University of KashanIranian Journal of Mathematical Chemistry2228-64896120150301On the Harmonic Index and Harmonic Polynomial of Caterpillars with Diameter Four4149904410.22052/ijmc.2015.9044ENM.IranmaneshDepartment of Mathematics, Yazd University, 89195741, Yazd, IranM.SaheliDepartment of Mathematics, Yazd University, 89195741, Yazd, IranJournal Article20150316The harmonic index H(G) , of a graph G is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in E(G), where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G. We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpillars withf diameter 4.https://ijmc.kashanu.ac.ir/article_9044_a7d924dff91432835709ad82ca38516a.pdf