Let M = {G / Kn,n ~ G ~ Kn V for some n 2:: 3} where V is the join operation. The author and N.K. Khachatrian proved that a connected graph G of order at least 3 is Hamiltonian if d(u) + d(v) 2:: IN(u) U N(v) U N(w)1 for each triple of vertices U,V,w with d(u,v) = 2 and w E N(u) n N(v) (where N( x) is the neighborhood of x). Here we prove that a graph G satisfying the above conditions is Hamilt...

The operator A(D) is called the radial component of D. Many special cases have been considered (see e.g. [1, §2], [4, §5], [5, §3], [7, §7 ], [8, Chapter IV, §§3-5]). Suppose now dv (resp. dw) is a positive measure on V (resp. W) which on any coordinate neighborhood is a nonzero multiple of the Lebesgue measure. Assume dg is a bi-invariant Haar measure on G. Given u E Cc (G X W) there exists [7...

where a and W satisfy (a1) a(t) is a continuous almost periodic function of t with a(t) ≥ a0 > 0 for all t ∈ R. (W1) There is a ξ ∈ R\{0} such that W ∈ C(R\{ξ},R). (W2) limx→ξW (x) = −∞. (W3) There is a neighborhood N of ξ and U ∈ C(N\{ξ},R) such that |U(x)| → ∞ as x→ ξ and |U ′(x)|2 ≤ −W (x) for x ∈ N\{ξ}, (W4) W (x) < W (0) = 0 if x 6= 0 and W ′′(0) is negative definite. (W5) There is a const...

The Threshold Dominating Set problem is that of determining for a graph G = (V, E) whether there is a subset V ′ ⊆ V of size k, such that for each vertex v ∈ V there are at least r elements of the closed neighborhood N [v] that belong to V ′. We consider the complexity of the problem parameterized by the pair (k, r). It is trivial to observe that this is hard for W [2]. It can also be easily sh...

We consider shape optimization problems governed by the unsteady Navier-Stokes equations by applying the method of mappings, where the problem is transformed to a reference domain Ωref and the physical domain is given by Ω = τ(Ωref) with a domain transformation τ ∈ W (Ωref). We show the Fréchet-differentiability of τ 7→ (v, p)(τ) in a neighborhood of τ = id under as low regularity requirements ...

Let G be a connected graph with vertex set V (G) and edge E(G). The open hop neighborhood of v ∈ is the NG(v, 2) = {w : dG(v, w) 2}, where denotes distance between w. A non-empty S ⊆ locating if NG(u, ∩ ̸= for every pair distinct vertices u, \ S. smallest cardinality G, denoted by lhn(G) called number G. This study focuses mainly on concept in graphs. Characterizations sets join corona two graph...

The neighborhood has been used as a sampling unit for exploring variations in health outcomes. In a variety of studies census tracts or ZIP codes have been used as proxies for neighborhoods because the boundaries are pre-defined units for which other data are readily available. However these spatial units can be arbitrary and do not account for social-cultural behaviors and identities that are ...

We can construct the neighborhood complex N(G), with vertices v1, ..., vn, from the graph G, in such a way that, for each vertex v of G, there is a simplex containing the vertex v along with its neighbouring vertices, corresponding to the directed edges v → w. To construct this, we can take each vertex in v1, ..., vn one by one, and construct the simplex along with it’s neighbors, each time. Th...

Different social groups tend to settle in different parts of cities leading over time to social segregation. Neighborhood obviously plays an important role in this process – and what constitutes neighborhood is a cognitive notion. In segregation analysis neighborhood borders are often drawn arbitrarily or simple assumptions are used to weight neighbor influences. Some authors have developed ide...