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...

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed 1 -injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no nite cover containing an embe...

The graph theory [1], traditionally covered in Discrete Mathematics and Algorithms courses [2, 3], introduces students to the modeling of various types of relations between entities in technical, physical, chemical, biological, and social systems and their complex dynamics. Graph representations are widely used in computer science (analyses of the flow of computation, data communication in netw...

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 2edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exa...