On harmonic index and diameter of quasi-tree graphs (Q2034960)

Summary: 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 \(uv\) of \(G\), where \(d_u\) is the degree of a vertex \(u\) in \(G\). In this paper, we show that \(H(G) \geq D(G)+5/3-(n/2)\) and \(H (G)\geq ((1/2) + (2/3 (n-2))) D(G)\), 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.