Large Chordal Rings for Given Diameter and Uniqueness Property of Minima

Abstract

This paper discusses the covering property and the Uniqueness Property of Minima (UPM) for linear forms in an arbitrary number of variables, with emphasis on the case of three variables (triple loop graph). It also studies the diameter of some families of undirected chordal ring graphs. We focus upon maximizing the number of vertices in the graph for given diameter and degree. We study the result in \([2]\), we find that the family of triple loop graphs of the form \(G(4k^2+2k+1; 1;2k+1; 2k^2)\) has a larger number of nodes for diameter \(k\) than the family \(G(3k^2+3k+1;1;3k+1;3k+2)\) given in \([2]\). Moreover, we show that both families have the Uniqueness Property of Minima.