Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=dp(x,y) and aquasi-axis if lim infd(x,y)/dp(x,y)>0 asx, y range over the vertex set of p anddp(x,y)?8. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if a is a translation whose directionD(a) is a thin end, then there exists an axis inD(a) andD(a-1) invariant under ar for somer not exceeding the maximum number of disjoint rays inD(a).The thinness ofD(a) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.
Authors:Raf Guns, Yu Xian Liu, and Dilruba Mahbuba
and related fields. In this paper, we study three centrality measures that are based on geodesics (shortest paths) between nodes: global Q-measures, local Q-measures, and betweenness centrality. Informally, these measures characterize different aspects
We characterize totally geodesic invariant submanifolds of C- and S-manifolds. We prove that the second fundamental form is parallel if and only if the submanifold of an S-manifold is totally geodesic. We show that this is not true for the C-manifolds.
In this paper we discuss the deformation retract of Buchdahi space into itself and onto a geodesics in Buchdahi space, we discuss also the deformation retract of Minkowski space into itself and onto a geodesics by using Lagrangian equations. We study also the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been discussed.
Authors:Alfonso Carriazo, Luis M. Fernández, and María Belén Hans-Uber
Summary We study some special types of slant submanifolds of S-manifolds related to the second fundamental form of the immersion: totally f-geodesic and f-umbilical, pseudo-umbilical and austere submanifolds. We also give several examples of such submanifolds.