Sudeep Stephen

Average Distance in Interconnection Networks via Reduction Theorems for Vertex-Weighted Graphs

Average distance is an important parameter for measuring the communication cost of computer networks. A popular approach for its computation is to first partition the edge set of a network into convex components using the transitive closure of the Djoković-Winkler’s relation and then to compute the average distance from the respective invariants of the components. In this paper we refine this idea further by shrinking the quotient graphs into smaller weighted graph called reduced graph, so that the average distance of the original graph is obtained from the reduced graphs. We demonstrate the significance of this technique by computing the average distance of butterfly and hypertree architectures. Along the way a computational error from [European J. Combin. 36 (2014) 71–76] is corrected.

On the metric dimension of circulant and Harary graphs

A metric generator is a set W of vertices of a graph G ( V , E ) such that for every pair of vertices u , v of G, there exists a vertex w ? W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. In this case the vertex w is said to resolve or distinguish the vertices u and v. The minimum cardinality of a metric generator for G is called the metric dimension. The metric dimension problem is to find a minimum metric generator in a graph G. In this paper, we make a significant advance on the metric dimension problem for circulant graphs C ( n , ? { 1 , 2 , ? , j } ) , 1 ≤ j ≤ ? n / 2 ? , n ? 3 , and for Harary graphs.

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.

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

On the Power Domination Number of de Bruijn and Kautz Digraphs

Let \(G=(V,A)\) be a directed graph, and let \(S\subseteq V\) be a set of vertices. Let the sequence \(S=S_0\subseteq S_1\subseteq S_2\subseteq \cdots \) be defined as follows: \(S_1\) is obtained from \(S_0\) by adding all out-neighbors of vertices in \(S_0\). For \(k\geqslant 2\), \(S_k\) is obtained from \(S_{k-1}\) by adding all vertices w such that for some vertex \(v\in S_{k-1}\), w is the unique out-neighbor of v in \(V\setminus S_{k-1}\). We set \(M(S)=S_0\cup S_1\cup \cdots \), and call S a power dominating set for G if \(M(S)=V(G)\). The minimum cardinality of such a set is called the power domination number of G. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs.

On the Partition Dimension of Circulant Graphs

For a vertex

Spectrum of Sierpiński triangles using MATLAB

v

Minimum rank and zero forcing number for butterfly networks

of a connected graph

Zero forcing in iterated line digraphs

G(V,E)