We correct few errors in the statement and proof of Theorem 4 [Soft Computing (24), 237–245 (2020)].

We introduce the notion of semibreak divisors on metric graphs and prove that every effective divisor class (of degree at most genus) has a representative. This appropriately generalizes break (in equal to genus). provide an algorithm efficiently compute such representatives. Semibreak tool establish some basic properties loci inside Picard groups graphs. are pure-dimensional polyhedral sets. a...

let $i$ be a proper ideal of a commutative semiring $r$ and let $p(i)$ be the set of all elements of $r$ that are not prime to $i$. in this paper, we investigate the total graph of $r$ with respect to $i$, denoted by $t(gamma_{i} (r))$. it is the (undirected) graph with elements of $r$ as vertices, and for distinct $x, y in r$, the vertices $x$ and $y$ are adjacent if and only if $x + y in p(i)...

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 and Z(R) the set of all zero divisors R. Γ(R) is said to divisor graph if x,y ∈ V (Γ(R)) = (x,y) E(Γ(R)) only x.y 0. In this paper, we determine total vertex irregularity strength graphs associated with rings ℤp2 × Zq for p,q prime numbers.

the concept of the bipartite divisor graph for integer subsets has been considered in [‎graph combinator., ‎ 26 (2010) 95--105‎.]. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $g...