Let Γ be a graph with vertex set V, and let b nonnegative integers. A subset C of V is called an (a,b)-regular in if every has exactly neighbors V∖C C. In particular, (0,1)-regular sets (1,1)-regular are perfect codes total Γ, respectively. group G said to there exists Cayley which admits as set. this paper we prove that, for any generalized dihedral or order 4p pq some primes p q, nontrivial s...

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronöı and later generalized by Koecher to arbitrary number fields. One knows that up to a natural “change of variables” equivalence, there are only finitely many perfect f...

In a graph $$\Gamma $$ with vertex set V, subset C of V is called an (a, b)-regular if every in has exactly neighbors and $$V\setminus C$$ b C, where are nonnegative integers. the literature, (0, 1)-regular sets known as perfect codes (1, total codes. this paper, we prove that, for any finite group G, non-trivial normal subgroup H G code some Cayley then pair integers $$0\leqslant a\leqslant |H...

A perfect covering of a v-set X is a set of subsets of X (called blocks), such that every pair of elements of X (points) occurs in a unique block. A perfect covering is also referred to in the literature as a pairwise balanced design or a finite linear space. The quantity of g’“‘(v) is defined to be the minimum number of blocks in a perfect covering of a v-set, in which the longest block has si...

E. Dawson Infonnation Security Research Centre Queensland University of Technology GPO Box 2434 Brisbane Queensland 400 1 Australia E.S.~ahmocKlian Department of ~athematical Sciences Sharif University of Technology PO Box 11365 9415 Tehran Iran Alan Rahilly Centre for Combinatorics Department of ~athematics University of Queensland Brisbane Queensland 4072 Australia Perfect threshold schemes w...

Let G be a connected, unipotent, perfect group scheme over an algebraically closed field of characteristic p > 0 and let E be a biextension (cf. [SGA7tI], exp. VII, no. 1) of G ×k G by the discrete group Qp/Zp. When E is skewsymmetric (cf. sec. 3.1), V. Drinfeld defined in [Dr] a certain metric group A (cf. appendix) associated to E. In this article, we prove a conjecture stated in [Dr] (cf. se...

In a graph G = (V; E), a set of vertices S is nearly perfect if every vertex in V ? S is adjacent to at most one vertex in S. Nearly perfect sets are closely related to 2-packings of graphs, strongly stable sets, dominating sets and eecient dominating sets. We say a nearly perfect set S is 1-minimal if for every vertex u in S, the set S ? fug is not nearly perfect. Similarly, a nearly perfect s...

Band structure of inorganic lead halide perovskites is substantially different from the band group IV, III–V and II–VI semiconductors. However, standard empirical tight-binding model with sp3d5s? basis gives nearly perfect fit calculated in density functional theory. The calculations ultrathin CsPbI3 layer show good agreement corresponding DFT calculations. parameters corrected for experimental...

An r-perfect code of a graph G = (V, E) is a set C ⊆ V such that the r-balls centered at vertices of C form a partition of V . It is proved that the direct product of Cm and Cn (r ≥ 1, m,n ≥ 2r+1) contains an r-perfect code if and only if m and n are each a multiple of (r + 1)2 + r2 and that the direct product of Cm, Cn, and C` (r ≥ 1, m,n, ` ≥ 2r +1) contains an r-perfect code if and only if m...

Let G = (V,E) be a weighted undirected graph with vertex set V , edge set E and a weight function d. Thus, d(u, v) denotes the weight of any edge (u, v) ∈ E. A matching M ⊆ E is a collection of edges such that every node in V is incident to at most one edge in M . The matching is perfect if every node in V is incident to exactly one edge in M . The cost of the matching is given by ∑ (u,v)∈M d(u...