A perfect r-code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is within distance r of exactly one vertex in the subset. We prove that the n-fold strong product of simple graphs has a perfect r-code if and only if each factor has a perfect r-code.

An r-perfect code of a graph G = (V, E) is a set C ⊆ V such that the r-balls centered at vertices of C form a partition of V . It is proved that the direct product of Cm and Cn (r ≥ 1, m,n ≥ 2r+1) contains an r-perfect code if and only if m and n are each a multiple of (r + 1)2 + r2 and that the direct product of Cm, Cn, and C` (r ≥ 1, m,n, ` ≥ 2r +1) contains an r-perfect code if and only if m...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

We examine the problem of detecting when a given sparse polynomial f is equal to hr for some other polynomial h and integer r ≥ 2. In this case we say f is a perfect power, and h is its r th root. We give randomized algorithms to detect when f is a perfect power, by repeatedly evaluating the polynomial in specially-chosen finite fields and checking whether the evaluations are themselves perfect...

Let R be a commutative ring with identity and M an R-module. The Scalar-Product Graph of M is defined as the graph GR(M) with the vertex set M and two distinct vertices x and y are adjacent if and only if there exist r or s belong to R such that x = ry or y = sx. In this paper , we discuss connectivity and planarity of these graphs and computing diameter and girth of GR(M). Also we show some of...