Cayley Sum Graph Research Articles

On regular sets in Cayley sum graphs

On regular sets in Cayley sum graphs

Subgroup total perfect codes in Cayley sum graphs

Subgroup total perfect codes in Cayley sum graphs

Some applications of eigenvalues of unitary Cayley graphs of matrix rings over finite fields

ABSTRACT In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then and are isomorphic. If this conjecture holds and , then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and are such rings. We examine the eigenvalues of and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple are mutually equienergetic non-isospectral and Ramanujan where is the unitary Cayley sum graph. We use the eigenvalues of the graph and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring being a product of matrix rings over local rings.

On the subgraphs of Cayley sum graphs

The Cayley sum graph CS(G,A) of an abelian group G with a square-free subset A of G as the connection set is the graph with vertex set G and two vertices x and y being adjacent if and only if x+y∈A. In this paper we obtain two results on the subgraphs of CS(G,A). The first is that every connected component of CS(G,A) is either a bipartite Cayley graph of a generalized dihedral group or a connected Cayley sum graph of a subgroup of G. The second is that CS(G,A) has a spanning subgraph isomorphic to a cubic bipartite Cayley graph of a generalized dihedral group if A is of cardinality at least 4. As a by-product of the second result, we prove that CS(G,A) admits a nowhere-zero 3-flow whenever it is of valency at least 4.

Cayley sum graphs and their applications to codebooks

Cayley sum graphs and their applications to codebooks

Subgroup Sum Graphs of Finite Abelian Groups

Let G be a finite abelian group, written additively, and H a subgroup of G. The subgroup sum graph \(\varGamma _{G,H}\) is the graph with vertex set G, in which two distinct vertices x and y are joined if \(x+y\in H{\setminus }\{0\}\). These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the prime sum graphs, in the case where \(H=pG\) for some prime number p. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph \(\varGamma ^+_{G,H}\), which we refer to as the extended subgroup sum graph, in which x and y are joined if \(x+y\in H\): the subgroup sum is obtained by removing from this graph the partial matching of edges having the form \(\{x,-x\}\) when \(2x\ne 0\). We study perfectness, clique number and independence number, connectedness, diameter, spectrum, and domination number of these graphs and their complements. We interpret our general results in detail in the prime sum graphs.

The Cayley sum graph of ideals of a semigroup

Let S be a regular semigroup, I(S) be the set of ideals of S and M be a subset of I(S). In this paper, we introduce an undirected Cayley graph of S, denoted by Гs,m with elements of I(S) as the vertex set, and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of M such that IK = J or JK = I. We study some basic properties of the graph Гs,m such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Гs,m.

Perfect Codes in Cayley Sum Graphs

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$.

Spectra of twists of Cayley and Cayley sum graphs

Let G be a finite group with |G|≥4 and S be a subset of G. Given an automorphism σ of G, the twisted Cayley graph C(G,S)σ is the graph with G as its set of vertices, and the neighbourhood of a vertex is obtained by applying σ to its neighbourhood in the Cayley graph of G with respect to S. If C(G,S)σ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from −1 and this bound depends only on its degree, the order of σ and the vertex Cheeger constant of C(G,S)σ. The twisted Cayley sum graph CΣ(G,S)σ is defined similarly and we establish an analogous result for it. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.

A Cheeger type inequality in finite Cayley sum graphs

Let G be a finite group with |G|⩾4 and S be a subset of G with |S|=d such that the Cayley sum graph C Σ (G,S) is undirected and connected. We show that the non-trivial spectrum of the normalised adjacency operator of C Σ (G,S) is controlled by its Cheeger constant and its degree. We establish an explicit lower bound for the non-trivial spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval - 1 + h Σ (G) 4 η , 1 - h Σ (G) 2 2d 2 , where h Σ (G) denotes the vertex Cheeger constant of the d-regular graph C Σ (G,S) and η=2 9 d 8 . Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the Cayley graph of finite groups.

