The vertex PI index PI (G) = SxyÎE (G) [nxy (x) + nxy (y)] is a distance-based molecular structure descriptor, where nxy (x) denotes the number of vertices which are closer to the vertex x than to the vertex y and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the Extremal graphs with the greatest and smallest vertex PI indices mong all cacti with a fixed number of vertices. As a consequence, we obtain the sharp bounds with corresponding Extremal cacti and extend a known result.

‎The first general multiplicative Zagreb index $P_1^a(G)$ is the product of the degree of each vertex $v$ in $G$‎, ‎raised to the power $a$ and the second general multiplicative Zagreb index $P_2^a(G)$ is the product of the degree of each vertex $v$ in $G$‎, ‎raised to the power $a$ times the degree of $v$‎, ‎where $a$ is a non-zero real number‎. ‎In this study‎, ‎we present bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs‎. ‎We also provide bounds for the first general multiplicative Zagreb index for trees‎. ‎Additionally‎, ‎we identify all the Extremal graphs for each bound mentioned as best as possible‎.

In 1991, McKay and Radziszowski proved that, however each 3-subset of a 13-set is assigned one of two colours, there is some 4-subset whose four 3-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey number known for hypergraphs. In this article, we nd all the Extremal colourings of the 3-subsets of a 12-set and list some of their properties. We also provide an answer to a question of Dudek, La Fleur, Mubayi and Rodl about the size-Ramsey numbers of hypergraphs.

The aim of this paper is to study orders over a valuation ring V with arbitrary rank in a central simple F-algebra Q. The relation between all of the orders is explained with a diagram. It is then shown that inside Bezout order, Extremal V-orders are precisely semi-hereditary. In the last section, the effect of Henselization on maximal and semi-hereditary orders is examined.

Let $ G $ be a finite simple graph. The Sombor index of $ G $ isdefined as $ \sum\nolimits_{uv\in E(G)} \sqrt{d_{u}^{2}+d_{v}^{2}} $where $d_{u}$ and $d_{v}$ represent the degrees of vertices $ u$ and$v$ in $ G $, respectively. The sum of the absolute values of theadjacency eigenvalues defines the energy of a graph. This paper aimsto enhance the current connections between the Sombor index and theenergy of graphs. Additionally, we provide some upper bounds for theSombor index of triangle-free, square-free, $K_r$-free andtripartite graphs in terms of order, size and minimum degree.

Recently, a novel topological index, Sombor index, was introduced by Gutman, defined as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}$ denotes the degree of vertex $u$. In this paper, we first determine the maximum Sombor index among cacti with $n$ vertices and $t$ cycles, then determine the maximum Sombor index among cacti with perfect matchings. We also characterize corresponding maximum cacti.