In this article, we study the degree-based topological indices in a random polyomino chain. The key purpose of this manuscript is to obtain the asymptotic distribution, expected value and variance for the degree-based topological indices in a random polyomino chain by using a martingale approach. Consequently, we compute the degree-based topological indices in a polyomino chain, hence some known results from the existing literature about polyomino chains are obtained as corollaries. Also, in order to apply the results, we obtain the expected value of several degree-based topological indices such as Sombor, Forgotten, Zagreb, atom-bond-connectivity, Randić and geometric-arithmetic index of a random polyomino chain.

Phytophthora is one of the most aggressive and worldwide extended phytopathogens that attack plants and trees. Its effects produce tremendous economical losses in agronomy and forestry since no effective fungicide exists. We propose to combine percolation theory with an intercropping sowing configuration as a non-chemical strategy to minimize the dissemination of the pathogen. In this work, we model a plantation as a square lattice where two types of plants are arranged in alternating columns or diagonals, and Phytophthora zoospores are allowed to propagate to the nearest and next-to-nearest neighboring plants. We determine the percolation threshold for each intercropping configuration as a function of the plant’s susceptibilities and the number of inoculated cells at the beginning of the propagation process. The results are presented as phase diagrams where crop densities that prevent the formation of a spanning cluster of susceptible or diseased plants are indicated. The main result is the existence of susceptibility value combinations for which no spanning cluster is formed even if every cell in the plantation is sowed. This finding can be useful in choosing a configuration and density of plants that minimize damages caused by Phytophthora. We illustrate the application of the phase diagrams with the susceptibilities of three plants with a high commercial value.