Albert William

Embedding of cycles and wheels into arbitrary trees

We estimate and characterize the edge congestion-sum measure for embeddings of various graphs such as cycles, wheels, and generalized wheels into arbitrary trees. All embedding algorithms apply an interesting general technique based on the consecutive label property. Our algorithms produce optimal values of sum of dilations and sum of edge-congestions in linear time.

Power domination in certain chemical structures

Let G ( V , E ) be a simple connected graph. A set S ? V is a power dominating set (PDS) of G, if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γ p ( G ) . In this paper, we establish a fundamental result that would provide a lower bound for the power domination number of a graph. Further, we solve the power domination problem in polyphenylene dendrimers, Rhenium Trioxide (ReO3) lattices and silicate networks.

Resolving-power dominating sets

A new concept resolving-power dominating sets is introduced.The problem is proven to be NP-complete.The resolving power domination number is studied for trees. For a graph G ( V , E ) that models a facility or a multi-processor network, detection devices can be placed at vertices so as to identify the location of an intruder such as a thief or fire or saboteur or a faulty processor. Resolving-power dominating sets are of interest in electric networks when the latter helps in the detection of an intruder/fault at a vertex. We define a set S ? V to be a resolving-power dominating set of G if it is resolving as well as a power-dominating set. The minimum cardinality of S is called resolving-power domination number. In this paper, we show that the problem is NP-complete for arbitrary graphs and that it remains NP-complete even when restricted to bipartite graphs. We provide lower bounds for the resolving-power domination number for trees and identify classes of trees that attain the lower bound. We also solve the problem for complete binary trees.

Minimum Cycle Covers of Butterfly and Benes Networks

Butterfly network is the most popular bounded-degree derivative of the hypercube network. The benes network consists of back-to-back butterflies. In this paper, we obtain the minimum vertex-disjoint cycle cover number for the odd dimensional butterfly networks and prove that it is not possible to find the same for the even dimensional butterfly networks and benes networks. Further we obtain the minimum edge-disjoint cycle cover number for butterfly networks.

On Laplacian Energy of Certain Mesh Derived Networks

of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called its spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, perfect matching etc. Laplacian spectrum is the multiset of eigenvalues of Laplacian matrix. The Laplacian energy of a graph is the sum of the absolute values of its Laplacian eigenvalues. In this paper we calculate the Laplacian energy of some grid based networks

Radio Antipodal Number of Certain Graphs

Let G = (V, E) be a graph with vertex set V and edge set E. Let diam(G) denote the diameter of G and d(u, v) denote the distance between the vertices u and v in G. An antipodal labeling of G with diameter d is a function f that assigns to each vertex u, a positive integer f(u), such that d(u,v) + |f(u) – f(v)| ≥ d, for all u, v ∈ V. The span of an antipodal labeling f is max{|f(u) – f (v)|:u, v ∈ V(G)}. The antipodal number for G, denoted by an(G), is the minimum span of all antipodal labelings of G. Determining the antipodal number of a graph G is an NP-complete problem. In this paper we determine the antipodal number of certain graphs.

VLSI layout of Benes networks

Abstract The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly–like architectures. We identify a new topological representation of Benes network. The significance of this representation is demonstrated by solving two problems, one related to VLSI layout and the other related to robotics. An important VLSI layout network problem is to produce the smallest possible grid area for realizing a given network. We propose an elegant VLSI layout of r-dimensional Benes networks using this representation. The area of this layout is O(22r ) whereas the lower bound for the area of the VLSI layout of Benes networks is O(22r ). This lower bound is estimated by applying Thompson result.

Packing Chromatic Number of Circular Fans and Mesh of Trees

The packing chromatic number

Shuffle Exchange Networks And Achromatic Labeling

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On Strong Metric Dimension Of Diametrically Ver-tex Uniform Graphs

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