A note on Steiner reciprocal degree distance

The concept of reciprocal degree distance [Formula: see text] of a connected graph [Formula: see text] was introduced in 2012. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. The [Formula: see text]-center Steiner reciprocal degree distance defined as [Formula: see text], where [Formula: see text] is the Steiner [Formula: see text]-distance of [Formula: see text] and [Formula: see text] is the degree of the vertex [Formula: see text] in [Formula: see text]. Motivated from Zhang’s paper [X. Zhang, Reciprocal Steiner degree distance, Utilitas Math., accepted for publication], we find the expression for [Formula: see text] of complete bipartite graphs. Also, we give a straightforward method to compute Steiner Gutman index and Steiner degree distance of path.

A Note on the Steiner k-Diameter of Tensor Product Networks

Given a graph [Formula: see text] and [Formula: see text], the Steiner distance [Formula: see text] is the minimum size among all connected subgraphs of [Formula: see text] whose vertex sets contain [Formula: see text]. The Steiner [Formula: see text]-diameter [Formula: see text] is the maximum value of [Formula: see text] among all sets of [Formula: see text] vertices. In this short note, we study the Steiner [Formula: see text]-diameters of the tensor product of complete graphs.

The Steiner Formula and the Polar Moment of Inertia for the Closed Planar Homothetic Motions in Complex Plane

The Steiner area formula and the polar moment of inertia were expressed during one-parameter closed planar homothetic motions in complex plane. The Steiner point or Steiner normal concepts were described according to whether rotation number was different from zero or equal to zero, respectively. The moving pole point was given with its components and its relation between Steiner point or Steiner normal was specified. The sagittal motion of a winch was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of winch and the moving arm of winch. The results obtained in the second section of this study were applied for this motion.

MESH FREE ESTIMATION OF THE STRUCTURE MODEL INDEX

The structure model index (SMI) is a means of subsuming the topology of a homogeneous random closed set under just one number, similar to the isoperimetric shape factors used for compact sets. Originally, the SMI is defined as a function of volume fraction, specific surface area and first derivative of the specific surface area, where the derivative is defined and computed using a surface meshing. The generalised Steiner formula yields however a derivative of the specific surface area that is – up to a constant – the density of the integral of mean curvature. Consequently, an SMI can be defined without referring to a discretisation and it can be estimated from 3D image data without need to mesh the surface but using the number of occurrences of 2×2×2 pixel configurations, only. Obviously, it is impossible to completely describe a random closed set by one number. In this paper, Boolean models of balls and infinite straight cylinders serve as cautionary examples pointing out the limitations of the SMI. Nevertheless, shape factors like the SMI can be valuable tools for comparing similar structures. This is illustrated on real microstructures of ice, foams, and paper.