An interesting class of orthogonal representations consists the so-called turn-regular ones, i.e., those that do not contain any pair reflex corners point to each other inside a face. For such representation H it is possible compute in linear time minimum-area drawing, drawing minimum area over all assignments vertex and bend coordinates H. In contrast, finding NP-hard if non-turn-regular. This...

This paper reports the characteristics of and mutual relationships between various combinatorial structures that give rise to certain imprimitive nonsymmetric three-class association schemes. Nontrivial relation graphs of an imprimitive symmetric 2-class association scheme are m ◦ Kr, (the union of m copies of the complete graph on r vertices) and its complement m ◦Kr, (the complete m-partite s...

Each agent of a finite population interacts strategically with each of his neighbours on a graph. All agents have the same pair of available actions. In every period, each agent chooses a particular action if at least a proportion p of his neighbours has chosen this action in the previous period. Contagion is said to occur if one action spreads from a particular group of agents to the entire po...

We use throughout the notation of [5], to which we refer for the basic theory of finite rank 3 permutation groups G. The solvable primitive rank 3 permutation groups have been determined by Foulser [4] and, independently, by Dornhoff [2]. Since rank 3 groups of odd order are solvable we assume that G has even order. Then the graphs JA and JT associated with the non-trivial orbitals A and T of G...

A map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the flags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integers p ≥ 3 and q ≥ 3 satisfying 1/p + 1/q ≤ 1/2, there is an orientable regular map ...

A coterie under an underlying set U is a family of subsets of U such that every pair of subsets has at least one element in common, but neither is a subset of the other. A coterie C under U is said to be nondominated (ND) if there is no other coterie D under U such that, for every Q ∈ C, there exists Q ∈ D satisfying Q ⊆ Q. We introduce the operation σ which transforms a ND coterie to another N...

Passive and Active Compensability Multicriteria ANalysis (PACMAN) is a multiple criteria methodology based on a decision maker oriented notion of compensation, called compensability. An important feature of PACMAN is a possible asymmetry of the connected decision procedure, since compensability is determined for each ordered pair of criteria, distinguishing the compensating criterion from the c...

Let Γ = (X, R) denote a distance-regular graph with distance function ∂ and diameter d ≥ 3. For 2 ≤ i ≤ d, by a parallelogram of length i, we mean a 4-tuple xyzu of vertices in X such that ∂(x, y) = ∂(z, u) = 1, ∂(x, u) = i, and ∂(x, z) = ∂(y, z) = ∂(y, u) = i − 1. Suppose the intersection number a1 = 0, a2 6= 0 in Γ. We prove the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial and con...

We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this linking step until a regular random graph, where all nodes have degree d, forms. We view this process as a multivariate aggregation process, and formally solv...

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1 + ε)-resistance sparsifier of size Õ(n/ε), and conjecture this bound holds for all graphs on n nodes. In comparison, spectral sparsification i...