We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a “bridge” between GLMY-theory and group homology theory, ...which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.
A canonical double cover B(X) of a graph X is the direct product of X and the complete graph K2 on two vertices. In order to answer the question when a canonical double cover of a given graph is a ...Cayley graph, in 1992 Marušič et al. introduced the concept of generalized Cayley graphs. In this paper this concept is generalized to a wider class of graphs, the so-called extended generalized Cayley graphs. It is proved that the canonical double cover of a connected non-bipartite graph X is a Cayley graph if and only if X is an extended generalized Cayley graph. This corrects an incorrectly stated claim in Discrete Math. 102 (1992), 279–285.
GCI-property of some groups Liao, Qianfen; Liu, Weijun
Applied mathematics and computation,
02/2023, Volume:
438
Journal Article
Peer reviewed
In this paper, firstly, we determine the local 2-GCI-property and 2−GCI-property of the cyclic group. Then, for the dihedral group D2n, we prove that it has local GCI-property if and only if n is an ...odd prime or 9. Further, the dihedral group D2n cannot have GCI-property. Moreover, we discuss the GCI-property of the elementary abelian group, the dicyclic group and the semi-dihedral group.
On normality of n-Cayley graphs Hujdurović, Ademir; Kutnar, Klavdija; Marušič, Dragan
Applied mathematics and computation,
09/2018, Volume:
332
Journal Article
Peer reviewed
Let G be a finite group and X a (di)graph. If there exists a semiregular subgroup G¯ of the automorphism group Aut(X) isomorphic to G with n orbits on V(X) then the (di)graph X is called an n-Cayley ...graph on G. If, in addition, this subgroup G¯ is normal in Aut(X) then X is called a normal n-Cayley graph on G.
In this paper the normalizers of semiregular subgroups of the automorphism group of a digraph are characterized. It is proved that every finite group admits a vertex-transitive normal n-Cayley graph for every n ≥ 2. For the most part the graphs are constructed as Cartesian product of graphs. It is proved that a Cartesian product of two relatively prime graphs is Cayley (resp. normal Cayley) if and only if the factor graphs are Cayley (resp. normal Cayley). In addition, the concept of graphical regular representations (GRRs) is generalized to n-GRR in a natural way, and it is proved that any group admitting a GRR also admits an n-GRR for any n ≥ 1.
The splitting field of a matrix associated with a graph is the smallest field extension of Q that contains all of its eigenvalues. The extension degree is called its algebraic degree. In this paper, ...by introducing a new characteristic vector for each normal subset of a finite group, we completely determine the splitting fields and algebraic degrees for the adjacency matrix and distance matrix of a normal Cayley graph, which generalize the main results of Godsil et al. and Lu et al. Moreover, we study the relation between the algebraic integrality of these two matrices, and generalize a main result of Huang and Li. Finally, for a normal mixed Cayley graph, we consider its Hermitian adjacency matrix and Hermitian adjacency matrix of the second kind, and we characterize their splitting fields and algebraic degrees, which generalize the main results of Huang et al. and Kadyan et al.
Compressed Cayley graph of groups Yari, Behnaz; Khashyarmanesh, Kazem; Afkhami, Mojgan
Indian journal of pure and applied mathematics,
03/2024
Journal Article
A graph is half-arc-transitive if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A half-arc-transitive graph is half-arc-regular if its full ...automorphism group acts regularly on its edges. A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-Cayley graph over a dihedral group is called bi-dihedrant. In this paper, it is shown that the smallest valency of half-arc-transitive bi-dihendrants is 6, and then a classification is given of connected half-arc-regular bi-dihedrants of valency 6. This work together with the result in 8, Theorem 6.7 completes the classification of edge-regular bi-dihedrants of valency 6.
We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square-free order are ...DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).
