In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.

In this paper, we present some inequalities for the Co-PI index involving the some topological indices, the number of vertices and edges, and the maximum degree. After that, we give a result for trees. In addition, we give some inequalities for the largest eigenvalue of the Co-PI MATRIX of G.

Rice cultivar Tarom Dilamani becauded a fragrance, flavor, cooking and marketing is a qualitative rice in Iran. This cultivar have high susceptibility against blast disease (Magnaporthe grisea). One of the important trouble producers of the Dilamani's rice cultivar is chemical control against blast disease and cause poisonous pollution of natural environment. The best manner in order to control this disease and avoid its damage is preparation and cultivation resistant cultivar. In this research we used classical breeding method to developing resistance cultivar. Female parent was qualitative Tarom Dilamani cultivar and male parent was near isogenic lines C101LAC and C101A51 that carry dominant resistant genes to blast Pi-1 and Pi-2, respectivly. Parents' seeds prepared, cultivated and propagated. When rice plant entered flowering stage, sensitive cultivar was confluenced with resistant lines' pollen. Hybrid seeds would cultivated next year. Plants cause selected according to morphological characters and virulences tests. Back cross was accomplished until four years until Bc4 generation. Plant inoculation was accomplished with dominant race of fungus cause blast at Agricultural Research Station of Sari Agricultural sciences and Natural Research University. Results of this research was transferred two resistance genes in sensitive Tarom Dilamani cultivar. The recovered cultivar showed resistance to races of fungus cause blast on region. Infection type of sensitivity (4 and 5) that was prior of accomplished of this research have converted to infection type of resistance (1 and 2). Morphology and quality of plant changed slightly, but was satisfactory prevention of damage due blast and increase grain yield.

In this paper, at first we mention to some results related to PI and vertex Co-PI indices and then we introduce the edge versions of Co-PI indices. Then, we obtain some properties about these new indices.

The PI index of a graph is given by , where is the number of equidistant vertices for the edge . Various topological indices of bicyclic graphs have already been calculated. In this paper, we obtained the exact value of the PI index of bicyclic graphs. We also explore the extremal graphs among all bicyclic graphs with respect to the PI index. Furthermore, we calculate the PI index of a cactus graph and determine the extremal values of the PI index among cactus graphs.

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.

Blast (Magnaporthe grisea) is the most important disease of rice in Mazandaran province, north of Iran, which causes severe damages annually on susceptible cultivars. Tarom Mahalli is a local aromatic cultivar with high cooking quality and marketability, but very susceptible to blast. To protect this cultivar and similar susceptible cultivars against rice blast, farmers have to use high amounts of chemicals every year. In the present study, an attempt was done to transfer rice blast resistance genes Pi -1 and Pi -2 from near isogenic lines C101LAC and C101A51 (as male parents) to Tarom Mahalli (as femate parent) to develop a resistant cultivar with high cooking cultivar. Necessary crosses and backcrosses were accomplished between parents and hybrids developed during three years were selected based on morphological, virulence and molecular markers (RG64 and C481). Selected plants were finally exposed to a mixture of races of the pathogen prevalent in Mazandaran. Based on the results of this investigation, several plants were selected that had morphological and physiological characteristics of Tarom Mahalli and carried Pi -1 and Pi -2 resistance genes. The infection types of the hybrids were 1-2, while those of the Tarom Mahalli were 4 and 5. Although some morphological and qualitative characteristics of the progenies were slightly different from those of Tarom Mahalli, but transferance of resistance genes gave it a satisfactory protection against the disease.

The vertex version of PI index is a molecular structure descriptor which is similar to vertex version of Szeged index. In this paper, we compute the vertex-PI index of TUC4C8(S), TUC4C8(R) and HAC5C7[r, p].