For a finite commutative ring $\mathbb{Z}_{n}$ with identity $1\neq 0$, the zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is simple connected having vertex set as of non-zero divisors, where two vertices $x$ and $y$ are adjacent if only $xy=0$. We find distance Laplacian spectrum graphs for different values $n$. Also, we obtain $n=p^z$, $z\geq 2$, in terms spectrum. As consequence, determine thos...

A fundamental theorem of Wilson states that, for every graph F , every sufficiently large F -divisible clique has an F -decomposition. Here a graph G is F -divisible if e(F ) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F -decomposition if the edges of G can be covered by edge-disjoint copies of F . We ext...

The field of mathematics that studies the relationship between algebraic structures and graphs is known as graph theory. It incorporates concepts from theory, which examines characteristics topology graphs, with those abstract algebra, deals such groups, rings, fields. If vertex set a G^ fully made up zero divisors modular ring Zn, said to be zero-divisor graph. products two vertices are equal ...

Let G = Γ(S) be a semigroup graph, i.e., zero-divisor graph of S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) {z∈V(G) | N (z) {x,y}}. Assume that there exist two y, vertex s∈C(x,y) and z such d (s,z) 3. This paper studies algebraic properties graphs Γ(S), giving some sub-semigroups ideals S. It constructs classes classifies all the property cases.

The tensor product, also called direct, categorical, or Kronecker product of graphs, is one of the least-understood graph products. In this paper we determine partial answers to the question given in the title, thereby significantly extending results of Broere and Hattingh (see [2]). We characterize completely those pairs of complete graphs whose tensor products are circulant. We establish that...

Given a graph whose edges are labeled by ideals of commutative ring R with identity, generalized spline is vertex labeling the elements such that difference labels on adjacent vertices lies in ideal associated to edge. The set splines has and an R-module structure. We study module structure where base greatest common divisor domain. give basis criteria for cycles, diamond graphs trees using det...

Let $G=(V,E)$ be a simple graph. A set $Ssubseteq V$ isindependent set of $G$,  if no two vertices of $S$ are adjacent.The  independence number $alpha(G)$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph $Gamma(R)$ and ideal-based zero-divisor graph $Gamma_I(R)$of a commutative ring $R$.