Obtaining coherent configurations on non-distance regular graphs

Abstract

Many regularly structured graphs, such as strongly regular graphs, have been extensively studied, and their spectral and algebraic properties are well documented in the literature. In contrast, the study of non-structured graphs remains limited, largely due to the difficulty of systematically constructing and analyzing them. In this paper, we explore whether simple graph operations—such as vertex deletion and switching—performed on strongly regular graphs can produce non-structured graphs, and examine how these operations affect their associated coherent configurations. We focus on two well-known families of strongly regular graphs: the Rook Graph R(n) and the Triangular Graph T(n). By applying specific graph modifications, we analyze the resulting adjacency algebras and track changes in their coherent configuration structure. Our experiments reveal that certain operations consistently result in configurations of fixed rank and algebraic patterns, suggesting underlying structure even within seemingly irregular graphs.