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Spectral properties of random graphs with fixed equitable partition
- Publication Year :
- 2023
-
Abstract
- We define a graph to be $S$-regular if it contains an equitable partition given by a matrix $S$. These graphs are generalizations of both regular and bipartite, biregular graphs. An $S$-regular matrix is defined then as a matrix on an $S$-regular graph consistent with the graph's equitable partition. In this paper we derive the limiting spectral density for large, random $S$-regular matrices as well as limiting functions of certain statistics for their eigenvector coordinates as a function of eigenvalue. These limiting functions are defined in terms of spectral measures on $S$-regular trees. In general, these spectral measures do not have a closed-form expression; however, we provide a defining system of polynomials for them. Finally, we explore eigenvalue bounds of $S$-regular graph, proving an expander mixing lemma, Alon-Bopana bound, and other eigenvalue inequalities in terms of the eigenvalues of the matrix $S$.<br />Comment: 24 pages, 3 figures
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2311.07675
- Document Type :
- Working Paper