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 \({ordered \;coloring\; problem}\) \([6]\), 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 \({rank}\) 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, \(F_R(G) = \text{max}\{I_R(f): f \text{ is a }(2,2)-{packing}\}\). 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.