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The question of finding expander graphs with strong vertex expansionproperties such as unique neighbor expansion and lossless expansion is centralto computer science. A barrier to constructing these is that strong notions ofexpansion could not be proven via the spectral expansion paradigm.  A very symmetric and structured family of optimal spectral expanders (i.e.,Ramanujan graphs) was constructed using number theory by Lubotzky, Phillips andSarnak, and was subsequently generalized by others. We call such graphs NumberTheoretic Ramanujan Graphs. These graphs are not only spectrally optimal, butalso posses strong symmetries and rich structure. Thus, it has been widelyconjectured that number theoretic Ramanujan graphs are lossless expanders, orat least unique neighbor expanders.  In this work we disprove this conjecture, by showing that there are numbertheoretic Ramanujan graphs that are not even unique neighbor expanders. This isdone by introducing a new combinatorial paradigm that we term the closed orbitmethod.  The closed orbit method allows one to construct finite combinatorial objectswith extermal substructures. This is done by observing that there existinfinite combinatorial structures with extermal substructures, coming from anaction of a subgroup of the automorphism group of the structure. The crux ofour idea is a systematic way to construct a finite quotient of the infinitestructure containing a simple shadow of the infinite substructure, whichmaintains its extermal combinatorial property.  Other applications of the method are to the edge expansion of numbertheoretic Ramanujan graphs and vertex expansion of Ramanujan complexes.Finally, in the field of graph quantum ergodicity we produce number theoreticRamanujan graphs with an eigenfunction of small support that corresponds to thezero eigenvalue. This again contradicts common expectations.

Combinatorics via Closed Orbits: Number Theoretic Ramanujan Graphs are not Unique Neighbor Expanders