The talk will summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schroedinger operators with random potential, quantum graph operators for trees with randomized edge lengths, and Schroedinger operators with radial quasi-periodic potentials. The results were derived jointly with R. Sims and S....

In this note, we prove that there exists a constant ρ such that any planar graph G of diameter ≤ 2R can be covered with at most ρ balls of radius R, a result conjectured by C. Gavoille, D. Peleg, A. Raspaud, and E. Sopena in 2001.

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold N , the set of mapping degrees D(M,N) is finite for any 3-manifold M , unless N is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere.

On January 18, 2022, around 1 million kilometers from Earth, five times the distance Earth to Moon, a large asteroid passed without harm Earth. Theoretically, however, event of falling into causing tsunami, is possible since there are over 27,000 near-Earth asteroids [1], and Earth’s surface covered in 71 percent by water. We introduce novel graph-grammar-based framework for tsunami simulations...

The monochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the ex...

An unpaired many-to-many -disjoint path cover (-DPC) of a graph  is a set of  disjoint paths joining  distinct sources and sinks in which each vertex of  is covered by a path. Here, a source can be freely matched to a sink. In this paper, we investigate unpaired many-to-many DPC's in a subclass of hpercube-like interconnection networks, called restricted HL-graphs, and show that every -d...

This short paper reports on the benefits that traversal queries over existing graph stores (such as RDF databases) can gain from a class of optimizations based on summaries. Summaries, also known as structural indexes, have been extensively covered in the literature (see [2] for a brief overview). Despite this, summary-based optimizations are not widely implemented. To make both graph traversal...