Executive Summary : | Expanders are a crucial family of graphs used in various fields such as Computer Science and Mathematics, error-correcting codes, complexity theory, derandomization of random algorithms, computational group theory analysis, and quantum cryptography. They are defined as graphs with two competing properties: sparseness (less edges) and well-connectivity (at least c edges between subsets S and S'). The isoperimetric number or Cheeger constant, or expansion ratio, is NP-hard to find, and higher h(G) is desirable for better expanders. Connected d-regular graphs satisfy both properties, making them a family of expander graphs. However, constructing such a family with a good expansion ratio is challenging. To construct expanders and Ramanujan graphs, we will construct expanders and Ramanujan graphs to provide robust network topologies and solve the consensus problem on the network. The consensus problem is widely studied in multi-agent systems (MASs) modeled as communication networks and is related to algebraic connectivity of the underlying network topology. Qualitative and numerical analysis of the produced communication networks will be performed using predefined structural and spectral measures and metrics, such as mean-first-passage-time, eigenratio, clustering coefficient, and average path length. Expanders offer a promising solution to the consensus problem in multi-agent systems (MASs), which is closely related to algebraic connectivity in the underlying network topology. |