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In the analysis of consensus problems for multi-agent systems affected bytime delays, the delay margin for a given protocol is heavily dependent on thecommunication topology being used. This document aims to answer the question ofwhat is the largest delay value under which convergence to consensus is stillpossible even if the communication topology is not known. To answer thisquestion we revisit the concept of most exigent eigenvalue, applying it to twodifferent consensus protocols for agents driven by second order dynamics. Weshow how the delay margin depends on the structure of the consensus protocoland the communication topology, and arrive to a boundary that guaranteesconsensus for any connected communication topology. The switching topologiescase is also studied and It is shown that for one protocol the stability of theindividual topologies is sufficient to guarantee consensus in the switchingcase, whereas for the other one it is not.

The Most Exigent Eigenvalue: Guaranteeing Consensus under an Unknown Communication Topology and Time Delays