For a graph G, let diam(G) denote the diameter of G. For any two vertices u and v in G, let d(u, v) denote the distance between u and v. A multi-level distance labeling (or distance labeling) for G is a function f that assigns to each vertex of G a non-negative integer such that for any vertices u and v, |f(u)− f(v)| ≥ diam(G) − dG(u, v) + 1. The span of f is the largest number in f(V ). The ra...

2.2 Distance between Two Vertices The distance between two vertices u and v is defined as the length of the shortest path connecting them. Remark 1. The number of paths connecting two vertices could be exponential. As shown in the below illustration, there are around 3 ”blocks” to choose from given that there are n vertices in total and each block takes around 3 vertices. The number of paths av...

16 other words, the ending point is not situated above the main diagonal x = y. When the path never goes above this diagonal, we simply call it an underdiagonal path. Shapiro's paths 8] are another interesting kind of paths. These paths are characterized by the fact that the northeast steps on the main diagonal may have diierent colours from the northeast steps on the other diagonals. In other ...

We aim here at introducing a new point of view of the Laplacian of a graph, Γ. With this purpose in mind, we consider L as a kernel on the finite space V (Γ), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties as maximum and energy principles and the equilibrium principle on any proper subset of V (Γ). If Γ is a proper se...

Given a polygonal path P with vertices p1, p2, . . . , pn ∈ Rd and a real number t ≥ 1, a path Q = (pi1 , pi2 , . . . , pik) is a t-distance-preserving approximation of P if 1 = i1 < i2 < . . . < ik = n and each straight-line edge (pij , pij+1) of Q approximates the distance between pij and pij+1 along the path P within a factor of t. We present exact and approximation algorithms that compute s...

Typical cryptography schemes are not well suited for low complexity types of equipment, e.g., Internet of things (IoT) devices, as they may need high power or impose high computational complexity on the device. Physical (PHY) layer security techniques such as beamforming (in multiple antennas systems)  are possible alternatives to provide security for such applications. In this paper, we consid...