(untitled) (European Journal of Pure and Applied Mathematics, 2010)
Let G = (V, E) be a given connected simple (p, q)-graph, and an arbitrary nonempty subset M V(G) of G and for each v V(G), define N^M_j[u] = {v M: d(u,v) = j}. Clearly, then N_j[u] = N^V(G)_j[u]. B. D. Acharya bda2 defined the M-eccentricity of u as the largest integer for which N^M_j[u] e and the p× (d_G+1) nonnegative integer matrix D^M_G = (|N^M_j[v_i]|),Â called the M-distance neighborhood pattern (or, M-dnp) matrix of G. The matrix D_G^*M is obtained from D_G^M by replacing each nonzero entry by 1. Clearly, f_M(u) = {j: N^M_j[u] e }. Hence, in particular, if f_M: u f_M(u) is an injective function, then the set M is a distance-pattern distinguishing set (or, a `DPD-set’ in short) of G and G is a dpd-graph. If f_M(u)-{0} is independent of the choice of u in G then M is an open distance-pattern uniform (or, ODPU) set of G. A study of these sets is expected to be useful in a number of areas of practical importance such as facility locationÂ hm and design of indices of `quantitative structure-activity relationships’ (QSAR) in chemistry bmg,dhr.Â This paper is a study ofÂ M-dnp matrices of a dpd-graph.