Integral Graph Spectrum and Energy of Interconnected Balanced Multi-star Graphs

Balanced multi-star graph K r (n) is a specialized type of graph formed by connecting apex vertices of star graphs to create a cohesive structure known as a clique.These graphs comprise r star graphs, where each star graph has an apex vertex connected to n pendant vertices.Balanced multistar graphs offer benefits in scenarios requiring equal distances between peripheral nodes, such as sensor networks, distributed computing, traffic engineering, telecommunications, supply chain management, and power distribution.The integral graph spectrum derived from the adjacency matrix of balanced multistar graphs holds significance across various domains.It aids in network analysis to understand connectivity patterns, facilitates efficient computation of structural properties through graph algorithms, and enables graph partitioning and community detection.Spectral graph theory assists in identifying connectivity patterns in network visualization, supports modeling biological networks in biomedical research, aids in generating personalized recommendations in recommendation systems and contributes to graph-based segmentation and scene analysis tasks in image processing.This paper aims to characterize the integral graph spectrum of balanced multi-star graphs K r (n) by focusing on spectral parameters of double-star graphs (r = 2), triple-star graphs (r = 3), and quadruple-star graphs (r = 4).This spectrum serves as an important tool across disciplines, providing insights into graph structure and facilitating tasks ranging from network analysis to computational biology and image processing.