In this paper, we describe two algorithms to identify the repeating subwords in a given partial word \( w_o = w_0[1,…,n] \). The first algorithm uses the suffix tree and the second algorithm uses the valency tree. Both algorithms take linear time to identify the repeating subwords of a partial word.

We present a class of Coded Petri net languages and study some algebraic properties. The purpose of introduction of this language is to bring out its usefulness in learning theory. We introduce an algorithm for learning a finite coded Petri net language and its running time is bounded by a polynomial function of given inputs.

In this present investigation, the authors obtain Fekete-Szegő’s inequality for certain normalized analytic functions \( f(z) \) defined on the open unit disk. As a special case of this result, Fekete-Szegő’s inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to give a generalization of the Fekete-Szegő inequalities obtained by Srivastava and Mishra and Ma and Minda.

This paper is mainly devoted to generate (special) (super) edge-magic labelings of graphs using matrices. Matrices are used in order to find lower bounds for the number of non-isomorphic (special) (super) edge-magic labelings of certain types of graphs. Also, new applications of graph labelings are discussed.

A well-designed interconnection network makes efficient use of scarce communication resources and is used in systems ranging from large supercomputers to small embedded systems on a chip. This paper deals with certain measures of vulnerability in interconnection networks. Let \( G \) be a non-complete connected graph and for \( S \subseteq V(G) \), let \( \omega(G – S) \) and \( m(G – S) \) denote the number of components and the order of the largest component in \( G – S \), respectively. The vertex-integrity of \( G \) is defined as

\[I(G) = \text{min}\{|S| + m(G – S) : S \subseteq V(G)\}.\]

A set \( S \) is called an \( I \)-set of \( G \) if \( I(G) = |S| + m(G – S) \). The rupture degree of \( G \) is defined by

\[r(G) = \text{max}\{\omega(G – S) – |S| – m(G – S) : S \subseteq V(G), \omega(G – S) \geq 2\}.\]

A set \( S \) is called an \( R \)-set of \( G \) if \( r(G) = \omega(G – S) – |S| – m(G – S) \). In this paper, we compute the rupture degree of complete binary trees and a class of meshes. We also study the relationship between an \( I \)-set and an \( R \)-set and find an upper bound for the rupture degree of Hamiltonian graphs.

In this paper, we establish the possibility of embedding a graph as an induced subgraph in an: elegant graph, harmonious graph, felicitous graph, cordial graph, odd-graceful graph, polychrome graph, and strongly c-harmonious graph, each with a given property, leading to prove the NP-completeness of some parameters like: chromatic number, clique number, domination number, and independence number
of these graphs.

This paper describes an approach based on modified invariant moments for recognition of multi-font English characters. The proposed method is independent of size and translation variations and shows better results under noisy conditions. The work treats isolated English characters which are normalized to a size of \( 33 \times 33 \) pixels and the image is thinned. As a pre-classification step, end points and Euler numbers have been estimated from this thinned image of the character. For size and translation invariance, the modified invariant moments suggested by Palaniappan have been evaluated. The system is trained for 7 different font styles with 364 images. A decision tree-based minimum distance nearest neighbor classifier has been adopted for classification. The system is tested for these seven fonts with various sizes of the characters between 8 to 72. A total of 7,280 character images are tested with this system and the success rate is found to be 99.65\%. The method shows encouraging results on multi-font/sized character images.

A Knowledge Based Document Management System (KBDMS) is proposed in this paper to organize, cluster, classify and discover free-text documents. Context sensitive information is discovered by means of word map, sentence map and paragraph map in an intelligent manner in this proposed system. A text learning procedure for the semantic retrieval of text documents is implemented using a hierarchy of self-organizing maps (SOM) and support vector machines (SVM). The hierarchical SOM generates histograms of paragraph maps based on the semantic similarity and these paragraph maps are trained using SVM for classification. The SVM also generates an index for each document given to it. The proposed system is scalable and capable of discovery of documents from a huge amount of free-text documents. It is tested over a maximum of 100,000 text documents with 75-80\% accuracy in the context-sensitive discovery of free-text documents.

