Let G be a graph of order n and let q(G) be the largest eigenvalue of the signless Laplacian of G. It is shown that if k > 2, n > 5k, and q(G) > n + 2k − 2, then G contains a cycle of length l for each l ∈ {3, 4, . . . , 2k + 2}. This bound on q(G) is asymptotically tight, as the graph Kk ∨Kn−k contains no cycles longer than 2k and q(Kk ∨ Kn−k) > n + 2k − 2− 2k(k − 1) n + 2k − 3 . The main resu...

We study the structure of graphs with high minimum degree conditions and given odd girth. For example, the classical work of Andrásfai, Erdős, and Sós implies that every n-vertex graph with odd girth 2k + 1 and minimum degree bigger than 2n 2k+1 must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the...

In this paper, we introduce the hyper-star graph HS(n, k) as a new interconnection network, and discuss its properties such as faulttolerance, scalability, isomorphism, routing algorithm, and diameter. A hyper-star graph has merits when degree × diameter is used as a desirable quality measure of an interconnection network because it has a small degree and diameter. We also introduce a variation...

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families congruences modulo arbitrary powers $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $k$, have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\mathrm{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}...

In this article, magnitude relation properties of Radix-2k SD number are discussed. Until now, the Radix-2k SD Number is proposed for the high-speed calculations for RSA Cryptograms. In RSA Cryptograms, many modulo calculations are used, and modulo calculations need a comparison between two numbers. In this article, we discussed about a magnitude relation of Radix-2k SD Number. In the first sec...

(a) [Problem 2.2 in DFT] For u 1 : For u 2 : For u 3 : Note that u 1 = u 2 2 = ∞ 0 1 · dt = ∞, while u ∞ = 1 and P ow(u) = lim T →∞ 1 2T T 0 1 · dt = 1 2. T −T u 2 (t)dt = T 0 u 2 (t)dt = T 0 |u(t)|dt = k−1 i=0 2 2i for 2 2k−1 ≤ T ≤ 2 2k , k ≥ 1. Hence, u 1 = u 2 = ∞ while u ∞ = 1. Finally: P ow(u) = lim T →∞ 1 2T T −T u 2 (t)dt =    lim k→∞ 1 2 2k k−1 i=0 2 2i = 1 4 when T = 2 2k−1 , k ≥ 1 ...