Subgroup perfect codes in Cayley sum graphs

Let $$\Gamma $$ be a graph with vertex set V. If a subset C of V is independent in $$\Gamma $$ and every vertex in $$V\setminus C$$ is adjacent to exactly one vertex in C, then C is called a perfect code of $$\Gamma $$. Let G be a finite group and let S be a square-free normal subset of G. The Cayley sum graph of G with respect to S is a simple graph with vertex set G and two vertices x and y are adjacent if $$xy\in S$$. A subset C of G is called a perfect code of G if there exists a Cayley sum graph of G which admits C as a perfect code. In particular, if a subgroup of G is a perfect code of G, then the subgroup is called a subgroup perfect code of G. In this paper, we give a necessary and sufficient condition for a non-trivial subgroup of an abelian group with non-trivial Sylow 2-subgroup to be a subgroup perfect code of the group. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. As an application, we classify the abelian groups whose every non-trivial subgroup is a subgroup perfect code. Moreover, we determine all subgroup perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.

The cayley sum graph of ideals of a lattice

The cayley sum graph of ideals of a lattice

Cayley sum graph of ideals of commutative rings

Let [Formula: see text] be a commutative ring, [Formula: see text] the set of all ideals of [Formula: see text] and [Formula: see text], a subset of [Formula: see text]. The Cayley sum graph of ideals of [Formula: see text], denoted by Cay[Formula: see text], is a simple undirected graph with vertex set is the set [Formula: see text] and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], for some [Formula: see text] in [Formula: see text]. In this paper, we study connectedness, Eulerian and Hamiltonian properties of Cay[Formula: see text]. Moreover, we characterize all commutative Artinian rings [Formula: see text] whose Cay[Formula: see text] is toroidal.

Cayley graphs of graded rings

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].

On subgraphs of random Cayley sum graphs

We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group G has edge density close to the expected one on every induced subgraph of size at least logc|G|, for any fixed c>1 and |G| large enough.

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay+(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z3 and Zn2n, n ≥ 1, where Zk is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.

Integral Cayley sum graphs and groups

Integral Cayley sum graphs and groups

Orthogonal Polarity Graphs and Sidon Sets

AbstractDetermining the maximum number of edges in an n‐vertex C4‐free graph is a well‐studied problem that dates back to a paper of Erdős from 1938. One of the most important families of C4‐free graphs are the Erdős‐Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose‐Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős‐Rényi orthogonal polarity graph. Using this isomorphism, we prove that the Petersen graph is a subgraph of every sufficiently large Erdős‐Rényi orthogonal polarity graph.

On Cayley Sum Graphs of Non-Abelian Groups

Let $$G$$G be a finite group and $$S$$S be a square-free normal subset of $$G$$G. The Cayley sum graph $$\mathrm{Cay}^+(G, S)$$Cay+(G,S) of $$G$$G with respect to $$S$$S is a simple graph whose vertex set is $$G$$G and two vertices $$g$$g and $$h$$h are joined by an edge if and only if $$g h \in S$$gh?S. In this paper, we obtain necessary and sufficient conditions on $$S$$S such that $$\mathrm{Cay}^+(G, S)$$Cay+(G,S) is connected. Also, we prove that if $$G$$G is a non-abelian group and $$|S|=3$$|S|=3, then $$\mathrm{Cay}^+(G, S)$$Cay+(G,S) is a connected integral graph if and only if $$G$$G is isomorphic to $$S_3$$S3, the symmetric group on $$3$$3 letters.

Cayley sum color and anti-circulant graphs

Let G be a finite group and α:G→R be a real-valued function on G. The Cayley sum color graph Cay+(G,α) is a complete directed graph with vertex set G where each arc (x,y)∈G×G is associated with a color α(xy). If α is the characteristic function on a subset S of G, then the Cayley sum graph Cay+(G,S) is obtained. The anti-circulant matrix associated to a vector v is an n×n matrix whose rows are given by iterations of the anti-shift operator acting on v. We note that a graph is a Cayley sum graph of a cyclic group if and only if it is an anti-circulant graph, a graph whose adjacency matrix is anti-circulant. In this paper, we obtain some results on the isomorphisms, connectivity and vertex transitivity about anti-circulant graphs. We find the spectrum of Cayley sum color graphs of abelian groups and as a result we compute the spectrum of real anti-circulant matrices.