Page 321 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 321
GROUPS, GRAPHS AND NETWORKS (MS-75)
Symmetries of bi-Cayley graphs
Jinxin Zhou, jxzhou@bjtu.edu.cn
Beijing Jiaotong University, China
A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with
exactly two orbits is called a bi-Cayley graph over H. Bi-Cayley graph is a natural generaliza-
tion of Cayley graph. A large body of research has been developed in recent years to explore
the symmetries of bi-Cayley graphs. Much of the work has been focused on the classification
and construction of bi-Cayley graphs with specific symmetry properties. In this lecture, I will
survey some recent results in this area.
Perfect 2-colourings of Cayley graphs
Sanming Zhou, sanming@unimelb.edu.au
The University of Melbourne, Australia
Let Γ = (V, E) be a graph. A partition π = {V1, . . . , Vm} of V is called an equitable partition
or a perfect m-colouring of Γ if there exists an m × m matrix (bij), called the quotient matrix
of π, such that every vertex in Vi has exactly bij neighbours in Vj. In particular, if {C, V \ C}
is a perfect 2-colouring of a d-regular graph Γ with quotient matrix 0d , then C is
1 d−1
called a perfect 1-code in Γ. In general, for an integer t ≥ 1, a perfect t-code in Γ is a subset C
of V such that every vertex of Γ is at distance no more than t to exactly one vertex in C. Perfect
t-codes in Hamming graph H(n, q) and in the Cartesian product of n copies of cycle Cq are
precisely q-ary perfect t-codes of length n under the Hamming and Lee metrics, respectively.
Thus perfect codes in Cayley graphs are a generalization of perfect codes in classical coding
theory.
I will talk about some recent and not-so-recent results on perfect 2-colourings of Cayley
graphs with an emphasis on perfect 1-codes in Cayley graphs.
319
Symmetries of bi-Cayley graphs
Jinxin Zhou, jxzhou@bjtu.edu.cn
Beijing Jiaotong University, China
A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with
exactly two orbits is called a bi-Cayley graph over H. Bi-Cayley graph is a natural generaliza-
tion of Cayley graph. A large body of research has been developed in recent years to explore
the symmetries of bi-Cayley graphs. Much of the work has been focused on the classification
and construction of bi-Cayley graphs with specific symmetry properties. In this lecture, I will
survey some recent results in this area.
Perfect 2-colourings of Cayley graphs
Sanming Zhou, sanming@unimelb.edu.au
The University of Melbourne, Australia
Let Γ = (V, E) be a graph. A partition π = {V1, . . . , Vm} of V is called an equitable partition
or a perfect m-colouring of Γ if there exists an m × m matrix (bij), called the quotient matrix
of π, such that every vertex in Vi has exactly bij neighbours in Vj. In particular, if {C, V \ C}
is a perfect 2-colouring of a d-regular graph Γ with quotient matrix 0d , then C is
1 d−1
called a perfect 1-code in Γ. In general, for an integer t ≥ 1, a perfect t-code in Γ is a subset C
of V such that every vertex of Γ is at distance no more than t to exactly one vertex in C. Perfect
t-codes in Hamming graph H(n, q) and in the Cartesian product of n copies of cycle Cq are
precisely q-ary perfect t-codes of length n under the Hamming and Lee metrics, respectively.
Thus perfect codes in Cayley graphs are a generalization of perfect codes in classical coding
theory.
I will talk about some recent and not-so-recent results on perfect 2-colourings of Cayley
graphs with an emphasis on perfect 1-codes in Cayley graphs.
319
