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.

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].

General formulas are obtained for the vertex Padmakar-Ivan index (PIv) of tetrathiafulvalene (TTF) dendrimer, whereby TTF units we are employed as branching centers. The PIv index is a Wiener-Szeged-like index developed very recently. This topological index is defined as the summation of all sums of nu (e) and nv (e), over all edges of connected graph G.

In this paper, at first we introduce a new index with the name Co-PI index and obtain some properties related this new index. Then we compute this new index for TUC4C8 (R) nanotubes.

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.

Benefiting from the great potential of the mathematical tool of fractional calculus, fractional order proportional integral (PI) controllers have been proposed as the next generations of popular PI controllers. On the other hand, the integral performance indices such as the integral square error (ISE), integral absolute error (IAE), integral time square error (ITSE), and integral time absolute error (ITAE) are commonly used in the design and evaluation of the performance of practical control systems. Considering these points, the present paper deals with the optimal tuning of the free parameters of fractional order PI controllers, to be used in control of first order plus dead time (FOPDT) processes in a unity negative control structure, on the basis of ISE performance index. Using the approach of ``tuning based on the implementable form of the controller'' instead of the approach of ``tuning based on the ideal form of the controller'' causes that no incompatibility is seen between the ideal behavior of the controller and the behavior of the implementable controller. Also, to avoid approximation error in the calculation of ISE based cost functions, algebraic relations have been used for analytically finding the values of ISE in the under study time delay control system. In fact, the main contribution of the paper is to propose simple tuning rules for implementable fractional order PI controllers with the aim of achieving an optimal control system in the viewpoint of the ISE performance criterion. These rules have been obtained by iteratively using the steepest descent (gradient descent) optimization algorithm with the constraint of optimizing the step size in each iteration. The proposed rules for tuning of the free parameters of implementable fractional order PI controllers yield in a control system whose performance is better than the control system with an optimal PI controller. This point has been successfully confirmed by some numerical and experimental examples.