The harmonic index of edge-semitotal graphs, total graphs and related sums

Abstract

For a connected graph G, there are several related graphs such as line graph L(G), subdivision graph S(G), vertex-semitotal graph R(G), edge-semitotal graph Q(G) and total graph T(G) [I. Gutman, B. Furtula, Ž. K. Vukićević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015), 5-16, W. Yan, B.-Y. Yang and Y. -N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett. 20 (2007), 290-295]. Let F be one of symbols S, R, Q or T. The F-sum G1+FG2 of two connected graphs G1 and G2 is a graph with vertex set (V (G1) U E(G1)) × V (G2) in which two vertices (u1, v1) and (u2, v2) of G1 +F G2 are adjacent if and only if [u1 = u2 _ V (G1) and u1v2 _ E(G2)] or [v1 = v2 and u1u2 2 E(F(G))] [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009), 794-803]. In this paper, we investigate the harmonic index of edge-semitotal graphs, total graphs and F-sum of graphs, where F = Q or T.