P-matrices and signed digraphs

2010/06/01 by Banaji, Murad, Rutherford, Carrie
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1006.0152

Abstract

We associate a signed digraph with a list of matrices whose dimensions permit them to be multiplied, and whose product is square. Cycles in this graph have a parity, that is, they are either even (termed e-cycles) or odd (termed o-cycles). The absence of e-cycles in the graph is shown to imply that the matrix product is a P0-matrix, i.e., all of its principal minors are nonnegative. Conversely, the presence of an e-cycle is shown to imply that there exists a list of matrices associated with the graph whose product fails to be a P0-matrix. The results generalise a number of previous results relating P- and P0-matrices to graphs.

Related

• The signed graphs with two eigenvalues unequal to ± 1
• On signed graphs with at most two eigenvalues unequal to ± 1
• Sign pattern matrices associated with cycle graphs that require algebraic positivity
• On Matrix Product Factorization of graphs
• No signed graph with the nullity η(G,σ)=|V(G)|-2m(G)+2c(G)-1