The Johnson graph J(d, r) is unique if (d, r) ≠ (2, 8)

AbstractWe use the classical Root Systems to show the Johnson graph J(d, r) (2 ⩽ 2d ⩽ r < ∞) is the unique distance-regular graph with its intersection numbers when (d, r) ≠ (2, 8). Since this exceptional case has been dealt with by Chang [6] this completes the characterization problem for Johnson graph

Corporate R&D and Firm Efficiency: Evidence from Europe’s Top R&D Investors

The main objective of this study is to investigate the impact of corporate R&D activities on firms' performance, measured by labour productivity. To this end, the stochastic frontier technique is applied, basing the analysis on a unique unbalanced longitudinal dataset consisting of 532 top European R&D investors over the period 2000–2005. R&D stocks are considered as pivotal input in order to control for their particular contribution to firm-level efficiency. Conceptually, the study quantifies the technical inefficiency of a given company and tests empirically whether R&D activities could explain the distance from the efficient boundary of the production possibility set, i.e. the production frontier. From a policy perspective, the results of this study suggest that – if the aim is to leverage companies' productivity – emphasis should be put on supporting corporate R&D in high-tech sectors and, to some extent, in medium-tech sectors. By contrast, supporting corporate R&D in the low-tech sector turns out to have a minor effect. Instead, encouraging investment in fixed assets appears vital for the productivity of low-tech industries. However, with regard to firms' technical efficiency, R&D matters for all industries (unlike capital intensity). Hence, the allocation of support for corporate R&D seems to be as important as its overall increase and an 'erga omnes' approach across all sectors appears inappropriate.corporate R&D, productivity, technical efficiency, stochastic frontier analysis

Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $\zeta^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $\zeta^* (T)\leq 2$ for every tree $T$ and give an upper bound on $\zeta^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $\zeta^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $\zeta^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $\zeta^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn

Shift Radix Systems - A Survey

Let $d\ge 1$ be an integer and ${\bf r}=(r_0,\dots,r_{d-1}) \in \mathbf{R}^d$. The {\em shift radix system} $\tau_\mathbf{r}: \mathbb{Z}^d \to \mathbb{Z}^d$ is defined by $$\tau_{{\bf r}}({\bf z})=(z_1,\dots,z_{d-1},-\lfloor {\bf r} {\bf z}\rfloor)^t \qquad ({\bf z}=(z_0,\dots,z_{d-1})^t).$$ $\tau_\mathbf{r}$ has the {\em finiteness property} if each ${\bf z} \in \mathbb{Z}^d$ is eventually mapped to ${\bf 0}$ under iterations of $\tau_\mathbf{r}$. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of $\tau_\mathbf{r}$ with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure

Bounds On (t,r)(t,r) Broadcast Domination of nn-Dimensional Grids

In this paper, we study at a variant of graph domination known as $(t, r)$ broadcast domination, first defined by Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a distance $d < t$ from the broadcast. A vertex is considered dominated if it receives $r$ total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as well as methods of computing them. Also, when $r \ge 2$ we describe a family of counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast domination.Comment: 15 pages, 4 figure

Universally Rigid Framework Attachments

A framework is a graph and a map from its vertices to R^d. A framework is called universally rigid if there is no other framework with the same graph and edge lengths in R^d' for any d'. A framework attachment is a framework constructed by joining two frameworks on a subset of vertices. We consider an attachment of two universally rigid frameworks that are in general position in R^d. We show that the number of vertices in the overlap between the two frameworks must be sufficiently large in order for the attachment to remain universally rigid. Furthermore, it is shown that universal rigidity of such frameworks is preserved even after removing certain edges. Given positive semidefinite stress matrices for each of the two initial frameworks, we analytically derive the PSD stress matrices for the combined and edge-reduced frameworks. One of the benefits of the results is that they provide a general method for generating new universally rigid frameworks.Comment: 16 pages, 4 figure

On the partition dimension of trees

Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $\Pi$ is the vector $r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $\Pi$ of $V$ is a \emph{resolving partition} of $G$ if different vertices of $G$ have different partition representations, i.e., for every pair of vertices $u,v\in V$, $r(u|\Pi)\ne r(v|\Pi)$. The \emph{partition dimension} of $G$ is the minimum number of sets in any resolving partition of $G$. In this paper we obtain several tight bounds on the partition dimension of trees