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...

(1) Regular control does not increase the generating power of 1restricted [d]K-iteration grammars provided that K ⊇ SYMBOL, and K is closed under isomorphism and under union with SYMBOL-languages. (2) Let Γ be a prequasoid closed under the regular operations. If K is a prequasoid [pseudoid], then H(Γ) ⊆ H(Γ,K) [η(Γ) ⊆ η(Γ,K)]. In particular we have H(Γ) ⊆ (Γ)ETOL and η(Γ) ⊆ (Γ)EDTOL. (3) Under ...

In this paper we give lower bounds on the size of (a, b) elementary trapping sets (ETSs) belonging to variable-regular LDPC codes with any girth, g, and irregular ones with girth 8, where a is the size, b is the number of degree-one check nodes and satisfy the inequality b a < 1. Our proposed lower bounds are analytical, rather than exhaustive search-based, and based on graph theories. The nume...

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph Γ is distance-regular and homogeneous. More precisely, Γ is edge-distance-regular if and only if it is bipartite distance-reg...

γ-Oryzanol, a natural mixture of ferulic acid esters of triterpene alcohols and sterols, are an important bioactive components present in rice bran oil. In light of the recent increase in the popularity of wild rice among consumers, and the possibility of a direct relationship between γ-oryzanol composition and its bioactivity, the oryzanol profile of major wild rice (Zizania palustris) grown i...

We present a simple, explicit construction of an infinite family F of bounded-degree ’unique-neighbor’ expanders Γ; i.e., there are strictly positive constants α and , such that all Γ = (X,E(Γ)) ∈ F satisfy the following property. For each subset S of X with no more than α|X| vertices, there are at least |S| vertices in X \ S that are adjacent in Γ to exactly one vertex in S. The construction o...

In the first part of the paper we show that the Busemann 1-compactification of the Siegel upper half plane of rank n: SHn = Sp(n, R)/Kn is the compactification as a bounded domain. In the second part of the paper we study certain properties of discrete groups Γ of biholomorphisms of SHn. We show that the set of accumulation points of the orbit Γ(Z) on the Shilov boundary of SHn is independent o...

Let Γ = Cay(G, S) and G ≤ X ≤ AutΓ. We say Γ is (X, 1)-regular Cayley graph if X acts regularly on its arcs. Γ is said to be corefree if G is core-free in some X ≤ Aut(Cay(G, S)). In this paper, we prove that if an (X, 1)-regular Cayley graph of valency 5 is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free ...

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...