On the bipartiteness constant and expansion of Cayley graphs

A spectral bound for vertex-transitive graphs

“…Later, a quantitative version of this fact was established by the first author [Bis19] and the same phenomenon with explicit bounds were also shown to hold true for variants of Cayley graphs, in prior works of the authors [BS21,BS20]. Recently, an improved bound for Cayley graphs has been given by Moorman-Ralli-Tetali [MRT20]. However, it is known that the above fact doesn't hold true for general regular graphs.…”

A spectral bound for vertex-transitive graphs and their spanning subgraphs

“…Under suitable hypothesis, there is an interplay between the expansions of these graphs as studied in [8]. Recently, an improved bound for Cayley graphs has been given by Moorman-Ralli-Tetali [24]. However, it is known that the above fact is not true for general regular graphs [24,Example 2.1].…”

“…Recently, an improved bound for Cayley graphs has been given by Moorman-Ralli-Tetali [24]. However, it is known that the above fact is not true for general regular graphs [24,Example 2.1]. This leads to the question that for which other subclass of regular graphs or even irregular graphs, does the above phenomenon occur.…”