.