Towards a better approximation for sparsest cut?

We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+\epsilon)$-approximation on such general family of graphs

On the Fiedler value of large planar graphs

The Fiedler value $\lambda_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $\lambda_{2\max}$, and we show the bounds $2+\Theta(\frac{1}{n^2}) \leq \lambda_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $\lambda_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and Its Application

Modularity of minor-free graphs

We prove that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, modularity tends to 1 as the number of edges tends to infinity.Comment: 7 pages, 1 figur

Graph Theory for Modeling and Analysis of the Human Lymphatic System

The human lymphatic system (HLS) is a complex network of lymphatic organs linked through the lymphatic vessels. We present a graph theory-based approach to model and analyze the human lymphatic network. Two different methods of building a graph are considered: the method using anatomical data directly and the method based on a system of rules derived from structural analysis of HLS. A simple anatomical data-based graph is converted to an oriented graph by quantifying the steady-state fluid balance in the lymphatic network with the use of the Poiseuille equation in vessels and the mass conservation at vessel junctions. A computational algorithm for the generation of the rule-based random graph is developed and implemented. Some fundamental characteristics of the two types of HLS graph models are analyzed using different metrics such as graph energy, clustering, robustness, etc

Higher-Order Cheeger Inequality for Partitioning with Buffers

We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph $G=(V,E)$. The buffered expansion of a set $S \subseteq V$ with a buffer $B \subseteq V \setminus S$ is the edge expansion of $S$ after removing all the edges from set $S$ to its buffer $B$. An $\varepsilon$-buffered $k$-partitioning is a partitioning of a graph into disjoint components $P_i$ and buffers $B_i$, in which the size of buffer $B_i$ for $P_i$ is small relative to the size of $P_i$: $|B_i| \le \varepsilon |P_i|$. The buffered expansion of a buffered partition is the maximum of buffered expansions of the $k$ sets $P_i$ with buffers $B_i$. Let $h^{k,\varepsilon}_G$ be the buffered expansion of the optimal $\varepsilon$-buffered $k$-partitioning, then for every $\delta>0$, $$h_G^{k,\varepsilon} \le O_\delta(1) \cdot \Big( \frac{\log k}{ \varepsilon}\Big) \cdot \lambda_{\lfloor (1+\delta) k\rfloor},$$ where $\lambda_{\lfloor (1+\delta)k\rfloor}$ is the $\lfloor (1+\delta)k\rfloor$-th smallest eigenvalue of the normalized Laplacian of $G$. Our inequality is constructive and avoids the ``square-root loss'' that is present in the standard Cheeger inequalities (even for $k=2$). We also provide a complementary lower bound, and a novel generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higher-order Cheeger inequalities and another recent Cheeger-type inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion.Comment: 45 page