Eigenvalue Interlacing of Bipartite Graphs and Construction of Expander Code using Vertex-split of a Bipartite Graph

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander codes are Error Correcting codes made up of bipartite expander graphs. In this paper, first we prove the interlacing of the eigenvalues of the adjacency matrix of the bipartite graph with the eigenvalues of the bipartite quotient matrices of the corresponding graph matrices. Then we obtain bounds for the second largest and second smallest eigenvalues. Since the graph is bipartite, the results for Laplacian will also hold for Signless Laplacian matrix. We then introduce a new method called vertex-split of a bipartite graph to construct asymptotically good expander codes with expansion factor $\frac{D}{2}<\alpha < D$ and $\epsilon<\frac{1}{2}$ and prove a condition for the vertex-split of a bipartite graph to be $k-$connected with respect to $\lambda_{2}.$ Further, we prove that the vertex-split of $G$ is a bipartite expander. Finally, we construct an asymptotically good expander code whose factor graph is a graph obtained by the vertex-split of a bipartite graph.Comment: 17 pages, 2 figure

Reachability Based Web Page Ranking Using Wavelets

AbstractA naïve approach has been made by applying the concept of reachability for web page ranking and implementing multi resolution analysis using Haar wavelet to order the web pages. In this article, page ranking has been done by developing a structured signal using in links, out links and reachability values of the web pages of network graphs. Using Haar wavelet, the page ranking is proposed and developed. The average and detailed coefficients of the input signal and the down sampling process provides the necessary page ranking of web pages. This approach does not involve any iterative technique, damping factor or initialization of the page ranks. In this paper, comparison between the original page rank, category-based page rank and the proposed approach have been made. The result reflects the role of paths between the pages in page rankings

A Simple Density with Distance Based Initial Seed Selection Technique for K Means Algorithm

Open issues with respect to K means algorithm are identifying the number of clusters, initial seed concept selection, clustering tendency, handling empty clusters, identifying outliers etc. In this paper we propose a novel and a simple technique considering both density and distance of the concepts in a dataset to identify initial seed concepts for clustering. Many authors have proposed different techniques to identify initial seed concepts; but our method ensures that the initial seed concepts are chosen from different clusters that are to be generated by the clustering solution. The hallmark of our algorithm is that it is a single pass algorithm that does not require any extra parameters to be estimated. Further, our seed concepts are one among the actual concepts and not the mean of representative concepts as is the case in many other algorithms. We have implemented our proposed algorithm and compared the results with the interval based technique of Fouad Khan. We see that our method outperforms the interval based method. We have also compared our method with the original random K means and K Means++ algorithms

NETWORK FLOW WITH FUZZY ARC LENGTHS USING HAAR RANKING

ABSTRACT Shortest path problem is a classical and the most widely studied phenomenon in combinatorial optimization. In a classical shortest path problem, the distance of the arcs between different nodes of a network are assumed to be certain. In some uncertain situations, the distance will be calculated as a fuzzy number depending on the number of parameters considered. This article proposes a new approach based on Haar ranking of fuzzy numbers to find the shortest path between nodes of a given network. The combination of Haar ranking and the well-known Dijkstra&apos;s algorithm for finding the shortest path have been used to identify the shortest path between given nodes of a network. The numerical examples ensure the feasibility and validity of the proposed method

Brief survey on divisor graphs and divisor function graphs

AbstractNumber theoretic graphs are one of the emerging fields in Graph theory. This article is a study on existing research results on Number theoretic graphs, especially on Divisor Graphs [Formula: see text], Divisor Function Graphs (DFGs) and Divisor Cayley Graphs (DCGs). We have provided a brief survey on the benchmark findings regarding the above-mentioned graphs

On computation of neighbourhood degree sum-based topological indices for zinc-based metal–organic frameworks

The permeable materials known as metal–organic frameworks (MOFs) have a large porosity volume, excellent chemical stability, and a unique structure that results from the potent interactions between metal ions and organic ligands. Work on the synthesis, architectures, and properties of various MOFs reveals their utility in a variety of applications, including energy storage devices with suitable electrode materials, gas storage, heterogeneous catalysis, and chemical assessment. A topological index, which is a numerical invariant, predicts the physicochemical properties of chemical entities based on the underlying molecular graph or framework. In this article, we consider two different zinc-based MOFs, namely zinc oxide and zinc silicate MOFs. We compute 14 neighbourhood degree sum-based topological indices for these frameworks, and the numerical and graphical representations of all the aforementioned 14 indices are made