Related papers: The unitary Cayley graph of upper triangular matri…

In this paper we completely describe the unital compressed commuting graph of the ring $\mathcal{M}_3(\mathrm{GF}(p))$ of $3 \times 3$ matrices over the finite prime field $\mathrm{GF}(p)$. To achieve this we combine methods from linear…

In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph $X=( Z_n, S)$ is a special graph as connection set $S \subseteq Z_n$…

Let $R$ be a commutative ring with identity. The involutory Cayley graph $\mathcal{G}(R)$ of $R$ is defined as the graph whose vertex set is the set of elements of $R$, where two vertices $a$ and $b$ are adjacent exactly when $(a-b)^2=1$.…

Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this…

A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…

As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…

The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes…

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…

In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and Alexander quandles. In particular, we…

Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The…

Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by…

A graph is called an integral graph when all eigenvalues of its adjacency matrix are integers. We study which Cayley graphs over a nonabelian group $$ T_{8n}=\left\langle a,b\mid a^{2n}=b^8=e,a^n=b^4,b^{-1}ab=a^{-1} \right \rangle $$ are…

The unitary Cayley graph, denoted $X_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. The quadratic unitary…

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in…

Let $\Gamma_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $\Gamma_n$ are studied, with any two such vertices adjacent if and only if they share (as…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…

We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph $G$,…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its automorphism group ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple…

Let R be a ring (not necessarily commutative ring) with identity. The clean graph Cl(R) of a ring R is a graph with vertices in the form of ordered pair (e; u), where e is an idempotent of the ring R and u is a unit of the ring R. Two…

A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory…