Certain Variants of Neighborhood Number of Wheel Related Graphs

Abstract

The concept of neighborhood number in a graph has broad applications across various fields. It quantifies how effectively a subset of vertices can collectively cover the entire network. For instance, in social network system the neighborhood number can be used to determine the sets of influential users whose immediate connections facilitate efficient information dissemination throughout the network. In a graph G, a subset S of V(G) is said to be neighborhood set (n-set), if the union of sub graphs of G induced by the closed neighbors of elements in S produces graph G. The minimum cardinality of a minimal neighborhood set of G is called the neighborhood number of a graph G, denoted by n(G). The structure of wheel graph is instrumental for designing, managing and analyzing systems where a central entity directly connects to all others, optimizing various practical tasks in real life. This paper aims to explore some variants of neighborhood sets and their dimensions of certain wheel related graphs.