Let S div ( n ) $S_{\rm div}(n)$ denote the set of permutations π such that for each 1 ⩽ j $1\leqslant \leqslant n$ either ∣ $j \mid \pi (j)$ or $\pi (j) j$ . These can also be viewed as vertex-disjoint directed cycle covers divisor graph D [ , ] $\mathcal {D}_{[1,n]}$ on vertices v … $v_1, \ldots v_n$ with an edge between i $v_i$ and $v_j$ if $i\mid i$ We improve recent results Pomerance by sh...

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literat...

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...

Let $A$ be a commutative ring with nonzero identity, and $1leq n

The algebraic polynomial plays a significant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices. index is utilized as tool study quantitative structure activity relationship (QSAR) structures property (QSPR) which correlate molecular its different properties activities. Graphs containing finite commutati...

We consider the common-divisor graph of the set of valencies of a naturally valenced scheme, where scheme is defined in the sense of P.-H. Zieschang. We prove structural results about this graph, and thus give restrictions on the set of natural numbers that can occur as the set of valencies of a naturally valenced scheme.