Page 324 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 324
SPECTRAL GRAPH THEORY (MS-46)
Playing with quaternions unit gain graphs
Maurizio Brunetti, mbrunett@unina.it
Università di Napoli "Federico II", Italy
A quaternion unit gain graph is a graph where a quaternion unit q is assigned to each oriented
edge eij, and the conjugate q¯ is assigned to eji. In a non-commutative context there exists
a “left" spectral theory and a “right" spectral theory. I will show how the latter, but not the
former, encapsulates some classical spectral results holding for ordinary graphs, signed graphs
and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency
and Laplacian matrix are also given, together with some explicit computations.
Complementary prisms and their spectra
Paula Carvalho, paula.carvalho@ua.pt
University of Aveiro, Portugal
Coauthors: Domingos M. Cardoso, Maria Aguieiras A. de Freitas, Cybele T. M. Vinagre
The complementary prism GG of a graph G is obtained from the disjoint union of the graph G
and its complement G defined on a copy of the vertex set of G, by adding an edge for each pair
vertices (v, v ), where v is in G and its copy v is in G. The Petersen graph C5C5 and the corona
of a complete graph KnKn, with n ≥ 2, are examples of complementary prisms. In this talk
we prove that the Petersen graph is the unique complementary prism which is strongly regular.
Furthermore, we compute the eigenpairs of adjacency, signless Laplacian and Laplacian matrix
of a complementary prism GG in terms of the eigenvalues of adjacency, signless Laplacian
and Laplacian matrix of G, respectively. In particular, to the complementary prisms of regular
graphs are given special attention.
Spectra and eigenspaces from regular partitions of Cayley (di)graphs of
permutation groups
Cristina Dalfo, cristina.dalfo@udl.cat
Universitat de Lleida, Spain
Coauthor: Miquel Àngel Fiol Mora
In this talk, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs
(that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that
every partition of the number n gives rise to a regular partition of the Cayley graph. By using
representation theory, we also obtain the complete spectra and the eigenspaces of the corre-
sponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues
and eigenvectors of such (di)graphs, based on their irreducible representations. As examples,
we apply this method to the pancake graphs P (n) and to a recent known family of mixed graphs
Γ(d, n, r) (having edges with and without direction). As a byproduct, the existence of perfect
codes in P (n) allows us to give a lower bound for the multiplicity of its eigenvalue −1.
322
Playing with quaternions unit gain graphs
Maurizio Brunetti, mbrunett@unina.it
Università di Napoli "Federico II", Italy
A quaternion unit gain graph is a graph where a quaternion unit q is assigned to each oriented
edge eij, and the conjugate q¯ is assigned to eji. In a non-commutative context there exists
a “left" spectral theory and a “right" spectral theory. I will show how the latter, but not the
former, encapsulates some classical spectral results holding for ordinary graphs, signed graphs
and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency
and Laplacian matrix are also given, together with some explicit computations.
Complementary prisms and their spectra
Paula Carvalho, paula.carvalho@ua.pt
University of Aveiro, Portugal
Coauthors: Domingos M. Cardoso, Maria Aguieiras A. de Freitas, Cybele T. M. Vinagre
The complementary prism GG of a graph G is obtained from the disjoint union of the graph G
and its complement G defined on a copy of the vertex set of G, by adding an edge for each pair
vertices (v, v ), where v is in G and its copy v is in G. The Petersen graph C5C5 and the corona
of a complete graph KnKn, with n ≥ 2, are examples of complementary prisms. In this talk
we prove that the Petersen graph is the unique complementary prism which is strongly regular.
Furthermore, we compute the eigenpairs of adjacency, signless Laplacian and Laplacian matrix
of a complementary prism GG in terms of the eigenvalues of adjacency, signless Laplacian
and Laplacian matrix of G, respectively. In particular, to the complementary prisms of regular
graphs are given special attention.
Spectra and eigenspaces from regular partitions of Cayley (di)graphs of
permutation groups
Cristina Dalfo, cristina.dalfo@udl.cat
Universitat de Lleida, Spain
Coauthor: Miquel Àngel Fiol Mora
In this talk, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs
(that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that
every partition of the number n gives rise to a regular partition of the Cayley graph. By using
representation theory, we also obtain the complete spectra and the eigenspaces of the corre-
sponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues
and eigenvectors of such (di)graphs, based on their irreducible representations. As examples,
we apply this method to the pancake graphs P (n) and to a recent known family of mixed graphs
Γ(d, n, r) (having edges with and without direction). As a byproduct, the existence of perfect
codes in P (n) allows us to give a lower bound for the multiplicity of its eigenvalue −1.
322
