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.

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.

The generalized atom-bond connectivity index of a graph G is denoted by ABCa(G) and defined as the sum of weights ((d(u)+d(v)-2)/d(u)d(v))aa$ over all edges uv∊ G. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, we compute sharp bounds for ABCa index for CACTI of order $n$ with fixed number of cycles and for CACTI of order $n$ with given number of pendant vertices. Furthermore, we identify all the CACTI that achieve the bounds.

A facial total-coloring of a plane graph G is a coloring of the vertices and edges such that no facially adjacent edges, no adjacent vertices, and no edge and its endvertices are assigned the same color. A facial total-coloring of G is odd if for every face f and every color c, either no element or an odd number of elements incident with f is colored by c. In this note we prove that every cactus forest with an outerplane embedding admits an odd facial total-coloring with at most 16 colors. Moreover, this bound is tight.

We propose a mathematical model, $SIM$ , that describes the dynamics of cochineal insect spread among CACTI and examines the effects of various control strategies. The model is analyzed for the existence and unique-ness of solutions, and we investigate the equilibrium points and stability of the system using both local and global stability analyses. By perform-ing numerical simulations in $MATLAB$, we validate our theoretical find-ings. Furthermore, we propose an optimal control strategy to minimize the cochineal population in CACTI fields. The optimal control problem is formulated using Pontryagin’s maximum principle, and the corresponding optimality system is solved iteratively. Our study compares three control strategies: cutting and burning infected CACTI, insecticide spraying, and a combined approach. The results demonstrate that the combined strategy is the most effective in reducing the cochineal population. This research provides valuable insights into managing cochineal infestations and offers practical recommendations for farmers to control the spread of these pests.

The edge Mostar index 𝑀 𝑜 𝑒 (𝐺 ) of a connected graph 𝐺 is defined as 𝑀 𝑜 𝑒 (𝐺 )=Σ 𝑒 =𝑢 𝑣 ∈ 𝐸 (𝐺 ) |𝑚 𝑢 (𝑒 |𝐺 )− 𝑚 𝑣 (𝑒 |𝐺 )|, where 𝑚 𝑢 (𝑒 |𝐺 )and 𝑚 𝑣 (𝑒 |𝐺 ) are, respectively, the number of edges of 𝐺 lying closer to vertex 𝑢 than to vertex 𝑣 and the number of edges of 𝐺 lying closer to vertex 𝑣 than to vertex 𝑢 . In this paper, we determine the extremal values of edge Mostar index of some graphs. We characterize extremal trees, unicyclic graphs and determine the extremal graphs with maximum and second maximum edge Mostar index among CACTI with size 𝑚 and 𝑡 cycles. At last, we give some open problems.

‎The additively weighted edge Mostar index is a topological index(TI) defined as an extension of the edge Mostar index‎. ‎In this paper‎, ‎we determine the extrema of the additively weighted edge Mostar index for trees‎. ‎Additionally‎, ‎we compute the lower bound and first four upper bounds of additively weighted edge Mostar index of unicyclic graphs and the upper bound for CACTI with a fixed number of cycles‎. ‎All the graphs attaining the bounds are characterized‎. ‎We also propose two conjectures on additively weighted edge Mostar index of bicyclic graphs‎.