Diagonal stability on cactus graphs

A square matrix E is said to be diagonally stable if there exists a diagonal matrix D >; 0 satisfying DE + E
^T
D <; 0. This notion has been instrumental in recent studies on stability of interconnected system models in communication and biological networks, in which the subsystems satisfy prescribed passivity properties and the matrix E combines this passivity information with the interconnection structure of the network. This paper presents a necessary and sufficient condition for diagonal stability when the digraph describing the structure of the matrix conforms to a “cactus” structure, which means that a pair of distinct simple circuits in the graph have at most one common vertex. In the special case of a single circuit, this diagonal stability test recovers the “secant criterion” that was recently derived for cyclic networks that commonly arise in biochemical reaction networks.