Page 327 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 327
SPECTRAL GRAPH THEORY (MS-46)
where G and G are the underlying graph of G˙ and its complement and r is the vertex degree of
G (and G˙ ).
We establish certain basic structural and spectral properties of such signed graphs, and sug-
gest a natural way to divide all SRSGs into five classes according to the relations among their
defining parameters, which allows us to better perceive their properties.
Next, we consider walk-regularity of SRSGs with a relatively small number of distinct
eigenvalues, belonging to some of those specified classes. In the end, we investigate the re-
lationship between SRSGs with three or four distinct eigenvalues and three-class symmetric
association schemes.
Systems of equiangular lines, Seidel matrices and adjacency matrices
Jack Koolen, koolen@ustc.edu.cn
University of Science and Technology of China, China
It is known that large systems of equiangular lines with common angle arccos 1/α are closely
related with Seidel matrices S with smallest eigenvalue −α such that S + αI has low rank. In
this talk I will introduce the notion of a switching root and show how we can use the switching
root to relate adjacency matrices and Seidel matrices. If time permits I will also discuss some
new maximal connected graphs with minimal eigenvalue −3, that is, any proper connected
supergraph of such graph has smallest eigenvalue less than −3. This is based on joint work with
Meng-Yue Cao (Beijing Normal University), Akihiro Munemasa (Tohoku University), Kiyoto
Yoshino (Tohoku University) and Brhane Gebremichel (University of Science and Technology
of China).
Isospectral magnetic graphs
Fernando Lledó, flledo@math.uc3m.es
University Carlos III, Madrid, Spain, and
Insitute for Mathematical Sciencies (ICMAT), Madrid, Spain
We present a new geometrical construction leading to an infinite collection of families of graphs,
where all the elements in each family are (finite) isospectral non-isomorphic graphs for the dis-
crete magnetic Laplacian with normalised weights (in particular for standard weights). The
construction is based on the notion of isospectral frames which, together with the s-partition
of a natural number r, define the isospectral families of graphs by contraction of distinguished
vertices. The isospectral frames have high symmetry and we use a spectral preorder of graphs
studied in [2,3] to control the spectral spreading of the eigenvalues under elementary perturba-
tions of the graph like vertex contraction and vertex virtualisation.
References
[1] J.S. Fabila-Carrasco, F. Lledó and O. Post, A geometric construction of isospectral mag-
netic graphs, 2021 (in preparation).
[2] J.S. Fabila-Carrasco, F. Lledó and O. Post, Spectral preorder and perturbations of discrete
weighted graphs, Math. Ann. 2020, DOI:https://doi.org/10.1007/s00208-020-02091-5
325
where G and G are the underlying graph of G˙ and its complement and r is the vertex degree of
G (and G˙ ).
We establish certain basic structural and spectral properties of such signed graphs, and sug-
gest a natural way to divide all SRSGs into five classes according to the relations among their
defining parameters, which allows us to better perceive their properties.
Next, we consider walk-regularity of SRSGs with a relatively small number of distinct
eigenvalues, belonging to some of those specified classes. In the end, we investigate the re-
lationship between SRSGs with three or four distinct eigenvalues and three-class symmetric
association schemes.
Systems of equiangular lines, Seidel matrices and adjacency matrices
Jack Koolen, koolen@ustc.edu.cn
University of Science and Technology of China, China
It is known that large systems of equiangular lines with common angle arccos 1/α are closely
related with Seidel matrices S with smallest eigenvalue −α such that S + αI has low rank. In
this talk I will introduce the notion of a switching root and show how we can use the switching
root to relate adjacency matrices and Seidel matrices. If time permits I will also discuss some
new maximal connected graphs with minimal eigenvalue −3, that is, any proper connected
supergraph of such graph has smallest eigenvalue less than −3. This is based on joint work with
Meng-Yue Cao (Beijing Normal University), Akihiro Munemasa (Tohoku University), Kiyoto
Yoshino (Tohoku University) and Brhane Gebremichel (University of Science and Technology
of China).
Isospectral magnetic graphs
Fernando Lledó, flledo@math.uc3m.es
University Carlos III, Madrid, Spain, and
Insitute for Mathematical Sciencies (ICMAT), Madrid, Spain
We present a new geometrical construction leading to an infinite collection of families of graphs,
where all the elements in each family are (finite) isospectral non-isomorphic graphs for the dis-
crete magnetic Laplacian with normalised weights (in particular for standard weights). The
construction is based on the notion of isospectral frames which, together with the s-partition
of a natural number r, define the isospectral families of graphs by contraction of distinguished
vertices. The isospectral frames have high symmetry and we use a spectral preorder of graphs
studied in [2,3] to control the spectral spreading of the eigenvalues under elementary perturba-
tions of the graph like vertex contraction and vertex virtualisation.
References
[1] J.S. Fabila-Carrasco, F. Lledó and O. Post, A geometric construction of isospectral mag-
netic graphs, 2021 (in preparation).
[2] J.S. Fabila-Carrasco, F. Lledó and O. Post, Spectral preorder and perturbations of discrete
weighted graphs, Math. Ann. 2020, DOI:https://doi.org/10.1007/s00208-020-02091-5
325
