On properties of cell matrices
The paper deals with some special classes of matrices, coming from star graphs (or claw graphs) used in graph theory and in biochemistry. A cell matrix \(D=(d_{ij})\) is associated with a vector \((\alpha _{i})\), where \( d_{ii}=0\) and \(d_{ij}=\alpha _{i}+\alpha _{j}\) for \(i\neq j\). Connecting \(k\) star graphs by inner vertices, one obtains a \(k\)-star graph. A Euclidean distance matrix (EDM) is a square \(n\) matrix having as entries the square distance of distances of \(n\) Euclidian points. A circum-Euclidean distance matrix (CEDM) imposes that the Euclidian points lie on a hypersphere. The authors prove that all cell matrices are (C)EDM, but only some special \(k\)-cell matrices are (C)EDM matrices. Some other interesting formulas concerning the calculus of a determinant of a symbolic matrix with polynomial entries are also proved.
scientific article; zbMATH DE number 5726769
scientific article