On Harmonic Index and Diameter of Quasi-Tree Graphs

Abstract

The harmonic index of a graph G ( H G ) is defined as the sum of the weights 2 / d u + d v for all edges u v of G , where d u is the degree of a vertex u in G . In this paper, we show that H G ≥ D G + 5 / 3 − n / 2 and H G ≥ 1 / 2 + 2 / 3 n − 2 D G , where G is a quasi-tree graph of order n and diameter D G . Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds.

The minimum harmonic index for bicyclic graphs with given diameter

“…In [7] the minimum and maximum harmonic indices for caterpillars with diameter 4 are computed. It is also showed that H(G) ≥ d + 5 3 − n 2 , where G is a quasi-tree graph of order n ≥ 4 and diameter d, except when G = U 1,1 5,3 or U 1,1 6,4 which are shown in Figure 1 [1].…”

“…By induction on n. First suppose n − d = 2. So if P ⊂ G be a diametrical path, then only one vertex of G is not in P. Note that two cycles of G should have a common vertex not in P and all other vertices of G should be in P, since every cycle has at least one vertex which is not in P. Therefore G ∈ B 3 and G = V r,s , is a quasi-tree graph introduced in [1]. The graph V r,s obtained by adding two paths of lengths r, s to two vertices of degree 2 of K − 4 , (see Figure 7).…”

“…The graph V r,s obtained by adding two paths of lengths r, s to two vertices of degree 2 of K − 4 , (see Figure 7). The authors showed [1], H(V r,s ) ≥ d +…”