On the Geodetic Domination Sombor Index of Graphs: Structural Characterization and Applications to Resilient IoT Networks

Abstract

In this paper a new invariant of the graph based on the Sombor index is proposed, which we call the Geodetic Domination Sombor Index (GDSO) that combines together geodetic domination degrees and the geometric structure which lies behind the Sombor index. “The geodetic domination degree determines how often a vertex appears in minimal geodetic dominating sets and is thus a genuine indicator of the importance of a vertex in the problem of shortest-path coverage and the above-mentioned network-control problem with domination in mind. The spectral properties of the geodetic domination Sombor matrix of fundamental graphs are analyzed as well as exact expressions for corresponding geodetic domination Sombor matrix are obtained, such as complete graphs, star graphs and complete bipartite graphs. Theoretical bounds are also laid out, and the extremal/or boundary cases are identified as well. To show the applicability of the proposed invariant, a resilient IoT-based environmental monitoring network application framework for achieving resilient GDSO is formulated, where the sensing positions and communication connections with environmental importance in the network can be identified through participation in optimal monitoring arrangements. This provides a graph-theoretical foundation to assess sensor placement for sensor-dense urban forests and other distributed ecological monitoring environments. As a result, the proposed index can be used as a global structural determinant and be used for structural descriptor binding when using shortest-path coverage and/or domination based control in critical infrastructure supervision, network reliability, environmental IoT systems, construction of molecular descriptors and complex system resilience, particularly when a shortest path solution is desired in addition to the domination control solution.