LetG= (V(G),E(G)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex setV(G) and the edge setE(G). The distanced(u, v) between two verticesu, vthat belong to the vertex set ofHis the shortest path between them. Ak-ordered partition of vertices is defined as β = {β₁, β₂, …, β_k}. If all distancesd(v, β_k) are finite for all verticesv∈V, then thek-tuple (d(v, β₁),d(v, β₂), …,d(v, β_k)) represents vertexvin terms of β, and is represented byr(v|β). If every vertex has a different presentation, thek-partition β is a resolving partition. The partition dimension of G, indicated bypd(G), is the minimalkfor which there is a resolvingk-partition ofV(G). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.

By means of the weight coefficients and the idea of introducing parameters, a discrete Hilbert-type inequality with the general homogeneous kernel and the intermediate variables is given. The equivalent form is obtained. The equivalent statements of the best possible constant factor related to some parameters, the operator expressions, and a few particular cases are considered.

By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality with the homogeneous kernel $\frac{1}{(x + y)^{\lambda}}\ (\lambda > 0)$1(x+y)λ(λ>0) involving two multiple upper-limit functions is obtained. The equivalent statements of the best possible constant factor related to the beta and gamma functions are considered. As applications, the equivalent forms and the case of a nonhomogeneous kernel are deduced. Some particular inequalities and the operator expressions are provided.

By means of the techniques of real analysis, applying some basic inequalities and formulas, a new reverse Mulholland’s inequality with one partial sum in the kernel is given. We obtain the equivalent conditions of the parameters related to the best value in the new inequality. As applications, we reduce to the equivalent forms and a few inequalities for particular parameters.

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo p^sq^t for all distinct primes p, q and s,t∈N. We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph Γ_E(R) are derived for R=Zpsqt.

In this paper, by virtue of the symmetry principle, applying the techniques of real analysis and Euler–Maclaurin summation formula, we construct proper weight coefficients and use them to establish a reverse extended Hardy–Hilbert’s inequality with multi-parameters. Then, we obtain the equivalent forms and some equivalent statements of the best possible constant factor related to several parameters. Finally, we illustrate how the obtained results can generate some new reverse Hardy–Hilbert-type inequalities.