The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

A new method for constructing infinite families of k-tight optimal double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. DLN has been widely used in the designing of local area networks and distributed systems. In this paper, a new method for constructing infinite families of k-tight optimal DLN is presented. For k = 0,1,…,40, the infinite families of k-tight optimal DLN can be constructed by the new method, where the number nk(t,a) of their nodes is a polynomial of degree 2 in t and contains a parameter a. And a conjecture is proposed.     

Procreating tiles of double commutative-step digraphs

Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L（l, h, x, y）, Aguil5 et al. define a discrete iteration L（p） = L（l ＋ 2p, h ＋ 2p, x ＋ p, y ＋ p）, p = 0, 1, 2,..., over L-shapes （equivalently over double commutative-step digraphs）, and obtain an orbit generated by L（l, h, x,y）, which is said to be a procreating k-tight tile if L（p）（p = 0, 1, 2, ~ ~ ~ ） are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L（l, h, x, y）, 0 ≤ y - x ≤ 2k ＋ 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches.     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is: for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained: (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

k-紧优有向双环网络无限族的构建

本文给出了一种方法用于构造k-紧优双环网络无限族(k≥1),并用此方法构造出了4族3-紧优无限族,3族新的4-紧比无限族,3族5-紧优无限族及2族6-紧优无限族．     

2族3 -紧优的有向双环网络无限族

该文给出一种寻找k -紧优的双环网络无限族(k>=0)的方法, 利用此方法得到了2族3 -紧优的有向双环网络无限族     

2族3-紧优的有向双环网络无限族

该文给出一种寻找k-紧优的双环网络无限族(k≥0)的方法,利用此方法得到了2族 3-紧优的有向双环网络无限族．     

20类新的2紧优双环网络无限族

本文利用双环网的L型瓦方法,给出了20类新的2紧优双环网无限族类.     

An optimal version of an inequality involving the third symmetric means

Let (GA) _n ^[k](a), A _n (a), G _n (a) be the third symmetric mean of k degree, the arithmetic and geometric means of a ₁, …, a _n (a _i > 0, i = 1, …, n), respectively. By means of descending dimension method, we prove that the maximum of p is k−1/n−1 and the minimum of q is n/n−1(k−1/k) ^k/n so that the inequalities {fx505-1} hold.     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010