Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...

The aim of our work is to study vertex-reinforced jump processes with super-linear weight function $w(t) = t^\alpha$ , for some $\alpha>1$. On any complete graph $G (V, E)$, we prove that there one vertex $v \in V$ such the total time spent at $v$ almost surely tends infinity while remaining vertices bounded.

By a Riemann function we mean $f:\mathbb{Z}^n\to\mathbb{Z}$ such that $f(\mathbf{d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and $d_1+\cdots+d_n+C$ constant, $C$, large. adding $1$ to the Baker-Norine rank of graph, one gets an equivalent function, similarly related functions.
 To each associate $W: \mathbb{Z}^n\to \mathbb{Z}$ via Möbius inversion call weight function. We g...