Incidence Graphs Of Projective Planes Research Articles

Generalized polygons and star graphs of cyclic presentations of groups

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that – via results of Edjvet and Vdovina – are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal. A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We obtain a classification of the concise cyclic presentations where the components of the corresponding star graph are generalized polygons. The classification reveals that both connected and disconnected star graphs are possible and that only generalized triangles (i.e. incidence graphs of projective planes) and regular complete bipartite graphs arise as the components. We list the presentations that arise in the Euclidean case and show that at most two of the corresponding groups are not SQ-universal (one of which is not SQ-universal, the other is unresolved). We obtain results that show that many of the SQ-universal groups are large.

The localization capture time of a graph

The localization game is a pursuit-evasion game analogous to Cops and Robbers, where the robber is invisible and the cops send distance probes in an attempt to identify the location of the robber. We present a novel graph parameter called the localization capture time, which measures how long the localization game lasts assuming optimal play. We conjecture that the localization capture time is linear in the order of the graph, and show that the conjecture holds for graph families such as trees and interval graphs. We study bounds on the localization capture time for trees and its monotone property on induced subgraphs of trees and more general graphs. We give upper bounds for the localization capture time on the incidence graphs of projective planes. We finish with new bounds on the localization number and localization capture time using treewidth.

Eulerian orientations and vertex-connectivity

It is well-known that every Eulerian orientation of an Eulerian 2k-edge-connected undirected graph is k-arc-connected. A long-standing goal in the area has been to obtain analogous results for vertex-connectivity. Levit, Chandran and Cheriyan recently proved in Levit et al. (2018) that every Eulerian orientation of a hypercube of dimension 2k is k-vertex-connected. Here we provide an elementary proof for this result.We also show other families of 2k-regular graphs for which every Eulerian orientation is k-vertex-connected, namely the even regular complete bipartite graphs, the incidence graphs of projective planes of odd order, the line graphs of regular complete bipartite graphs and the line graphs of complete graphs.Furthermore, we provide a simple graph counterexample for a conjecture of Frank attempting to characterize graphs admitting at least one k-vertex-connected orientation.

How many zombies are needed to catch the survivor on toroidal grids?

In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is at least 1/2.This variant of the game was recently investigated for several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and grids Pn□Pn. However, unfortunately, still very little is known for toroidal grids Cn□Cn: the zombie number of Cn□Cn is at least n/(ωlog⁡n), where ω=ω(n) is any function going to infinity as n→∞, and no upper bound is known except a trivial bound of O(n2log⁡n). In this note, we provide an approach that gives an embarrassing bound of O(n2) but it is possible that (with more careful, deterministic, argument) it might actually give a bound of O(n3/2). On the other hand, by analyzing a specific strategy for the survivor, it seems that one could slightly improve the lower bound to n/ω. In any case, we are far away from understanding this intriguing question.

Projective Networks: Topologies for Large Parallel Computer Systems

The interconnection network comprises a significant portion of the cost of large parallel computers, both in economic terms and power consumption. Several previous proposals exploit large-radix routers to build scalable low-distance topologies with the aim of minimizing these costs. However, they fail to consider potential unbalance in the network utilization, which in some cases results in suboptimal designs. Based on an appropriate cost model, this paper advocates the use of networks based on incidence graphs of projective planes, broadly denoted as Projective Networks. Projective Networks rely on generalized Moore graphs with uniform link utilization and encompass several proposed direct (PN and demi-PN) and indirect (OFT) topologies under a common mathematical framework. Compared to other proposals with average distance between 2 and 3 hops, these networks provide very high scalability while preserving a balanced network utilization, resulting in low network costs.

A probabilistic version of the game of Zombies and Survivors on graphs

We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighboring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighboring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is at least 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids.

Regular graphs with maximal energy per vertex

We study the energy per vertex in regular graphs. For every k⩾2, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs of projective planes of order k−1 or, in case k=2, the disjoint union of triangles and hexagons. For every k, we also construct k-regular subgraphs of incidence graphs of projective planes for which the energy per vertex is close to the upper bound. In this way, we show that this upper bound is asymptotically tight.

On orienting graphs for connectivity: Projective planes and Halin graphs

Nash-Williams proved that the edges of a k-edge connected (undirected) graph can be oriented such that the resulting directed graph is ⌊k2⌋-edge connected. A long-standing goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity, element connectivity, and hypergraph edge connectivity. We focus on two special classes of graphs, namely, incidence graphs of projective planes and (generalized) Halin graphs, and we prove some analogs of Nash-Williams’ result for these classes.

Reflection group of the quaternionic Lorentzian Leech lattice

We study a second example of the phenomenon studied in the article “The complex Lorentzian Leech lattice and the bimonster.” We find 14 roots in the automorphism group of the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram given by the incidence graph of projective plane over F 2 . We prove that the reflections in these roots generate the automorphism group of L. The investigation is guided by an analogy with the theory of Weyl groups. There is a unique point in the quaternionic hyperbolic space fixed by the “diagram automorphisms” that we call the Weyl vector. The unit multiples of the 14 roots forming the diagram are the analogs of the simple roots. The 14 mirrors perpendicular to the simple roots are the mirrors that are closest to the Weyl vector.

An Extremal Characterization of the Incidence Graphs of Projective Planes

Let G be a 4-cycle free, bipartite graph on 2n vertices with partitions of equal cardinality n. Let c6(G) denote the number of cycles of length 6 in G. We prove that for n ≥ 3, c6(G) ≤ \(\frac{1} {3}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)(n - r_n ) \), where \(r_n = \frac{1} {2} + \frac{{\sqrt {4n - 3} }} {2} \) , with equality if and only if G is the incidence point-line graph of a projective plane.