A study on energy sum of dominating sets in East Indian states

Abstract

This article explores the spectral properties of dominating sets in a map network for the East Indian states, with a focus on computing various energy sums using the eigen values of matrices depending on the Laplacian. Dominating sets—subsets of vertices that ensure each node of a graph is either included in or close to a node in the set—are critical for network optimization, resource allocation, and regional planning. This study uses three types of Laplacian matrices: The Laplacian Matrix, the Laplacian Matrix of Vertex Dominance, and the Laplacian Matrix of Edge Domination. The structural and dominating characteristics of the graph are characterized by calculating the eigen values for those matrices and examining the energy sums associated with them. The results are confirmed using computational coding in the MATLAB application, ensuring correctness and providing a consistent framework for spectral graph study. The findings contribute to our understanding of the network's resilience, connectivity, and optimization potential, as well as give important details for East India's growth in infrastructure and regional planning. This paper explains why spectral graph theory can be used to investigate map-based networks and provides a versatile approach for future research in related disciplines.