Indian Journal of Science and Technology

Abstract

Objectives: This study introduces the concept of distance average degree independent resolving sets and its dimensions for identity graphs of finite groups and order prime graphs of finite groups as a theorem in detail. Methods: The methodology involves first constructing the subset from the given graph, then finding the distance between each and every vertice of the graph and the subset, and also finding the average of the distance such that each and every vertice should receive a distinct code. The minimum number of vertices that satisfy this condition, their cardinality, is called the dimension. And then check various conditions like adjacency, equitability, independence, etc. in order to find various types of dimensions for the considered graphs. Findings: In this article, the concept of distance average degree resolving sets has been found, and further distance average degree independent resolving sets have been found, with their corresponding dimensions also found for algebraic graphs like identity graphs of finite groups and order prime graphs of finite groups. Novelty: The novelty in this article is the concept of distance-average-degree independent resolving sets. Particularly, this concept has been applied to the algebraic graphs, namely the identity graph and the order prime graph of a finite group. In addition to that, the concept of distance has been added to this resolving sets area, which is totally new to this area of graph theory. AMS Subject Classification: 05C12, 05C50.

Keywords: Resolving set, Independent set, Distance average degree independent resolving set, Identity graphs, Order prime graphs