scholarly journals Which Cayley Graphs are Integral?

10.37236/211 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
A. Abdollahi ◽  
E. Vatandoost
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Trivial Group ◽  
Vertex Set

Let $G$ be a non-trivial group, $S\subseteq G\setminus \{1\}$ and $S=S^{-1}:=\{s^{-1} \;|\; s\in S\}$. The Cayley graph of $G$ denoted by $\Gamma(S:G)$ is a graph with vertex set $G$ and two vertices $a$ and $b$ are adjacent if $ab^{-1}\in S$. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs.

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
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.

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

Cayley graphs of graded rings

2018 ◽  
Vol 17 (06) ◽  
pp. 1850116
Author(s):  
Saadoun Mahmoudi ◽  
Shahram Mehry ◽  
Reza Safakish
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Graded Rings ◽  
Graded Ring ◽  
Total Graph ◽  
Hamming Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Cayley Sum Graph

Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].

Automorphisms of a Family of Cubic Graphs

2013 ◽  
Vol 20 (03) ◽  
pp. 495-506 ◽  
Author(s):  
Jin-Xin Zhou ◽  
Mohsen Ghasemi
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

CAYLEY GRAPHS OF PARTIALLY ORDERED SETS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250184 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
ZAHRA BARATI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Cayley Graph ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Cayley Graphs ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Basic Properties ◽  
Vertex Set

In this paper, we introduce the Cayley graph of a partially ordered set (poset). Let (P, ≤) be a poset, and let S be a subset of P. We define the undirected Cayley graph of P, denoted by Cay (P, S), as a graph with vertex-set P and edge-set E consisting of those sets {x, y} such that y ∈ {x, s}ℓ or x ∈ {y, s}ℓ for some s ∈ S, where for a subset T of P, Tℓ is the set of all x ∈ P such that x ≤ t, for all t ∈ T. We study some basic properties of Cay (P, S) such as connectivity, diameter and girth.

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

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.

scholarly journals Isomorphisms of Cayley digraphs of Abelian groups

1998 ◽  
Vol 57 (2) ◽  
pp. 181-188 ◽  
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.

Isomorphisms of bi-Cayley graphs on Dihedral groups

2020 ◽  
Vol 12 (04) ◽  
pp. 2050051
Author(s):  
Majid Arezoomand ◽  
Afshin Behmaram ◽  
Mohsen Ghasemi ◽  
Parivash Raeighasht
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Vertex Set

For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] the bi-Cayley graph BCay[Formula: see text] of [Formula: see text] with respect to [Formula: see text] is the bipartite graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bi-Cayley graph BCay[Formula: see text] is called a BCI-graph if for any bi-Cayley graph BCay[Formula: see text], [Formula: see text] implies that [Formula: see text] for some [Formula: see text] and [Formula: see text]. A group [Formula: see text] is called a [Formula: see text]-BCI-group if all bi-Cayley graphs of [Formula: see text] with valency at most [Formula: see text] are BCI-graphs. In this paper, we characterize the [Formula: see text]-BCI dihedral groups for [Formula: see text]. Also, we show that the dihedral group [Formula: see text] ([Formula: see text] is prime) is a [Formula: see text]-BCI-group.

scholarly journals Polynomials of unitary Cayley graphs

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 2079-2086
Author(s):  
Milan Basic ◽  
Aleksandar Ilic
Keyword(s):  
Cayley Graph ◽  
Spanning Trees ◽  
Cayley Graphs ◽  
Vertex Set ◽  
Unitary Cayley Graph ◽  
Minimal Spanning Trees

The unitary Cayley graph Xn has the vertex set Zn = {0,1,2,..., n-1} and vertices a and b are adjacent, if and only if gcd(a-b,n) = 1. In this paper, we present some properties of the clique, independence and distance polynomials of the unitary Cayley graphs and generalize some of the results from [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. In addition, using some properties of Laplacian polynomial we determine the number of minimal spanning trees of any unitary Cayley graph.

scholarly journals Wiener and Hyper–Wiener Indices of Unitary Addition Cayley Graphs

2019 ◽  
Vol 8 (2S3) ◽  
pp. 131-132
Keyword(s):  
Graph Theory ◽  
Cayley Graph ◽  
Topological Index ◽  
Cayley Graphs ◽  
Wiener Index ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Shortest Distance ◽  
Vertex Set

A topological index is a number associated to a graph. In chemical graph theory the Wiener index of a graph G, denoted by W(G) is the sum of the distance between all (unordered) pairs of vertices of G. That is, W(G) = ,where d (ui , uj) is the shortest distance between the vertices. ui and uj .The Hyper-Wiener Index WW(G) is the generalization of the Wiener index. The Hyper- Wiener Index of a graph G is, WW (G) = .The unitary addition Cayley graph Gn has a vertex set Zn = {0, 1,…, n-1} and the vertices u and v are adjacent if gcd (u+v,n) =1. In this paper Wiener index and Hyper Wiener indices of Unitary addition Cayley graph Gn is computed

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