A labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \( (x, y) \) is the absolute value of the difference of the labels of \( x \) and \( y \). We say that a labeling of the vertices of a graph of order \( n \) is minimally \( k \)-equitable if the vertices are labeled with \( 1, 2, \ldots, n \) and in the induced labeling of its edges, every label either occurs exactly \( k \) times or does not occur at all. In this paper, we prove that Butterfly and Benes networks are minimally \( 2^r \)-equitable, where \( r \) is the dimension of the networks.

A \((2,2)\) packing on a graph \(G\) is a function \(f: V(G) \to \{0, 1, 2\}\) with \(f(N[v]) \leq 2\) for all \(v \in V(G)\). For a function \(f: V(G) \to \{0,1,2\}\), the Roman influence of \(f\), denoted by \(I_R(f)\), is defined to be \(I_R(f) = (|V_1|+|V_2|) + \sum_{v\in V_2} deg(v)\). The efficient Roman domination number of \(G\), denoted by \(F_R(G)\), is defined to be the maximum of \(I_R(f)\) such that \(f\) is a \((2,2)\)-packing. That is, \(F_R(G) = \text{max}\{I_R(f): f \text{ is a }(2,2)-{packing}\}\). A \((2,2)\)-packing \(F_R(G)\) with \(F_R(G) = I_R(f)\) is called an \(F_R(G)\)-function. A graph \(G\) is said to be efficiently Roman dominatable if \(F_R(G) = n\), and when \(F_R(G) = n\), an \(F_R(G)\)-function is called an efficient Roman dominating function. In this paper, we focus our study on certain graphs which are efficiently Roman dominatable. We characterize the class of \(2 \times m\) and \(3 \times m\) grid graphs, trees, unicyclic graphs, and split graphs which are efficiently Roman dominatable.

The aim of this article is focused on developing an efficient algorithm for simulating Cellular Neural Network arrays (CNNs) using numerical integration techniques. The role of the simulator is that it is capable of performing raster simulation for any kind as well as any size of input image. It is a powerful tool for researchers to investigate the potential applications of CNN. This article proposes an efficient pseudo code for exploiting the latency properties of Cellular Neural Networks along with well known numerical integration algorithms. Simulation results and comparison have also been presented to show the efficiency of the numerical integration algorithms. It is observed that the Runge-Kutta (RK) sixth order algorithm outperforms well in comparison with the Explicit Euler, RK-Gill and RK-fifth order algorithms.

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.