UPD Mathematicians Investigate Intriguing Higher-Dimensional Graphs

Graphs are collections of points and edges which afford a mathematical means of exploring networks and connections. Certain families of graphs, called association schemes, provide a flexible framework where several mathematical areas intersect. Drs. Jose Maria Balmaceda and Dom Vito Briones of the UP Diliman Institute of Mathematics are exploring a higher-dimensional analogue of association schemes by obtaining the parameters that govern its behavior.

In this three-dimensional structure called an association scheme on triples (AST), an edge connects 3 points together instead of two. This modification gives ASTs complicated structures, extending known connections of certain graphs with other areas of mathematics in ways that have yet to be fully explored. In an attempt to shed light on these properties, the authors focused on a particular family of ASTs: those which allow swapping pairs of points in a certain way.

Leveraging the advantage brought on by this symmetry, the authors determined the number of ways the edges of these ASTs intersect in a certain manner. As a consequence of these computations, they found that certain families of ASTs can be represented as 3-dimensional matrices that interact in a particularly nice way.

Drs. Balmaceda and Briones’ research findings further the study of the nascent area of association schemes on triples, with potential applications in fields such as graph theory and coding theory.

Link to the study: https://amc-journal.eu/index.php/amc/article/view/2897

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