The directed vertex leafage of a chordal graph G is the smallest integer k such that G is the intersection graph of subtrees of a rooted directed tree where each subtree has at most k leaves. In this note, we show how to find in time O(kn) an optimal colouring, a maximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k ...

We introduce the path resistance method for lower bounds on the smallest nontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits; it uses clique embeddings to produce lower bounds on λ2 and star embeddings to produce lower bounds on the smallest Rayleigh quotient when there is a zero Dirichlet boundary condition. The met...

Shortest path problems are a well-studied class of problems in theoretical computer science. One particularly applicable type of shortest path problem is to find the geodesic shortest path on a terrain. This type of algorithm finds the shortest path between two points that stays on the surface of a terrain. The most popular methods for finding such a shortest path involve a variant of Dijkstra’...

We show that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. The next smallest one is the Meyerhoff manifold.

in the exterior domain Ω(1) := {x ∈ RN : ‖x‖ > 1}, where p > 1, x = (x1, . . . ,xN ) ∈ RN , N ≥ 2, Du= (∂u/∂x1, . . . ,∂u/∂xN ), ‖x‖ is usual Euclidean norm in RN . Throughout this paper we will assume that (A1) p ∈ C loc(Ω(1)), 0 < μ < 1, and p > 1 constant, (A2) A = (Aij(x))N×N is a real symmetric positive define matrix function with Aij ∈ C 1+μ loc (Ω(1)), i, j = 1, . . . ,N , and 0 < μ < 1....

We report the draft genome sequence of Commensalibacter papalotli strain MX01, isolated from the intestines of an overwintering monarch butterfly. The 2,332,652-bp AT-biased genome of C. papalotli MX01 is the smallest genome for a member of the Acetobacteraceae family and provides the first evidence of plasmids in Commensalibacter.