For a homogeneous space G/P , where P is a parabolic subgroup of a complex semisimple group G, an explicit Kähler–Einstein metric on it is constructed. The Einstein constant for the metric is 1. Therefore, the intersection number of the first Chern class of the holomorphic tangent bundle of G/P coincides with the volume of G/P with respect to this Kähler–Einstein metric, thus enabling us to com...

Let G = (V, E) be a connected graph and let d(u, v) denote the distance between vertices u, v ∈ V. A metric basis for G is a set B ⊆ V of minimum cardinality such that no two vertices of G have the same distances to all points of B. The cardinality of a metric basis of G is called the metric dimension of G, denoted by dim(G). In this paper we determine the metric dimension of the circulant grap...

The Farrell-Jones warping deformation is a powerful geometric construction that has been crucial in the proofs of many important contributions to the theory of manifolds of negative curvature. In this paper we study this construction in depth, in a more general setting, and obtain explicit quantitative results. The results in this paper are key ingredients in the problem of smoothing Charney-Da...

A resolving set of a graph G is a set S ⊆ V (G), such that, every pair of distinct vertices of G is resolved by some vertex in S. The metric dimension of G, denoted by β(G), is the minimum cardinality of all the resolving sets of G. Shamir Khuller et al. [10], in 1996, proved that a graph G with β(G) = 2 can have neither K5 nor K3,3 as its subgraph. In this paper, we obtain a forbidden subgraph...

A metric basis is a set W of vertices of a graph G(V,E) such that for every pair of vertices u, v of G, there exists a vertex w ∈ W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. The minimum cardinality of a metric basis for G is called the metric dimension. A pair of vertices u, v is said to be strongly resolved by...

Let G be a connected graph. A vertex w strongly resolves a pair u, v of vertices of G if there exists some shortest u− w path containing v or some shortest v − w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set for G is called the strong metric dimen...