Abstract
For a connected graph G of order n, the harmonic index of G is the sum of weights 2d(u)+d(v) over all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. We prove that H(G)≥χDP(G)−22+2(χDP(G)−1)n+χDP(G)−2+2(n−χDP(G))n, and this bound is sharp for all n and 2≤χDP(G)≤n, where χDP(G) is the DP-chromatic number of G. This generalizes the previous lower bounds on H(G). Moreover, we also determine the tree with minimum harmonic index among trees in 𝒯n,l, where 𝒯n,l is the set of trees of order n with a given segment sequence l.
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