Distance Neighbourhood Pattern Matrices in a Graph

Authors

  • Germina Kizhekekunnel Augustine Reader in Mathematics
  • Alphy Joseph
  • Sona Jose

Keywords:

Distance-pattern distinguishing sets, Distance neighborhood pattern matrix

Abstract

Let G=(V,E) be a given connected simple (p,q)-graph, and an arbitrary nonempty subset MV(G) of G and for each vV(G), define NjM[u]={vM:d(u,v)=j}. Clearly, then Nj[u]=NjV(G)[u]. B. D. Acharya ~\cite{bda2} defined the M{\it -eccentricity} of u as the largest integer for which NjM[u] and the p×(dG+1) nonnegative integer matrix DGM=(|NjM[vi]|),  called the M-{\it distance neighborhood pattern} (or, M{\it -dnp}) {\it matrix} of G. The matrix DGM is obtained from DGM by replacing each nonzero entry by 1. Clearly, fM(u)={j:NjM[u]}. Hence, in particular, if fM:ufM(u) is an injective function, then the set M is a \emph{distance-pattern distinguishing set} (or, a `DPD-set' in short) of G and G is a dpd-graph. If fM(u){0} is independent of the choice of u in G then M is an {\it open distance-pattern uniform} (or, ODPU) {\it set} of G. A study of these sets is expected to be useful in a number of areas of practical importance such as facility location  ~\cite{hm} and design of indices of `quantitative structure-activity relationships' (QSAR) in chemistry ~\cite{bmg,dhr}.  This paper is a study of  M-dnp matrices of a dpd-graph.

Author Biography

  • Germina Kizhekekunnel Augustine, Reader in Mathematics

    Head, Mathematics Research Centre(Kannur university), Mary Matha Arts & Science CollegeMary Matha Arts & Science College, Mananthavady.

    Reader in Mathematics

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Published

2010-09-02

Issue

Section

Discrete Mathematics

How to Cite

Distance Neighbourhood Pattern Matrices in a Graph. (2010). European Journal of Pure and Applied Mathematics, 3(4), 748-764. https://www.ejpam.com/index.php/ejpam/article/view/524