Integrity Degree in graphs

The concepts of integrity play a crucial role in graph theory. In this study, we introduce a novel parameter called the integrity degree of a graph. The integrity degree of a vertex u is the minimum cardinality of a minimal integrity set that contains u. Using this concept, this study relates several inequalities involving the integrity degree with other established graph parameters. A new class of graphs called d I – k regular graphs are defined. The integrity degree is investigated for various classes of graphs such as complete graphs, paths, cycles, wheel graphs, complete bipartite graphs, book graphs, windmill graphs, star graphs. The computation and illustration of integrity degree of the vertices of petersen graph is provided. The integrity degree of vertices in binary trees and binomial trees are computed. An algorithm for finding the integrity degree of every vertex of a graph using the adjacency matrix is presented. Also an application of integrity degree of vertices in communication network is discussed.