The purpose of this paper is to construct the membership functions of performance measures in bulk arrival queuing systems with arrival rate and service rate being fuzzy numbers. Thus, this paper develops the parametric programming approach to derive the membership functions of the steady-state performance measures in bulk arrival queuing systems with varying batch size. On the basis of a cut representation and extension principle, a parametric programming is formulated to describe the family of crisp bulk arrival queues. The performance measures are expressed by membership functions rather than crisp values, which completely conserve the fuzziness of input information when some data of bulk arrival queuing systems are ambiguous.

In order to establish the mathematical basis for connections between molecular structures and physicochemical properties of chemical compounds, some topological indices have been put forward. Among them, the Wiener index is one of the most important topological indices. The sum of distances of all pairs of vertices in a connected graph is known as Wiener index or Wiener number. All structural formulas of chemical compounds are molecular graphs where vertices represent the set of atoms and edges represent chemical bonds. A graph is said to be detour saturated if the addition of any edge results in an increased greatest path length. The characteristic graph of a given benzenoid graph consists of vertices corresponding to hexagonal rings of the graph; two vertices are adjacent if and only if the corresponding rings share an edge. A benzenoid graph is called Cata-condensed if its characteristic graph is a tree. In this paper, we derive Wiener indices for characteristic graphs of benzenoid graphs in the form of hexagonal rings, which are detour-saturated trees.

A vertex \( v \in V(G) \) is said to be a self vertex switching of \( G \) if \( G \) is isomorphic to \( G^v \), where \( G^v \) is the graph obtained from \( G \) by deleting all edges of \( G \) incident to \( v \) and adding all edges incident to \( v \) which are not in \( G \). Two vertices \( u \) and \( v \) in \( G \) are said to be interchange similar if there exists an automorphism \( \alpha \) of \( G \) such that \( \alpha(u) = v \) and \( \alpha(v) = u \). In this paper, we give a characterization for a cut vertex in \( G \) to be a self vertex switching where \( G \) is a connected graph such that any two self vertex switchings, if they exist, are interchange similar.

Pavel Hrnciar and Alfonz Havier \([6]\) introduced a clever idea of transferring labeled pendant edges incident at a vertex of a graceful tree to some other suitable vertex of that tree, so that another graceful tree is obtained. This idea is further explored in this paper to generate graceful lobsters from a graceful caterpillar with \( n \) edges.

In this paper, we study the prime filters of a bounded pseudocomplemented semilattice. We extend some of the results of \([3]\) to pseudocomplemented semilattices. It is observed that the set of all prime filters \( \mathcal{P} \) of a pseudocomplemented semilattice \( S \) is a topology, and it is \( T_0 \) and compact. We also obtain some necessary and sufficient conditions for the subspace of maximal filters to be normal.

A node ranking problem is also called an \emph{ordered coloring problem} \([6]\), which labels a graph \( G = (V, E) \) with \( C: V \to \{1, 2, \ldots, k\} \) such that for every path between any two nodes \( u \) and \( v \), with \( C(u) = C(v) \), there is a node \( w \) on the path with \( C(w) > C(u) = C(v) \). The value \( C(v) \) is called the \emph{rank} or color of the node \( v \). Node ranking is the problem of finding the minimum \( k \) such that the maximum rank in \( G \) is \( k \). There are two versions of the node ranking problem: off-line and on-line. In the off-line version, all the vertices and edges are given in advance. In the on-line version, the vertices are given one by one in an arbitrary order (say \( v_1, v_2, \ldots, v_n \)) and only the edges of the induced subgraph \( \langle\{v_1, v_2, \ldots, v_i\}\rangle_G \) are known when the rank of \( v_i \) has to be chosen. This paper establishes the node ranking number of complete \( r \)-partite graphs for the off-line version and gives a tight bound for the on-line version with the algorithms to accomplish them in linear time.

Embeddings capabilities play a vital role in evaluating interconnection networks. Wirelength is an important measure of an embedding. As far as the most versatile architecture, the hypercube, is concerned, only approximate estimates of the wirelength of various embeddings are available. This paper presents an optimal embedding of the hypercube into a new architecture called \( k \)-cube necklace, which minimizes wirelength. In addition, this paper gives an exact formula for the minimum wirelength of the hypercube into \( k \)-cube necklace and thereby we solve completely the wirelength problem of the hypercube into \( k \)-cube necklace.

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, \(F_R(G) = \text{max}\{I_R(f): f \text{ is a }(2,2)-\emph{packing}\}\). 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.