Cayley Graphs of Order 27p Are Hamiltonian

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

Download Full-text

Integral Cayley Graphs over Abelian Groups

The Electronic Journal of Combinatorics ◽

10.37236/353 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 17

Author(s):

Walter Klotz ◽

Torsten Sander

Keyword(s):

Finite Group ◽

Abelian Group ◽

Boolean Algebra ◽

Cayley Graph ◽

Cayley Graphs ◽

Additive Group ◽

Cyclic Groups ◽

Vertex Set ◽

A Finite Group ◽

Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

Download Full-text

On the spectrum of Cayley graphs related to the finite groups

Filomat ◽

10.2298/fil1720419g ◽

2017 ◽

Vol 31 (20) ◽

pp. 6419-6429 ◽

Cited By ~ 1

Author(s):

Modjtaba Ghorbani ◽

Farzaneh Nowroozi-Larki

Keyword(s):

Finite Group ◽

Cayley Graph ◽

Finite Groups ◽

Cayley Graphs ◽

Prime Numbers ◽

A Finite Group

Let G be a finite group of order pqr where p > q > r > 2 are prime numbers. In this paper, we find the spectrum of Cayley graph Cay(G,S) where S ? G \ {e} is a normal symmetric generating subset.

Download Full-text

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

Journal of Algebra and Its Applications ◽

10.1142/s0219498819500130 ◽

2019 ◽

Vol 18 (01) ◽

pp. 1950013

Author(s):

Alireza Abdollahi ◽

Maysam Zallaghi

Keyword(s):

Finite Group ◽

Cayley Graph ◽

Finite Groups ◽

Closed Subset ◽

Cayley Graphs ◽

Character Sums ◽

Main Question ◽

Vertex Set ◽

A Finite Group ◽

Positive Integers

Let [Formula: see text] be a group and [Formula: see text] an inverse closed subset of [Formula: see text]. By a Cayley graph [Formula: see text], we mean the graph whose vertex set is the set of elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. A group [Formula: see text] is called a CI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for some automorphism [Formula: see text] of [Formula: see text]. A finite group [Formula: see text] is called a BI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for all positive integers [Formula: see text], where [Formula: see text] denotes the set [Formula: see text]. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 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.

Download Full-text

Isomorphisms of Cayley digraphs of Abelian groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700031579 ◽

1998 ◽

Vol 57 (2) ◽

pp. 181-188 ◽

Cited By ~ 2

Author(s):

Cai Heng Li

Keyword(s):

Finite Group ◽

Positive Integer ◽

Cayley Graph ◽

Prime Divisor ◽

Open Problem ◽

Abelian Groups ◽

Cayley Graphs ◽

Vertex Set ◽

A Finite Group

For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

Download Full-text

Random Cayley Graphs are Expanders: a Simple Proof of the Alon–Roichman Theorem

The Electronic Journal of Combinatorics ◽

10.37236/1815 ◽

2004 ◽

Vol 11 (1) ◽

Cited By ~ 11

Author(s):

Zeph Landau ◽

Alexander Russell

Keyword(s):

Finite Group ◽

Cayley Graph ◽

Group Algebra ◽

Simple Proof ◽

Cayley Graphs ◽

Random Variables ◽

Irreducible Representations ◽

Original Proof ◽

A Finite Group ◽

Second Eigenvalue

We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting $c_\varepsilon \log |G|$ elements, independently and uniformly at random, from a finite group $G$ has expected second eigenvalue no more than $\varepsilon$; here $c_\varepsilon$ is a constant that depends only on $\varepsilon$. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the $\log |G|$ term may be replaced with $\log D$, where $D \leq |G|$ is the sum of the dimensions of the irreducible representations of $G$, and (iii.) it establishes the result above with a smaller constant $c_\varepsilon$.

Download Full-text

Which Haar graphs are Cayley graphs?

The Electronic Journal of Combinatorics ◽

10.37236/5240 ◽

2016 ◽

Vol 23 (3) ◽

Cited By ~ 5

Author(s):

István Estélyi ◽

Tomaž Pisanski

Keyword(s):

Finite Group ◽

Abelian Group ◽

Cayley Graph ◽

Regular Graph ◽

Abelian Groups ◽

Cayley Graphs ◽

Equivalent Condition ◽

Dihedral Groups ◽

A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$.

Download Full-text

The Energy of Cayley Graphs for Symmetric Groups of Order 24

ASM Science Journal ◽

10.32802/asmscj.2020.sm26(1.22) ◽

2020 ◽

pp. 1-6

Author(s):

Amira Fadina Ahmad Fadzil ◽

Nor Haniza Sarmin ◽

Ahmad Erfanian

Keyword(s):

Finite Group ◽

Symmetric Group ◽

Cayley Graph ◽

Adjacency Matrix ◽

Cayley Graphs ◽

Symmetric Groups ◽

Finite Graph ◽

Energy Of Graph ◽

A Finite Group

A Cayley graph of a finite group G with respect to a subset S of G is a graph where the vertices of the graph are the elements of the group and two distinct vertices x and y are adjacent to each other if xy−1 is in the subset S. The subset of the Cayley graph is inverse closed and does not include the identity of the group. For a simple finite graph, the energy of a graph can be determined by summing up the positive values of the eigenvalues of the adjacency matrix of the graph. In this paper, the graph being studied is the Cayley graph of symmetric group of order 24 where S is the subset of S4 of valency up to two. From the Cayley graphs, the eigenvalues are calculated by constructing the adjacency matrix of the graphs and by using some properties of special graphs. Finally, the energy of the respected Cayley graphs is computed and presented. Keywords: energy of graph; cayley graph; symmetric groups

Download Full-text

The Energy of Cayley Graphs for a Generating Subset of the Dihedral Groups

MATEMATIKA ◽

10.11113/matematika.v35.n3.1115 ◽

2019 ◽

Vol 35 (3) ◽

Author(s):

Amira Fadina Ahmad Fadzil ◽

Nor Haniza Sarmin ◽

Ahmad Erfanian

Keyword(s):

Finite Group ◽

Cayley Graph ◽

Adjacency Matrix ◽

Dihedral Group ◽

Cayley Graphs ◽

Dihedral Groups ◽

Specific Subset ◽

A Finite Group

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a group G with respect to the subset S is a graph where its vertices are the elements of G and two vertices a and b are connected if ab^(−1) is in the subset S. The energy of a Cayley graph is the sum of all absolute values of the eigenvalues of its adjacency matrix. In this paper, we consider a specific subset S = {b, ab, . . . , a^(n−1)b} for dihedral group of order 2n, where n is greater or equal to 3 and find the Cayley graph with respect to the set. We also calculate the eigenvalues and compute the energy of the respected Cayley graphs. Finally, the generalization of the energy of the respected Cayley graphs is found.

Download Full-text

Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups

ISRN Algebra ◽

10.5402/2011/428959 ◽

2011 ◽

Vol 2011 ◽

pp. 1-6

Author(s):

Mehdi Alaeiyan ◽

Siamak Firouzian ◽

Mohsen Ghasemi

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Cayley Graph ◽

Cayley Graphs ◽

Partial Answer ◽

Dihedral Groups ◽

Undirected Edge ◽

A Finite Group ◽

Edge Transitive

A Cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts transitively on edges. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. This is a partial answer to the question of Praeger (1999).

Download Full-text