PI Index of Generalized Gear Graph

Let G be a simple connected graph, e = uv is an edge of the connecting u and v in G, the PI index is defined as ■ The distance from a vertex to an edge is taken as the minimum distance between the given point and the two endpoints of that edge. The generalized gear graph is a graph obtained from the conical graph with a vertex added between each pair adjacent vertices of the cycles, which has a graceful symmetry. Let PI（C^*） be the PI index of generalized gear graph C^*. In this paper, the symmetry of generalized gear graph was used to obtain a method to calculate the number of edges that is equidistant from two ends of an edge and classifies its edges. By using this method, we found the edges that did not contribute to PI（C^*）, and then estimated the PI index of the generalized gear graph, which provided a clue for the study of the PI index of some special graphs.