Research On The Diameter Of Double Loop Networks

Let n, s₁ and s₂ be three positive integers, where s₁ < s₂ < n, gcd(n, s₁, s₂) = 1, G(n; s₁, s₂) be a double loop network with n nodes, its vertex-set isV = Z_n = {0,1,2,…,n-1},its edge(arc)-set isE = {i→ i + s₁(mod n), i →i + s₂(mod n), …, i→i + s_p(mod n) | i ∈ Z_n}. s₁ and s₂ being its steps, and d(n; s₁, s₂) being its diameter. Letd(n) = min{d(n; s₁, s₂) | s₁ < s₂ < n}, d₁(n) = min{d(n; 1, s) | 1 < s < n}. It is known that d₁(n) ≥ d(n) ≥ 「(3n)^1/2(?) - 2 = lb(n). Ifd(n; s₁, s₂) = d(n) = lb(n) + k, k≥0,then G(n; s₁, s₂) is called a k-tight optimal double loop network. If d₁(n) > d(n) = lb(n) + k, then the integer n is called singular (k-tight) integer. In this thesis, we do the following works:(1) give new infinite families of k tight optimal double loop networks G(n; 1, s), the number of whose nodes being n(t, a) = 3t² + (2i -1)t + B(a), where i = 1, 2, 3, k = 0,1,2,…,20;(2) present a method for generating infinite families of singular k-tight integers and generate such ones for k = 1,2, …,20; for singular k tight integers n, we consider the difference of d₁(n) — d(n), where k = 1,2, …, 7;(3) give the increasement of average distanse when an arc is the only failed component.