Journal of Advances in Mathematics and Computer Science

We introduce the spectral entropy topological index S(G), defined as the Shannon entropy of the squared adjacency eigenvalues of a simple graph G. This construction converts the adjacency spectrum into a probability distribution and yields a concise measure of global structural complexity. We establish general bounds for S(G), prove strict positivity for every nonempty simple graph, and compute the index explicitly for some classical families, including stars, complete bipartite graphs, and complete graphs, together with spectral expressions for paths and cycles. These examples illustrate how S(G) distinguishes between centralized, regular, and irregular structures. We also provide heuristic observations for Erd˝os–R´enyi random graphs and demonstrate that the index has natural applications in computer science, including network anomaly detection, robustness analysis in distributed and wireless sensor networks, graph based data mining, and the analysis of web graphs and blockchain style peer to peer networks. In these domains, S(G) serves as an interpretable indicator of decentralization, heterogeneity, and structural diversity.