Image compression is the key technology in the development of various multimedia applications. Vector quantization is a universal and powerful technique to compress a data sequence, such as speech or image, resulting in some loss of information. In VQ, minimization of Mean Square Error (MSE) between code book vectors and training vectors is a non-linear problem. Traditional LBG types of algorithms used for designing the codebooks for Vector Quantizer converge to a local minimum, which depends on the initial code book. Memetic algorithms (MAs) are population-based meta-heuristic search approaches that have been receiving increasing attention in the recent years. These algorithms are inspired by models of natural systems that combine the evolutionary adaptation of a population with individual learning within the lifetimes of its members. It has shown to be successful and popular for solving optimization problems. In this paper, we present a new approach to vector quantization based on memetic algorithm. Simulations indicate that vector quantization based on memetic algorithm has better performance in designing the optimal codebook for Vector Quantizer than conventional LBG algorithm. The Peak Signal to Noise Ratio (PSNR) is used as an objective measure of reconstructed image quality.

Let \( G = (V, E) \) be a connected simple graph. Let \( u, v \in V(G) \). The detour distance, \( D(u, v) \), between \( u \) and \( v \) is the distance of a longest path from \( u \) to \( v \). E. Sampathkumar defined the detour graph of \( G \), denoted by \( D(G) \), as follows: \( D(G) \) is an edge-labelled complete graph on \( n \) vertices, where \( n = |V(G)| \), the edge label for \( uv \), \( u, v \in V(K_n) \), being \( D(u, v) \). Any edge-labelled complete graph need not be the detour graph of a graph. In this paper, we characterize detour graphs of a tree. We also characterize graphs for which the detour distance sequences are given.

Let \( M = \{v_1, v_2, \ldots, v_n\} \) be an ordered set of vertices in a graph \( G \). Then \( (d(u,v_1), d(u,v_2), \ldots, d(u,v_n)) \) is called the \( M \)-coordinates of a vertex \( u \) of \( G \). The set \( M \) is called a metric basis if the vertices of \( G \) have distinct \( M \)-coordinates. A minimum metric basis is a set \( M \) with minimum cardinality. The cardinality of a minimum metric basis of \( G \) is called minimum metric dimension. This concept has wide applications in motion planning and in the field of robotics. In this paper we provide bounds for minimum metric dimension of certain classes of enhanced hypercube networks.

In this paper, we use a genetic algorithm and direct a hill-climbing algorithm in choosing differences to generate solutions for difference triangle sets. The combined use of the two algorithms optimized the hill-climbing method and produced new improved upper bounds for difference triangle sets.

The covering problem in the \( n \)-dimensional \( q \)-ary Hamming space consists of the determination of the minimal cardinality \( K_q(n, R) \) of an \( R \)-covering code. It is known that the sphere covering bound can be improved by considering decompositions of the underlying space, leading to integer programming problems. We describe the method in an elementary way and derive about 50 new computational and theoretical records for lower bounds on \( K_q(n, R) \).

For any graph \( G = (V, E) \), \( D \subseteq V \) is a global dominating set if \( D \) dominates both \( G \) and its complement \( \overline{G} \). The global domination number \( \gamma_g(G) \) of a graph \( G \) is the fewest number of vertices required of a global dominating set. In general,\(
\max\{\gamma(G), \gamma(\overline{G})\} \leq \gamma_g(G) \leq \gamma(G) + \gamma(\overline{G}),\) where \( \gamma(G) \) and \( \gamma(\overline{G}) \) are the respective domination numbers of \( G \) and \( \overline{G} \). We show that when \( G \) is a planar graph, \(\gamma_g(G) \leq \max\{\gamma(G) + 1, 4\}.\)

Given an acyclic digraph \( D \), we seek a smallest sized tournament \( T \) having \( D \) as a minimum feedback arc set. The reversing number of a digraph is defined to be \(r(D) = |V(T)| – |V(D)|.\)

We use integer programming methods to obtain new results for the reversing number where \( D \) is a power of a directed Hamiltonian path. As a result, we establish that known reversing numbers for certain classes of tournaments actually suffice for a larger class of digraphs.

A directed covering design, \( DC(v, k, \lambda) \), is a \( (v, k, 2\lambda) \) covering design in which the blocks are regarded as ordered \( k \)-tuples and in which each ordered pair of elements occurs in at least \( \lambda \) blocks. Let \( DE(v, k, \lambda) \) denote the minimum number of blocks in a \( DC(v, k, \lambda) \). In this paper, the values of the function \( DE(v, k, \lambda) \) are determined for all odd integers \( v \geq 5 \) and \( \lambda \) odd, with the exception of \( (v, \lambda) = (53, 1), (63, 1), (73, 1), (83, 1) \). Further, we provide an example of a covering design that cannot be directed.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). For a labeling \( f: V(G) \to A = \{0, 1\} \), define a partial edge labeling \( f^*: E(G) \to A \) such that, for each edge \( xy \in E(G) \),\(f^*(xy) = f(x) \quad \text{if and only if} \quad f(x) = f(y).\) For \( i \in A \), let \(\text{v}_f(i) = |\{ v \in V(G) : f(v) = i \}|\) and \(\text{e}_{f^*}(i) = |\{ e \in E(G) : f^*(e) = i \}|.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(
|\text{v}_f(0) – \text{v}_f(1)| \leq 1.\) If a friendly labeling \( f \) induces a partial labeling \( f^* \) such that \(|\text{e}_{f^*}(0) – \text{e}_{f^*}(1)| \leq 1,\)then \( G \) is said to be balanced. In this paper, a necessary and sufficient condition for balanced graphs is established. Using this result, the balancedness of several families of graphs is also proven.

Expanding upon a comment by P. A. Leonard [9], we exhibit \(\mathbb{Z}\)-cyclic patterned-starter based whist tournaments for \(q^2\) players, where \(g = 4k + 3\) is prime; the cases \(3 < q < 200\) are included herein, with data for \(200 < q < 5,000\) available electronically.