Algebraic Graph Theory Seminar - Mónica Andrea Reyes Quiroz (19.4.2024)

Friday 19.4.2024 at 13:00, Lecture room M/IX (online too)

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Mónica Andrea Reyes Quiroz:
Spectral properties of token graphs

Abstract:
In graph theory, constructing token graphs has recently gained attention. The k-token graph F_k(G) of a graph G represents k-subsets of vertices from G, where two vertices are adjacent if their symmetric difference forms an edge in G. In particular, F_k(K_n) yields the Johnson graph J(n,k), a distance-regular graph crucial in coding theory. This talk will discuss the adjacency and Laplacian spectrum of F_k(G) in relation to G. For instance, when G is walk-regular, an exact value for the spectral radius ρ (or maximum eigenvalue) of F_k(G) is obtained. Moreover, we deal with some properties of the Laplacian matrices of the k-token graph of G and its complement. Additionally, the commutativity of these matrices is analyzed. Expanding on these findings, a "local" algebra is introduced, closely linked with the Bose-Mesner algebra of Johnson graphs J(n,k). Besides, a "global" algebra contains the local algebra with Laplacian and adjacency matrices of the k-token graphs for any G on n vertices.

Those of you who are not able to attend in person or who are still uncertain about the safety of attending in person are welcome to attend via MS Teams. In either case, we hope to see as many of you as possible (either in person or virtually) at our Friday gatherings.

 

 

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