Diameter Bounds for Planar Graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight

An inequality between the diameter and the inverse dual degree of a tree

Let T be a nontrivial tree with diameter D(T) and radius R(T). Let I(T) be the inverse dual degree of T which is defined to be , where for uV(T). For any longest path P of T, denote by a(P) the number of vertices outside P with degree at least 2, b(P) the number of vertices on P with degree at least 3 and distance at least 2 to each of the end-vertices of P, and c(P) the number of vertices adjacent to one of the end-vertices of P and with degree at least 3. In this note we prove that . As a corollary we then get with equality if and only if T is a path of at least four vertices. The latter inequality strengthens a conjecture made by the program Graffiti.postprin

Graph Clustering using Effective Resistance

$\def\vecc#1{\boldsymbol{#1}}$We design a polynomial time algorithm that for any weighted undirected graph G = (V, E,\vecc w) and sufficiently large $\delta > 1$, partitions $V$ into subsets $V_1, \ldots, V_h$ for some $h\geq 1$, such that $\bullet$ at most $\delta^{-1}$ fraction of the weights are between clusters, i.e. $$w(E - \cup_{i = 1}^h E(V_i)) \lesssim \frac{w(E)}{\delta};$$ $\bullet$ the effective resistance diameter of each of the induced subgraphs $G[V_i]$ is at most $\delta^3$ times the average weighted degree, i.e. $$\max_{u, v \in V_i} \mathsf{Reff}_{G[V_i]}(u, v) \lesssim \delta^3 \cdot \frac{|V|}{w(E)} \quad \text{ for all } i=1, \ldots, h.$$ In particular, it is possible to remove one percent of weight of edges of any given graph such that each of the resulting connected components has effective resistance diameter at most the inverse of the average weighted degree. Our proof is based on a new connection between effective resistance and low conductance sets. We show that if the effective resistance between two vertices $u$ and $v$ is large, then there must be a low conductance cut separating $u$ from $v$. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design

Marrow-Derived Stem Cell Motility in 3D Synthetic Scaffold Is Governed by Geometry Along With Adhesivity and Stiffness

Author Manuscript 2012 May 21.Design of 3D scaffolds that can facilitate proper survival, proliferation, and differentiation of progenitor cells is a challenge for clinical applications involving large connective tissue defects. Cell migration within such scaffolds is a critical process governing tissue integration. Here, we examine effects of scaffold pore diameter, in concert with matrix stiffness and adhesivity, as independently tunable parameters that govern marrow-derived stem cell motility. We adopted an “inverse opal” processing technique to create synthetic scaffolds by crosslinking poly(ethylene glycol) at different densities (controlling matrix elastic moduli or stiffness) and small doses of a heterobifunctional monomer (controlling matrix adhesivity) around templating beads of different radii. As pore diameter was varied from 7 to 17 µm (i.e., from significantly smaller than the spherical cell diameter to approximately cell diameter), it displayed a profound effect on migration of these stem cells—including the degree to which motility was sensitive to changes in matrix stiffness and adhesivity. Surprisingly, the highest probability for substantive cell movement through pores was observed for an intermediate pore diameter, rather than the largest pore diameter, which exceeded cell diameter. The relationships between migration speed, displacement, and total path length were found to depend strongly on pore diameter. We attribute this dependence to convolution of pore diameter and void chamber diameter, yielding different geometric environments experienced by the cells within. Bioeng. 2011; 108:1181–1193(National Institute of General Medical Sciences (U.S.) (NRSA Fellowship GM083472)National Institutes of Health (U.S.) (National Institute of General Medical Sciences (U.S.) Cell Migration Consortium Grant GM064346)National Science Foundation (U.S.) (CAREER CBET-0644846