Let M be a planar centrally symmetric convex body. If H is an a‰ne regular hexagon of the smallest (the largest) possible area inscribed in M, then M contains (respectively, the interior of M does not contain) an additional pair of symmetric vertices of the a‰neregular 12-gon TH whose every second vertex is a vertex of H. Moreover, we can inscribe in M an octagon whose three pairs of opposite v...

Let G be a connected finite graph. The average distance μ(G) of G is the average of the distances between all pairs of vertices of G. For a positive integer k a k-packing of G is a subset S of the vertex set of G such that the distance between any two vertices in S is greater than k. The k-packing number βk(G) of G is the maximum cardinality of a k-packing of G. We prove upper bounds on the ave...

An O (mn log n) algorithm is proposée to détermine a point of a network with m arcs and n vertices which minimizes the variance of the weighted distances to all vertices.

The vertex distance complement (VDC) matrix \(\textit{C}\), of a connected graph \(G\) with set consisting \(n\) vertices, is real symmetric \([c_{ij}]\) that takes the value \(n - d_{ij}\) where \(d_{ij}\) between vertices \(v_i\) and \(v_j\) for \(i \neq j\) 0 otherwise. spectrum subdivision join, \(G_1 \dot{\bigvee} G_2\) edge join \underline{\bigvee} regular graphs \(G_1\) \(G_2\) in terms ...

Let P be an H-polytope in R with vertex set V . The vertex centroid is defined as the average of the vertices in V . We first prove that computing the vertex centroid of an H-polytope, or even just checking whether it lies in a given halfspace, are #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an ǫ distance from it and prove this problem to...

The eccentricity e(u) of a vertex u in a digraph G is the maximum distance from u to any other vertex in G. A vertex v is an eccentric vertex of u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G, but with an arc from u to v in ED(G) if and only if v is an eccentric vertex of u in G. Given a positive integer k, the k iterated ...

Solids dispersed in a drying drop migrate to the (pinned) contact line. This migration is caused by outward flows driven by the loss of the solvent due to evaporation and by geometrical constraint that the drop maintains an equilibrium surface shape with a fixed boundary. Here, in continuation of our earlier paper, we theoretically investigate the evaporation rate, the flow field, and the rate ...

For a graph G = (V , E), a subset D V ðGÞ is said to be distance two-dominating set in G if for each vertex u 2 V D, there exists a vertex v 2 D such that dðu; vÞ 2. The minimum cardinality of a distance two-dominating set in G is called a distance twodomination number and is denoted by c2ðGÞ. In this note we obtain various upper bounds for c2ðGÞ and characterize the classes of graphs attaining...

The betweenness centrality of a vertex of a graph is the portion of shortest paths between all pairs of vertices passing through that vertex. We study selected general properties of this invariant and its relations to distance parameters (diameter, mean distance); also, there are studied properties of graphs whose vertices have the same value of betweenness centrality.

We study the behavior of dynamic programming methods for the tree edit distance problem, such as [5,13]. We show that those two algorithms may be described as decomposition strategies. We introduce the general framework of cover strategies, and we provide an exact characterization of the complexity of cover strategies. This analysis allows us to define a new tree edit distance algorithm, that i...