arXiv subjects
Publications and source records attributed to Mugdha Mahesh Pokharanakar.
We show, under a certain assumption, that the Cheeger constant of a connected Cayley graphon is bounded from above in terms of its bipartiteness ratio, and also in terms of the gap between the top of the spectrum of its Laplacian and $2$. Using this result, we prove that if a connected Cayley graphon $C(G,γ)$ is bipartite, then there exists an index two subgroup $H$ of $G$ that avoids the support of $γ$.
The higher-order Cheeger inequalities were established for graphs by Lee, Oveis Gharan and Trevisan. We prove analogous inequalities for graphons in this article.
We introduce the notion of bipartiteness ratio for graphons. We prove the dual Cheeger-Buser inequality for graphons, which relates the gap between $2$ and the top of the spectrum of the Laplacian of a graphon with its bipartiteness ratio. The dual Cheeger-Buser inequality was established by Trevisan and Bauer-Jost for graphs. Our result is an analog of that for graphons.