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

Abstract

The F-sum G 1 + F G 2 of two connected graphs G 1 and G 2 is a graph with vertex set (V (G 1) ∪ E(G 1)) × V (G 2) in which two vertices (u 1 , v 1) and (u 2 , v 2) of G 1 + F G 2 are adjacent if and only if [u 1 = u 2 ∈ V (G 1) and v 1 v 2 ∈ E(G 2)] or [v 1 = v 2 and u 1 u 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 .

The harmonic index of graphs based on some operations related to the lexicographic product

“…For example, Zhong [16] gave the minimum and maximum values of the harmonic index for simple connected graphs and trees and characterized the corresponding extremal graphs. Recently, Onagh investigated the harmonic index of product graphs, subdivision graphs, t-subdivision graphs, vertex-semitotal graphs, edge-semitotal graphs, total graphs and F-sum of graphs, where F ∈ {S, S t , R, Q, T} [8][9][10]. More results on the harmonic index can be found in [3,13,14,17,18].…”

The harmonic index of product graphs

“…Shwetha et al [9] derived expressions for the harmonic index of the join, corona product, Cartesian product, composition and symmetric difference of graphs. Recently, Onagh investigated the harmonic index of subdivision graph S(G), t-subdivision graph S t ðGÞ, vertex-semitotal graph R(G), edge-semitotal graph Q(G), total graph T(G) and F-sum of graphs, where F 2 fS; S t ; R; Q; Tg [5][6][7]. More results on the harmonic index can been found in [1][2][3][10][11][12].…”