Spectral properties of matrices over generalized topological vector spaces

In this paper, we investigate the spectral behavior of matrices constructed over generalized topological vector spaces (GTVS) using a framework inspired by random geometric graphs and random matrix theory. By assigning random weights to adjacency matrices and systematically varying the connection radius, we analyze transitions in both topological and spectral properties across different graph sizes. We observe that key topological indices the number of nonisolated vertices V×​(G), the Randić index R(G), and the harmonic index H(G) are strongly correlated with spectral measures such as the Shannon entropy S(G), the inverse participation ratio (IPR), and the eigenvalue spacing ratio rrr. Crucially, all these measures scale with a universal parameter ξ=ε/ε∗, where ε∗∼n−1/2. This scaling collapse suggests a unifying mechanism governing spectral complexity in matrix models defined on GTVS.