On Isomorphisms of Vertex-transitive Graphs

The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups

Degree distributions for a class of Circulant graphs

We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group \C{A}. Using these characterizations, we find degree distribution polynomials for weak equivalence of some graphs including 1) circulant graphs of prime power order, 2) circulant graphs of order $4p$, 3) circulant graphs of square free order and 4) Cayley graphs of order $p$ or $2p$. As an application, we find an enumeration formula for the number of weak equivalence classes of circulant graphs of prime power order, order $4p$ and square free order and Cayley graphs of order $p$ or $2p$

Approximating Cayley diagrams versus Cayley graphs

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a spanning tree in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this subtree is a Hamiltonian cycle, but convergence is meant in a stronger sense. These latter are related to whether having a Hamiltonian cycle is a testable graph property.Comment: 8 pages, 1 figur

Isomorphisms of Cayley graphs on nilpotent groups

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. For the special case where the groups are abelian, these results were previously proved by A.A.Ryabchenko and C.Loeh, respectively.Comment: 12 pages, plus 7 pages of notes to aid the referee. One of our corollaries is already known, so a reference to the literature has been adde

Automorphism groups of circulant graphs -- a survey

A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic automorphism.Comment: 16 pages, 0 figures, LaTeX fil

Amalgamation and Symmetry: From Local to Global Consistency in The Finite

Amalgamation patterns are specified by a finite collection of finite template structures together with a collection of partial isomorphisms between pairs of these. The template structures specify the local isomorphism types that occur in the desired amalgams; the partial isomorphisms specify local amalgamation requirements between pairs of templates. A realisation is a globally consistent solution to the locally consistent specification of this amalgamation problem. This is a single structure equipped with an atlas of distinguished substructures associated with the template structures in such a manner that their overlaps realise precisely the identifications induced by the local amalgamation requirements. We present a generic construction of finite realisations of amalgamation patterns. Our construction is based on natural reduced products with suitable groupoids. The resulting realisations are generic in the sense that they can be made to preserve all symmetries inherent in the specification, and can be made to be universal w.r.t. to local homomorphisms up to any specified size. As key applications of the main construction we discuss finite hypergraph coverings of specified levels of acyclicity and a new route to the lifting of local symmetries to global automorphisms in finite structures in the style of Herwig-Lascar extension properties for partial automorphisms.Comment: A mistake in the proposed construction from [arXiv:1211.5656], cited in Theorem 3.21, was discovered by Julian Bitterlich. This version relies on the new approach to this construction as presented in the new version of [arXiv:1806.08664

Testing isomorphism of central Cayley graphs over almost simple groups in polynomial time

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).Comment: 20 page

The Cayley isomorphism property for the group C25×CpC^5_2\times C_p

A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_2^5\times C_p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$. Together with the previously obtained results, this implies that a group $G$ of order $32p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$ and $G\cong C_2^5\times C_p$.Comment: 19 pages. arXiv admin note: text overlap with arXiv:2003.08118, arXiv:1912.0883

A solution of an equivalence problem for semisimple cyclic codes

In this paper we propose an efficient solution of an equivalence problem for semisimple cyclic codes

Non-abelian finite groups whose character sums are invariant but are not Cayley isomorphism

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^\alpha=T$ for some automorphism $\alpha$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_\nu^S=M_\nu^T$ for all positive integers $\nu$, where $M_\nu^S$ denotes the set $\big\{\sum_{s\in S}\chi(s) | \chi(1)=\nu, \chi \text{ is a complex irreducible character of } G \big\}$. It was asked by L\'aszl\'o Babai [\textit{J. Combin. Theory Ser. B}, {\bf 27} (1979) 180-189] if every finite group is a BI-group; various examples of finite non BI-groups are presented in [\textit{Comm. Algebra}, {\bf 43} (12) (2015) 5159-5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However it is known that there are finite abelian non CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders $20$ and $42$ which are not CI-groups. We also list all BI-groups of orders up to $30$