On the Core Structure of Distance-Regular Graphs: Homomorphism Matrices and Spectral Conditions

Mohammadesmail Nikfar¹

1) Ph.D. Graduate in Mathematics, Doctor of Philosophy (Ph.D.) in Pure Mathematics, Mathematics Teacher, District 4 Department of Education, Tehran, Iran

IOI Code : ABEB-ACABDC
Publication : 4th International Conference on Recent Advances in Engineering, Innovation and Technology - Belgium(eitconf.com)
Abstract :
    This paper investigates the structural and algebraic conditions under which distance-regular graphs are core-complete, that is, either isomorphic to their cores or possessing a complete graph as their core. Motivated by Roberson’s homomorphism-matrix framework, which established the Cameron–Kazanidis conjecture for strongly regular graphs, we extend this approach to distance-regular graphs of arbitrary diameter d. We formulate a generalized theory of homomorphism matrices that exploits the spectral structure and intersection parameters characteristic of this class of graphs. In particular, we establish necessary conditions on the cosine sequences associated with distance-regular graphs that permit graph endomorphisms into subgraphs of strictly smaller diameter e<d. These conditions yield a rigidity theorem for primitive distance-regular graphs: if the subgraph induced by the vertices at maximum distance from a fixed vertex is connected, then every retraction onto a subgraph of diameter d must be an automorphism. This result extends the corresponding theorem for strongly regular graphs, recovered as the special case d=2. To assess the applicability of the theoretical framework, we present computational investigations of distance-regular graphs of diameter three. Our results indicate that no antipodal, non-bipartite distance-regular graph of diameter three with valency at most 50 admits an endomorphism onto a subgraph of diameter two. Furthermore, we identify numerous intersection arrays associated with primitive distance-regular graphs of diameter three that satisfy core-completeness. The analysis combines computational techniques with established methods from algebraic graph theory, particularly the theory of association schemes and spectral idempotents, thereby providing a unified framework for studying graph cores through spectral and combinatorial constraints.

Keywords : algebraic graph theory distance regular graphs graph cores graph homomorphisms endomorphisms homomorphism matrices cosine sequences association schemes spectral idempotents intersection arrays