<p style="text-align: justify;">Let <em>R</em> be a finite commutative ring with identity and <em>P</em> prime ideal of <em>R</em>. The vertex set is <em>R - </em>{0} two distinct vertices are adjacent if their product in <em>P</em>. This graph called the denoted by Γ<sub>P</sub>. relationship among ideal, zero-diviso...

Consider the (p,q) simple connected graph . The sum absolute values of spectrum quotient matrix a make up graph's energy. objective this study is to examine energy identity graphs and zero-divisor commutative rings using group theory, applications. In study, derived from few classes ring R are examined.

In this paper, we prove that for any positive integers k, n with k ≥ 2, the graph P k n is a divisor graph if and only if n ≤ 2k + 2, where P k n is the k power of the path Pn. For powers of cycles we show that C n is a divisor graph when n ≤ 2k + 2, but is not a divisor graph when n ≥ 2k + bk2 c+ 3, where C k n is the k th power of the cycle Cn. Moreover, for odd n with 2k + 2 < n < 2k + bk2 c...

A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2,... | |} V such that an edge uv is assigned the label 1 if either ( ) | ( ) f u f v or ( ) | ( ) f v f u and the label 0 if ( ) ( ) f u f v , then number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a divisor cordial labeling is called a divisor cordial ...

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...

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is with vertex set $Z(R)\setminus \{0\}$ which the all nonzero elements and two distinct vertices $x$ $y$ are adjacent if only $xy=0.$ In this paper, we characterize rings whose graphs outerplanar graphs. Further, establish planar index, index finite