We introduce the Laplacian eigenpolytopes ("L-polytopes") associated to a simple undirected graph G, investigate how they change under basic operations such as taking the union, join, complement, line graph and cartesian product of graphs, and show how several "famous" polytopes arise as L-polytopes of "famous" graphs.
Eigenpolytopes have been previously introduced by Godsil, who studied them in
detail in the context of distance-regular graphs. Our focus on the Laplacian matrix,
as opposed to the adjacency matrix of G, permits simpler proofs and descriptions of
the result of operations on not necessarily distance-regular graphs. Additionally, it
motivates the study of new operations on polytopes, such as the Kronecker product.
Thus, we open the door to a detailed study of how combinatorial properties of G are reflected in its L-polytopes. Subsequent papers will use these tools to construct interesting polytopes from interesting graphs, and vice versa.
CitacióPadrol, A.; Pfeifle, J. Graph operations and Laplacian eigenpolytopes. A: Jornadas de Matemática Discreta y Algorítmica. "VII Jornadas de Matemática Discreta y Algorítmica". Castro Urdiales: 2010, p. 505-516.